Averaged controllability for random evolution partial differential equations
aa r X i v : . [ m a t h . O C ] J un Averaged controllability for RandomEvolution Partial Differential Equations
Qi L¨u ∗ and Enrique Zuazua †‡ Abstract
We analyze the averaged controllability properties of random evolution Partial Dif-ferential Equations.We mainly consider heat and Schr¨odinger equations with random parameters, al-though the problem is also formulated in an abstract frame.We show that the averages of parabolic equations lead to parabolic-like dynam-ics that enjoy the null-controllability properties of solutions of heat equations in anarbitrarily short time and from arbitrary measurable sets of positive measure.In the case of Schr¨odinger equations we show that, depending on the probabilitydensity governing the random parameter, the average may behave either as a conser-vative or a parabolic-like evolution, leading to controllability properties, in average, ofvery different kind.
Key Words . random evolution Partial Differential Equations, averaged controllability,averaged observability, Schr¨odinger equation, heat equation.
We analyze the problem of controlling systems with randomly depending coefficients inthe context of evolution Partial Differential Equations (PDEs). More precisely, we considerthe problem of averaged controllability which consists, roughly, of controlling the averageddynamics, with respect to the random parameters. This problem was introduced and solvedin [57] in the context of finite dimensional systems, where the same issue was also formulatedfor PDE.When the dynamics of the state is governed by a pair of random operators (determiningthe free dynamics and the control operators, respectively), generally speaking, controllingthe system would require to know the actual value of the random parameters. But this is ∗ School of Mathematics, Sichuan University, Chengdu, 610064, China ( [email protected]). † BCAM - Basque Center for Applied Mathematics, Mazarredo, 14. E-48009 Bilbao-Basque Country-Spain. ‡ Ikerbasque, Basque Foundation for Science, Mara Diaz de Haro 3, E-48013 Bilbao-Basque Country-Spain( [email protected]). ( y t + α · ∇ y = 0 in R d × [0 , ∞ ) ,y (0) = y in R d . (1.1)Here y ∈ L ( R d ) and α ( · ) : Ω → R d is a d -dimensional standard normally distributedrandom variable, with the probability density ρ ( α ) = 1(2 π ) d e − | α | for α ∈ R d . The solution to (1.1) reads y ( x, t, ω ; y ) = y ( x − tα ) for ( x, t ) ∈ R d × [0 , ∞ ) . Then, the mathematical expectation or averaged state˜ y ( x, t ) △ = Z Ω y ( x, t, ω ; y ) d P ( ω ) = 1(2 π ) d Z R d y ( x − αt ) e − | α | dα = 1(2 π ) d t d Z R d y ( z ) e − ( x − z )22 t dz solves the following heat equation ˜ y t − t ∆˜ y = 0 in R d × [0 , ∞ ) , ˜ y (0) = y in R d , (1.2)2amely, ϕ ( x, t ) = ˜ y ( x, √ t ) solves ( ϕ t − ∆ ϕ = 0 in R d × [0 , ∞ ) ,ϕ (0) = y in R d . (1.3)The fact that averages of transport-like equations may enjoy enhanced regularity prop-erties, first discovered in [1], is well known in different contexts. There are some differentpresentations of this smoothing properties, referred to as “averaging lemmas” in the contextof kinetic equations (see [9, 21, 22] for example). These smoothing properties are usefulwhen proving existence of solutions of linear and nonlinear kinetic equations [6, 12, 13, 39].The proofs of these “averaging lemma” usually employ the Fourier transform in all space-velocity-time variables. In [9], the authors gave a proof that only uses the Fourier transformwith respect to x and v , thus leading to a Fourier representation of the average, evolving intime, which is very similar in spirit to the method we employ in this paper. Note howeverthat, for control purposes, we need to identify the nature of the evolution associated withthe averaged variable, and not only its smoothing properties. Remark 1.1
The above computation shows that one can get diffusion processes by averaginga simple random convection process with respect to its velocity. This is also well known froma different perspective, in the context of chaotic and stiff oscillatory systems, that can beregarded as the characteristic systems of transport equations (see [11, 19, 45]).
This example shows that, by averaging, the solution of a random transport equation maylead to a solution of a heat-like equation and, consequently, that time-reversible systems maybecome strongly irreversible through averaging. Furthermore, this occurs with the normallydistributed random variable which is ubiquitous in nature due to the central limit theorem,which states that the mean of many independent random variables drawn from the samedistribution is distributed approximately normally, irrespective of the form of the originaldistribution. Accordingly, in the real world, physical quantities that are expected to be thesum of many independent variables, such as measurement errors, often have a distributionvery close to the normal distribution (see [10]).The “smoothing by averaging” effect mentioned above has important consequences froma control theoretical point of view as well. Indeed, while the transport equation (1.1),for a given value of the random variable α (which determines the velocity of propagationof waves), enjoys the property of exact controllability in finite time, proportional to thetravel time of characteristics to get to the control set (the boundary or an open subset ofthe domain), the averaged heat dynamics is controllable to zero (or any other sufficientlysmooth target) in an arbitrarily short time and from any subset of the domain where thedynamics evolves, without any geometric condition on the support of the controls, involvingthe propagation of characteristics. Accordingly, through averaging, we encounter on a singlemodel, with randomly depending coefficients, the classical dichotomy arising in the contextof controllability of hyperbolic versus parabolic systems (see [56]).This paper is devoted to systematically addressing these questions in the context of heatand Schr¨odinger equations. Our aim here is not, by any means, to systematically addressall the possible scenarios but simply to highlight some of the most fundamental phenomena3llustrating how, the existing tools for the analysis of the controllability of PDE, can beemployed in this averaged context too. It is important to highlight, however, that theaveraged states do not obey a PDE, not even a semigroup. The dynamics can howeverbe represented in such a way that its main controllability properties can be identified, byanalogy, with some of the main well-known models, and analyzed by similar techniques.In particular, we shall show that the averages of heat equations lead to heat-like dynamicsthat enjoy the null-controllability properties of solutions of heat equations in an arbitrar-ily short time and from arbitrary measurable sets of positive measure. In order to provethese results we employ classical techniques based on Carleman inequalities and the Fourierexpansion of solutions on the basis of the eigenfunctions of the Laplacian generating thedynamics.In the case of Schr¨odinger equations we show that, depending on the probability density,the average may behave either as a conservative or a heat-like evolution, leading to con-trollability properties, in average, of very different kind. When the obtained average is ofparabolic nature the techniques above, employed to treat the control properties of parabolicaverages, can be applied. However, when the average behaves rather in a conservative waywe employ specific techniques for the control of wave-like equations.The paper is organized as follows. In section 2 we present all these problems in anabstract setting in which different relevant PDE models enter naturally. In Section 3, westudy the null and approximate averaged controllability problems for a class of random heatequations. In Section 4, we study the null and exact averaged controllability problem fora class of random Schr¨odinger equations. In Section 5 we give some further comments andopen problems. Let
T > E ⊂ [0 , T ] be a Lebesgue measurable set with positive Lebesgue measure.Let H and U be two Hilbert spaces. Let V ⊂ H be a Hilbert space which is dense in H .Denote by V ′ the dual space of V with respect to the pivot space H . Let (Ω , F , P ) be aprobability space. Let { A ( ω ) } ω ∈ Ω be a family of linear operators satisfying the followingconditions:1. A ( · ) ∈ L (Ω; L ( D ( A ) , H ));2. A ( ω ) : D ( A ) → H generates a C -semigroup { S ( t, ω ) } t ≥ on both H and V for all ω ∈ Ω;3. S ( t, · ) y ∈ L (Ω; V ) for all y ∈ V and t ∈ [0 , T ].Let B ( · ) ∈ L (Ω; L ( U, V )).Consider the following linear control system (cid:26) y t ( t ) = A ( ω ) y ( t ) + χ E ( t ) B ( ω ) u ( t ) in (0 , T ] ,y (0) = y , (2.1)where y ∈ V and u ( · ) ∈ L ( E ; U ) is the control.4n what follows, we denote by y ( · , ω ; y ) the solution to (2.1), which is the state of thesystem. Although the initial datum y ∈ V and the control u ( · ) are independent of thesample point ω , the state y ( t, ω ; y ) of the system depends on ω nonlinearly.According to the setting above, for a.e. ω ∈ Ω, there is a solution y ( · , ω ; y ) ∈ C ([0 , T ]; V )and the expectation or averaged state R Ω y ( · , ω ; y ) d P ( ω ) ∈ C ([0 , T ]; V ) . We introduce the following notions of averaged controllability for the system (2.1):
Definition 2.1
System (2.1) is said to fulfill the property of exact averaged controllabilityor to be exactly controllable in average in E with control cost C > if given any y , y ∈ V ,there exists a control u ( · ) ∈ L ( E ; U ) such that | u | L ( E ; U ) ≤ C ( | y | V + | y | V ) (2.2) and the average of solutions to (2.1) satisfies Z Ω y ( T, ω ; y ) d P ( ω ) = y . (2.3) Remark 2.1
The notion of exact averaged controllability was first introduced in [57]. Afull characterization was also given in the finite-dimensional setting. In [40], this issue wasdiscussed for systems involving finitely-many linear parametric wave equations.
Definition 2.2
System (2.1) fulfills the property of null averaged controllability or is nullcontrollable in average in E with control cost C if given any initial datum y ∈ V , thereexists a control u ∈ L ( E ; U ) such that | u | L ( E ; U ) ≤ C | y | V (2.4) and the average of the solutions to (2.1) satisfies Z Ω y ( T, ω ; y ) d P ( ω ) = 0 . (2.5) Definition 2.3
System (2.1) fulfills the property of approximate averaged controllability oris approximately controllable in average in E if given any y , y ∈ V and ε > , there existsa control u ε ∈ L ( E ; U ) such that the average of solutions to (2.1) satisfies (cid:12)(cid:12)(cid:12) Z Ω y ( T, ω ; y ) d P ( ω ) − y (cid:12)(cid:12)(cid:12) V < ε. Remark 2.2
As in the finite dimensional context ([57]), we can also consider the averagedcontrol problem with random initial data, i.e., y ∈ L (Ω; V ) . Nevertheless, according toRemark A.1, this does not lead to any essential new difficulty. Thus, for the sake of simplicityof the presentation, we only deal with the case where y is independent of ω . These notions are motivated by the problem of controlling a dynamics governed by a pairof random operators ( A ( ω ) , B ( ω )), where the effective value of the parameter ω is unknown.Then, one aims at choosing a control, independent of the unknown ω , to act optimally in5n averaged sense, making a robust compromise of all the possible realizations of the systemfor the various possible values of the sample point ω . Similar problems can be considered inthe case where the initial datum to be controlled depends on ω too.We have introduced the notions of exact/null/approximate averaged controllability in theframework of random evolution equations but similar concepts make sense for parametrizedevolution equations (see [57] for example). In that context it is sufficient to replace theexpectation by a weighted average of the parameter-depending controlled states. Remark 2.3
In the present context of randomly depending operators, the classical subordi-nation properties of some control properties with respect to the others, that are classical fora given system, have to be addressed more carefully. Of course, averaged exact controllabil-ity implies the averaged null and approximate controllability properties as well. But, when A and B are independent of ω and A generates a C -group, exact controllability is also aconsequence of null controllability. However, the later may fail when considering averagedcontrollability properties, as shown in the example in Remark 4.6 below. Remark 2.4
For parametric control systems one can also consider the problems of simulta-neous controllability and ensemble controllability, which concern the possibility of controllingall states with respect to different parameters simultaneously by one single control. We referthe readers to [37, 38] and [8, 35] for an introduction to these notions, respectively. Of coursethe property of averaged controllability we consider here is weaker than these other ones sincewe only deal with the average of the states with respect to the parameters. But, as we shallsee, averaged controllability properties may be achieved in situations where simultaneous andensemble controllability are impossible. For instance, let us consider the following controlsystem: (cid:26) y t ( t ) = Ay ( t ) + B ( ω ) u ( t ) in (0 , T ] ,y (0) = y ∈ R , (2.6) where A = (cid:18) (cid:19) , B ( ω ) = B or B for B = (cid:18) (cid:19) . By Theorem 1 in [57], we know that the system (2.6) is null averaged controllable.But system (2.6) is not simultaneous null controllable. Otherwise there would exist a u ∈ L (0 , T ) such that e AT y + Z T e A ( T − s ) Bu ( s ) ds = e AT y + 2 Z T e A ( T − s ) Bu ( s ) ds = 0 , which implies that y = 0 . Remark 2.5
The connection between averaged and simultaneous controllability was analyzedin [40] through the following optimal control problem:Minimize J k ( u ) △ = 12 | u | L (0 ,T ; U ) + k Z Ω | y ( T ) − y | d P ( ω )6 or all u ∈ n u ∈ L (0 , T ; U ) : The corresponding solution y satisfies that Z Ω y ( T, ω ) d P ( ω ) = y o . In that paper it is proved that for every k , J k ( · ) has a unique minimizer u k ( · ) and that,if H and U are finite dimensional and the system (2.1) is simultaneously controllable, then { u k } ∞ k =1 weakly converges to a simultaneous control. Remark 2.6
In the particular case that A is independent of ω , the averaged controllabilityproblems can be reduced to the classical controllability ones by setting ¯ B = Z Ω B ( ω ) d P ( ω ) , ¯ y ( t ) = Z Ω y ( t, ω ; y ) d P ( ω ) . Then we have that (cid:26) ¯ y t ( t ) = A ¯ y ( t ) + χ E ( t ) ¯ Bu ( t ) in (0 , T ] , ¯ y (0) = y . (2.7) The exact (resp. null, approximate) averaged controllability problems of (2.1) are equivalentto the exact (resp. null, approximate) controllability problem of (2.7) . Remark 2.7
Random evolution equations can be used to model lots of uncertain physicalprocesses (see [49, 50, 51] for example). Several notions of controllability have been introducedbut, as far as we know, all of them concern driving the state to a given destination bya control depending on ω (see [27, 43, 44] and the references therein). The property ofaveraged controllability is, however, independent of the specific realization of ω . Following the classical approach to deal with controllability problems, we introduce theadjoint system, which also depends on the parameter ω : (cid:26) − z t ( t ) = A ∗ ( ω ) z ( t ) in [0 , T ) ,z ( T ) = z , (2.8)where z ∈ V ′ .Note that, in this adjoint system the initial value (at time t = T ) is taken to be inde-pendent of ω . This is due to the fact that, although y ( T, ω ; y ) depends on ω , its average,which belongs to V , is of course independent of ω . Then, to deal with the averaged stateit is sufficient to use as test functions, adjoint states departing from configurations that areindependent of ω . This is the reason we choose the final datum of (2.8) to be independentof ω .As dual notions of the properties of averaged controllability above we introduce thefollowing three concepts of averaged observability. Definition 2.4
System (2.8) is exactly averaged observable or exactly observable in averagein E if there is a constant C > such that for any z ∈ V ′ , | z | V ′ ≤ C Z T χ E ( t ) (cid:12)(cid:12)(cid:12) Z Ω B ( ω ) ∗ z ( t, ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) U dt. (2.9)7 efinition 2.5 System (2.8) is null averaged observable or null observable in average in E if there is a constant C > such that for any z ∈ H , (cid:12)(cid:12)(cid:12) Z Ω z (0 , ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) V ′ ≤ C Z T χ E ( t ) (cid:12)(cid:12)(cid:12) Z Ω B ( ω ) ∗ z ( t, ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) U dt. (2.10) Definition 2.6
System (2.8) is said to satisfy the averaged unique continuation property in E if the fact that χ E R Ω B ( ω ) ∗ z ( · , ω ; z ) d P ( ω ) = 0 in L (0 , T ; U ) implies that z = 0 . The adjoint system (2.8) being exactly ( resp. null) averaged observable means that onecan estimate the norm of the average of the final ( resp. initial) data of the adjoint system,out of partial measurements done on the averages of the adjoint states, with respect to ω .These concepts have their own interest when dealing with the observation of random systems.The actual realization of the system depending on ω being unknown, it is natural to addressthe problem based on the measurements done on averages.The weakest notion of averaged observability under consideration is averaged uniquecontinuation. System (2.8) satisfies the averaged unique continuation property when itsstate can be uniquely determined by the partial measurements done on the mathematicalexpectation. It is a natural generalization of the unique continuation property of evolutionequations.The average of the adjoint state, being represented by R Ω z ( t, ω ; z ) d P ( ω ), does not satisfythe semigroup property and it is not a solution to an evolution equation. Thus, one can notdirectly employ the existing results on the observability of PDEs to establish the averagedobservability of the adjoint system (2.9). However, as we shall see, by carefully analyzingand identifying the dynamics generated by the averages of the adjoint states, we shall beable to apply the existing PDE techniques.In this paper, we mainly consider the case that A ( ω ) = α ( ω ) A and B ( ω ) = B , where A generates a C -semigroup on H , α ( ω ) is a random variable and B ∈ L ( U, H ). We will showthat the controllability properties of the system (2.1) depend on the choice of the randomvariable. We only consider the following commonly used ones:1.
Uniformly distributed random variable , with probability density function ρ ( · ) on [ a, b ],where 0 < a < b , α ( · ), given by ρ ( α ) = b − a , if α ∈ [ a, b ] , , if α ∈ ( −∞ , a ) ∪ ( b, ∞ ) . Exponentially distributed random variable , with probability density function ρ ( · ) givenby ρ ( α ) = ( e − ( α − c ) , if α ≥ c, , if α < c, (2.11)for a given positive number c . In what follows, for simplicity, we choose c = 1 in (2.11).3. Standard normally distributed random variable , with probability density function of ρ ( · ) reads ρ ( α ) = 1 √ π e − α for α ∈ R . A random variable with standard Laplace distribution , with density function ρ ( · ) givenby ρ ( α ) = 12 e −| α | for α ∈ R . A random variable with standard Chi-squared distribution , with density function ρ ( · )given by ρ ( α ; k ) = α k − e − α k Γ( k ) , if α ≥ , , if α < , where k ≥ k ) = R ∞ α k − e − α dα .6. A random variable with standard Cauchy distribution , with density function ρ ( · ) givenby ρ ( α ) = 1 π (1 + α ) for α ∈ R . In this section, we study the null averaged controllability problem for random heat equa-tions. Of course, the property of averaged exact controllability has to be excluded becauseof the very strong regularizing effect of heat equations, that is preserved by averaging. Thus,we focus on the properties of the null and approximate averaged controllability.Let
T > G ⊂ R d ( d ∈ N ) be a bounded domain with C boundary ∂G and considerthe following controlled random heat equation: y t − α ∆ y = χ G × E u in G × (0 , T ) ,y = 0 on ∂G × (0 , T ) ,y (0) = y in G. (3.1)Here y ∈ L ( G ), and G and E are subsets of G and [0 , T ] respectively, where the controlsare being applied. The constant diffusivity α : Ω → R + is assumed to be a random variable.In this section, we make the following assumptions on G and E :( A1 ) G ⊂ G is a nonempty open subset.( A2 ) E ⊂ [0 , T ] is a Lebesgue measurable set with positive measure.We first analyze the dynamics of the mathematical expectation (average) of the solutionsof the random heat equations under consideration. This will be done using Fourier seriesexpansions and will give us an intuition for the averaged observability and controllabilityresults to be expected and that will be proved below.Consider first the following uncontrolled random heat equation: ˆ y t − α ∆ˆ y = 0 in G × (0 , T ) , ˆ y = 0 on ∂G × (0 , T ) , ˆ y (0) = ˆ y in G. (3.2)9e decompose solutions in Fourier series on the basis of the eigenfunctions of the Dirichletlaplacian. To be more precise, consider the unbounded linear operator A ∆ on L ( G ) givenas ( D ( A ∆ ) = H ( G ) ∩ H ( G ) ,A ∆ u = − ∆ u, for any u ∈ D ( A ∆ ) . Let us denote by { λ j } ∞ j =1 (with 0 < λ < λ ≤ · · · ) the eigenvalues of A ∆ and let { e j } ∞ j =1 bethe corresponding eigenfunctions such that | e j | L ( G ) = 1 for j ∈ N .We assume that the initial datum takes the form ˆ y = P ∞ j =1 ˆ y ,j e j ∈ L ( G ). The averagedstate can also be described in Fourier series as follows, distinguishing the probability densitiesabove. Case 1 . If α ( · ) is a uniformly distributed random variable on [ a, b ], where 0 < a < b ,then, Z Ω ˆ y ( x, t, ω ; ˆ y ) d P ( ω ) = 1 b − a Z ba ∞ X i =1 ˆ y ,j e − λ j αt e j dα = 1 b − a ∞ X j =1 λ j t ˆ y ,j (cid:0) e − λ j at − e − λ j bt (cid:1) e j . (3.3) Remark 3.1
The values of a uniformly distributd random variable, which is a relevant onein practice, are uniformly distributed over an interval, i.e., all points in the interval areequally likely. It models the random phenomenon with “equally possible outcomes”. Whenthere is no any a priori knowledge for α other than α ( ω ) ∈ [ a, b ] for all ω ∈ Ω , this is thebest possible choice. More details can be found in [26]. Case 2.
When α ( · ) is the exponentially distributed random variable, Z Ω ˆ y ( x, t, ω ; ˆ y ) d P ( ω ) = Z ∞ e − ( α − ∞ X j =1 ˆ y ,j e − λ j αt e j dα = ∞ X j =1 λ j t + 1 ˆ y ,j e − λ j t e j . (3.4) Remark 3.2
The exponentially distributed random variable is one of the most importantrandom variables that can be used to describe the time between events in a process wherethey occur continuously and independently at a constant average rate.
In both cases, the mathematical expectation of the solution of the parameter-dependingheat equation evolves according to a heat-like dynamics. The representation of the averageson the basis of eigenfunctions exhibits the exponential decay and smoothing effects that areprototypical of heat-like problems.Accordingly, the following null averaged controllability result holds.
Theorem 3.1
Let ( A1 ) and ( A2 ) hold. Assume that, either α ( · ) is a uniformly or expo-nentially distributed random variable. Then, the system (3.1) is null controllable in averagewith control u ( · ) ∈ L (0 , T ; L ( G )) . Further, there is a constant C > such that | u | L (0 ,T ; L ( G )) ≤ C | y | L ( G ) . (3.5)10 emark 3.3 In the context of heat equations, the null controllability result with controlssupported in measurable sets is related to the bang-bang property of the time optimal controlproblems with constrained controls (see [48, 52] for example).More precisely, let us consider the following heat equation: ˜ y t − ∆˜ y = χ G ˜ u in G × [0 , + ∞ ) , ˜ y = 0 on ∂G × [0 , + ∞ ) , ˜ y (0) = ˜ y in G. (3.6) Here the initial state ˜ y ∈ L ( G ) and the control is assumed to belong to the constrained setof admissible controls ˜ u ∈ U M △ = { ˜ u ∈ L ∞ (0 , + ∞ ; L ( G )) : | ˜ u | L ( G ) ≤ M a.e. t ∈ [0 , + ∞ ) } for some M > . Let e T ∗ be the min { ˜ t : ˜ y (˜ t ; ˜ u, ˜ y ) = 0 , ˜ u ∈ U M } . A control u ∗ ∈ U M suchthat ˜ y ( e T ∗ ; ˜ u, ˜ y ) = 0 is called a time optimal control and satisfies the bang-bang property if | ˜ u ( t ) | L ( G ) = M for a.e. t ∈ [0 , T ∗ ] . The proof of this fact requires the null controllabilitywith controls supported in measurable sets (see [52]).The same bang-bang problem can be formulated in the context of averaged null controlla-bility we are considering here. However, the techniques in [52] do not seem to apply becausethe averages of the heat processes under consideration do not satisfy the semigroup property. To prove Theorem 3.1, as in the deterministic frame, we introduce the following adjointsystem: z t + α ∆ z = 0 in G × (0 , T ) ,z = 0 on ∂G × (0 , T ) ,z ( T ) = z in G, (3.7)where z ∈ L ( G ). As mentioned above in the abstract setting, the initial data (at time t = T ) of the adjoint system are assumed to be independent of the random parameter.According to Theorem A.2, we only need to prove that (3.7) is null observable in average,which is a corollary of the following result. Theorem 3.2
Let ( A1 ) and ( A2 ) hold. Assume that α ( · ) is either an uniformly distributedor an exponentially distributed random variable. Then, there exists a constant C > suchthat for any y ∈ L ( G ) it holds that (cid:12)(cid:12)(cid:12) Z Ω z (0 , ω ; y ) d P ( ω ) (cid:12)(cid:12)(cid:12) L ( G ) ≤ C Z E (cid:12)(cid:12)(cid:12) Z Ω z ( t, ω ; y ) d P ( ω ) (cid:12)(cid:12)(cid:12) L ( G ) dt. (3.8)An immediate corollary of Theorem 3.2 is as follows: Corollary 3.1
Let ( A1 ) and ( A2 ) hold. Assume that α ( · ) is either an uniformly distributedor an exponentially distributed random variable. Then the system (3.7) is null observable inaverage. roof : From H¨older’s inequality, we have that Z E (cid:12)(cid:12)(cid:12) Z Ω z ( t, ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) L ( G ) dt ≤ (cid:16) Z E dt (cid:17) (cid:16) Z E (cid:12)(cid:12)(cid:12) Z Ω z ( t, ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) L ( G ) dt (cid:17) ≤ p m ( E ) (cid:16) Z E (cid:12)(cid:12)(cid:12) Z Ω z ( t, ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) L ( G ) dt (cid:17) . This, together with the inequality (3.8), implies that the system (3.7) is null observable inaverage.To prove Theorem 3.2, we adopt the strategy developed in [46] in the context of the heatequation, using spectral decompositions. The null averaged observability inequality is builtin an iterative manner. More precisely, we decompose the set E into an infinite sequenceof connective (in time) subsets in which an increasing number of Fourier components of theaverage of the solution is observed with uniform observability constants. By iteration, thefinal datum is observed. To apply this strategy, we need to use classical results on how todivide E into an infinite sequence of subsets of positive Lebesgue measure. We also needto know how to observe a finite number of the Fourier components of the average of thesolution. These ingredients are given in the following two lemmas. Lemma 3.1 [48, Proposition 2.1] Let E ⊂ [0 , T ] be a measurable set of positive Lebesguemeasure m ( E ) . Let ℓ be a density point of E . Then for each a > , there exists an ℓ ∈ ( ℓ, T ) such that the sequence { ℓ k } ∞ k =1 , given by ℓ k +1 = ℓ + ℓ − ℓa k , (3.9) satisfies m ( E ∩ ( ℓ k +1 , ℓ k )) ≥ ℓ k − ℓ k +1 . (3.10) Lemma 3.2 [41, Theorem 1.2] There is a constant C > such that for any r > and { a j } λ j ≤ r ⊂ C , (cid:16) X λ j ≤ r | a j | (cid:17) ≤ C e C √ r (cid:16) Z G (cid:12)(cid:12)(cid:12) X λ j ≤ r a j e j ( x ) (cid:12)(cid:12)(cid:12) dx (cid:17) . (3.11)We are now in conditions to proceed to the proof of Theorem 3.2. Proof of Theorem 3.2 : We only give a proof for the case where α ( · ) is an exponentiallydistributed random variable. The proof for that α ( · ) is a uniformly distributed randomvariable is very similar.Put ˜ z ( x, t ) = R Ω z ( x, T − t, ω ; z ) d P ( ω ). Let z ,j = h z , e j i L ( G ) . Then, similar to thecomputation of (3.4), we have that˜ z ( x, t ) = ∞ X j =1 λ j t + 1 z ,j e − λ j t e j . We only need to prove that (cid:12)(cid:12)(cid:12) ∞ X j =1 z ,j e − λ j T e j (cid:12)(cid:12)(cid:12) L ( G ) ≤ C Z E | ˜ z ( x, t, ω ) | L ( G ) dt. (3.12)12ote that, in the right hand side of this inequality, the observation is done in the L ( E ; L ( G ))-norm. Thus, the result is even stronger than the one we actually need, in which the obser-vation is done in L ( E ; L ( G )). As a consequence of the inequality above we shall prove,the controls we shall build will belong to L ∞ ( E ; L ( G )).Let X r = span { e j } λ j ≤ r for each r >
0. For any ξ ∈ L ( G ), we put S ( t, ξ ) = ∞ X j =1 λ j t + 1 ξ j e − λ j t e j , (3.13)where ξ j = h ξ, e j i L ( G ) . Then we see that (cid:12)(cid:12) S ( t, ξ ) (cid:12)(cid:12) L ( G ) ≤ (cid:12)(cid:12) S ( s, ξ ) (cid:12)(cid:12) L ( G ) , for 0 ≤ s ≤ t ≤ T, (3.14)and (cid:12)(cid:12) S ( t, ξ ) (cid:12)(cid:12) L ( G ) ≤ e − r ( t − s ) | S ( s, ξ ) | L ( G ) , for all ξ ∈ X ⊥ r and 0 ≤ s ≤ t ≤ T. (3.15)Let ℓ be a density point for E . By Lemma 3.1, for a given a >
1, there exists a sequence { ℓ k } ∞ k =1 satisfying (3.9) and (3.10).We now define a sequence of subsets { E k } ∞ k =1 of (0 , T ) in the following way: E k △ = n t − ℓ k − ℓ k +1 t ∈ E ∩ (cid:16) ℓ k +1 + ℓ k − ℓ k +1 , ℓ k (cid:17)o , for k ∈ N . (3.16)Clearly, E k ⊂ ( ℓ k +1 , ℓ k +1 + ( ℓ k − ℓ k +1 )). From (3.10), we have that m ( E k ) = m (cid:16) E ∩ (cid:16) ℓ k +1 + ℓ k − ℓ k +1 , ℓ k (cid:17)(cid:17) = m (cid:16) E ∩ h ( ℓ k +1 , ℓ k ) \ (cid:16) ℓ k +1 , ℓ k +1 + ℓ k − ℓ k +1 (cid:17)i(cid:17) ≥ m ( E ∩ ( ℓ k +1 , ℓ k )) − ℓ k − ℓ k +1 ≥ ℓ k − ℓ k +1 . (3.17)Let b > a be a positive number such that ba > C + 6 ln(12 C a )( a − ℓ − ℓ ) . (3.18)Set r k = b k . From (3.14), we have that for any ξ ∈ X r k , Z ℓ k +1 + ( ℓ k − ℓ k +1 ) ℓ k +1 χ E k ( t ) (cid:12)(cid:12)(cid:12) S (cid:16) ℓ k +1 + 56 ( ℓ k − ℓ k +1 ) , ξ (cid:17)(cid:12)(cid:12)(cid:12) L ( G ) dt ≤ Z ℓ k +1 + ( ℓ k − ℓ k +1 ) ℓ k +1 χ E k ( t ) (cid:12)(cid:12) S ( t, ξ ) (cid:12)(cid:12) L ( G ) dt. ℓ k − ℓ k +1 (cid:12)(cid:12)(cid:12) S (cid:16) ℓ k +1 + 56 ( ℓ k − ℓ k +1 ) , ξ (cid:17)(cid:12)(cid:12)(cid:12) L ( G ) ≤ m ( E k ) (cid:12)(cid:12)(cid:12) S (cid:16) ℓ k +1 + 56 ( ℓ k − ℓ k +1 ) , ξ (cid:17)(cid:12)(cid:12)(cid:12) L ( G ) ≤ Z ℓ k +1 + ( ℓ k − ℓ k +1 ) ℓ k +1 χ E k ( t ) (cid:12)(cid:12) S ( t, ξ ) (cid:12)(cid:12) L ( G ) dt ≤ C e C √ r k Z ℓ k +1 + ( ℓ k − ℓ k +1 ) ℓ k +1 χ E k ( t ) (cid:12)(cid:12) S ( t, ξ ) (cid:12)(cid:12) L ( G ) dt. (3.19)Let z = z + z , where z ∈ X r k and z ∈ X ⊥ r k . Taking ξ = S (cid:16) ℓ k − ℓ k +1 , z (cid:17) in (3.19), we getthat ℓ k − ℓ k +1 | S ( ℓ k , z ) | L ( G ) ≤ Z ℓ k +1 + ( ℓ k − ℓ k +1 ) ℓ k +1 χ E k ( t ) (cid:12)(cid:12)(cid:12) S (cid:16) t + ℓ k − ℓ k +1 , z (cid:17)(cid:12)(cid:12)(cid:12) L ( G ) dt ≤ C e C √ r k Z ℓ k +1 + ( ℓ k − ℓ k +1 ) ℓ k +1 χ E k ( t ) (cid:12)(cid:12)(cid:12) S (cid:16) t + ℓ k − ℓ k +1 , z (cid:17)(cid:12)(cid:12)(cid:12) L ( G ) dt ≤ C e C √ r k Z ℓ k ℓ k +1 + ℓk − ℓk +16 χ E k (cid:16) t − ℓ k − ℓ k +1 (cid:17) | S ( t, z ) | L ( G ) dt. (3.20)By (3.16), we have that χ E k (cid:16) t − ℓ k − ℓ k +1 (cid:17) = χ E ( t ) , for any t ∈ (cid:16) ℓ k +1 + ℓ k − ℓ k +1 , ℓ k (cid:17) . (3.21)Combining (3.14), (3.15), (3.20) and (3.21), we find that ℓ k − ℓ k +1 | S ( ℓ k , z ) | L ( G ) ≤ C e C √ r k Z ℓ k ℓ k +1 + ℓk − ℓk +16 χ E ( t ) | S ( t, z ) | L ( G ) dt ≤ C e C √ r k Z ℓ k ℓ k +1 + ℓk − ℓk +16 χ E ( t ) (cid:0) | S ( t, z ) | L ( G ) + | S ( t, z ) | L ( G ) (cid:1) dt ≤ C e C √ r k Z ℓ k ℓ k +1 + ℓk − ℓk +16 χ E ( t ) | S ( t, z ) | L ( G ) dt + C e C √ r k ( ℓ k − ℓ k +1 ) (cid:12)(cid:12)(cid:12) S (cid:16) ℓ k +1 + ℓ k − ℓ k +1 , z (cid:17)(cid:12)(cid:12)(cid:12) L ( G ) ≤ C e C √ r k Z ℓ k ℓ k +1 χ E ( t ) | S ( t, z ) | L ( G ) dt + C e C √ r k ( ℓ k − ℓ k +1 ) e − ℓk − ℓk +16 r k (cid:12)(cid:12) S ( ℓ k +1 , z ) (cid:12)(cid:12) L ( G ) . (3.22)14herefore, we obtain that ℓ k − ℓ k +1 | S ( ℓ k , z ) | L ( G ) ≤ C e C √ r k Z ℓ k ℓ k +1 χ E ( t ) | S ( t, z ) | L ( G ) dt + C e C √ r k ( ℓ k − ℓ k +1 ) e − ℓk − ℓk +16 r k (cid:12)(cid:12) S ( ℓ k +1 , z ) (cid:12)(cid:12) L ( G ) + ℓ k − ℓ k +1 | S ( ℓ k , z ) | L ( G ) ≤ C e C √ r k Z ℓ k ℓ k +1 χ E ( t ) | S ( t, z ) | L ( G ) dt + C e C √ r k ( ℓ k − ℓ k +1 ) e − ℓk − ℓk +16 r k (cid:12)(cid:12) S ( ℓ k +1 , z ) (cid:12)(cid:12) L ( G ) + ℓ k − ℓ k +1 e − r k ( ℓ k − ℓ k +1 ) | S ( ℓ k +1 , z ) | L ( G ) . (3.23)Thus, it holds that ℓ k − ℓ k +1 | S ( ℓ k , z ) | L ( G ) ≤ C e C √ r k Z ℓ k ℓ k +1 χ E ( t ) | S ( t, z ) | L ( G ) dt +( ℓ k − ℓ k +1 ) e − ℓk − ℓk +16 r k (cid:0) C e C √ r k + 1 (cid:1)(cid:12)(cid:12) S ( ℓ k +1 , z ) (cid:12)(cid:12) L ( G ) . (3.24)This concludes that ℓ k − ℓ k +1 C e C √ r k | S ( ℓ k , z ) | L ( G ) − C e C √ r k + 1 C e C √ r k ( ℓ k − ℓ k +1 ) e − ℓk − ℓk +16 r k | S ( ℓ k +1 , z ) | L ( G ) ≤ Z ℓ k ℓ k +1 χ E ( t ) | S ( t, z ) | L ( G ) dt. (3.25)By summing the inequality (3.25) from k = 1 to k = ∞ , we obtain that ℓ − ℓ C e C √ r | S ( ℓ , z ) | L ( G ) + ∞ X k =1 f k | S ( ℓ k +1 , z ) | L ( G ) ≤ Z T χ E ( t ) | S ( t, z ) | L ( G ) dt, (3.26)where f k = ℓ k +1 − ℓ k +2 C e C √ r k +1 − C e C √ r k + 1 C e C √ r k ( ℓ k − ℓ k +1 ) e − ℓk − ℓk +16 r k , k = 1 , , · · · From (3.18) and r k = b k , we have that f k ≥ k = 1 , , · · · This, together with (3.26), deduces that | S ( ℓ , z ) | L ( G ) ≤ C e C √ r ℓ − ℓ Z E | S ( t, z ) | L ( G ) dt. (3.27)Since ℓ < T , we can find a constant C > C λ j ℓ ≥ e − λ j ( T − ℓ ) for every j ∈ N . (3.28)15rom (3.27) and (3.28), we get that (cid:12)(cid:12)(cid:12) ∞ X j =1 z ,j e − λ j T e j (cid:12)(cid:12)(cid:12) L ( G ) ≤ C Z E | S ( t, z ) | L ( G ) dt. (3.29)This completes the proof.As an easy corollary of Theorem 3.2, we have the following result. Theorem 3.3
Let ( A1 ) and ( A2 ) hold. System (3.1) is approximately controllable in av-erage, provided that α ( · ) is a uniformly distributed or an exponentially distributed randomvariable.Proof : According to Theorem A.3, we only need to prove that the unique continuationproperty in E is satisfied.Assume that z = 0 in G × E . From Theorem 3.2, we obtain that (cid:12)(cid:12)(cid:12) Z Ω z (0 , ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) L ( G ) ≤ C Z E (cid:12)(cid:12)(cid:12) Z Ω z ( t, ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) L ( G ) dt. Hence, R Ω z (0 , ω ; z ) d P ( ω ) = 0. On the other hand, if z ,j = R G z e j dx ,0 = Z Ω z (0 , ω ; z ) d P ( ω ) = Z ∞ e − ( α − ∞ X j =1 z ,j e − λ j αT e j dα = ∞ X j =1 λ j T + 1 z ,j e − λ j T e j . (3.30)Since { e j } ∞ j =1 is an orthonormal basis of L ( G ), it follows that λ j T +1 z ,j e − λ j T = 0 for all j ∈ N . Thus, we get that z ,j = 0 for all j ∈ N , which implies that z = 0. In this section, we study the null and exact averaged controllability problems for a classof random Schr¨odinger equations of the form y t − iα ∆ y = χ G × E u in G × (0 , T ) ,y = 0 on ∂G × (0 , T ) ,y (0) = y in G. (4.1)Here α ( · ) : Ω → R is a random variable, the initial datum y belongs to L ( G ) and G is asuitable subdomain of G .In this time-reversible setting the Schr¨odinger equation is well-posed whatever the signof α is, contrary to the heat equation. Thus, we have more choices for the random variable α ( · ). 16he average of the solutions to random Schr¨odinger equations may lead to very differentdynamics, depending on the random variable under consideration. To see this, let us firstconsider the Schr¨odinger system in the absence of control: ˆ y t − iα ∆ˆ y = 0 in G × (0 , T ) , ˆ y = 0 on ∂G × (0 , T ) , ˆ y (0) = ˆ y in G. (4.2)Here ˆ y = P ∞ j =1 ˆ y ,j e j ∈ L ( G ). Case 1 . When α ( · ) is a uniformly distributed random variable on [ a, b ], where a, b ∈ R ,then, Z Ω ˆ y ( x, t, ω ; ˆ y ) d P ( ω ) = 1 b − a Z ba ∞ X i =1 ˆ y ,j e − iλ j αt e j dα = 1 b − a ∞ X j =1 iλ j t ˆ y ,j (cid:0) e − iλ j at − e − iλ j bt (cid:1) e j . (4.3) Case 2 . When α ( · ) is an exponentially distributed random variable, then Z Ω ˆ y ( x, t, ω ; ˆ y ) d P ( ω ) = Z ∞ e − ( α − ∞ X j =1 ˆ y ,j e − iλ j αt e j dα = ∞ X j =1 iλ j t + 1 ˆ y ,j e − iλ j t e j . (4.4) Case 3 . For the normally distributed random variable α ( · ) we have, Z Ω ˆ y ( x, t, ω ; ˆ y ) d P ( ω ) = 1 √ π Z ∞−∞ e − α ∞ X j =1 ˆ y ,j e − iλ j αt e j dα = ∞ X j =1 ˆ y ,j e − λ j t e j . (4.5) Case 4 . When α ( · ) is a random variable with Laplace distribution we have Z Ω ˆ y ( x, t, ω ; ˆ y ) d P ( ω ) = 12 Z ∞−∞ e −| α | ∞ X j =1 ˆ y ,j e − iλ j αt e j dα = ∞ X j =1
11 + λ j t ˆ y ,j e − iλ j t e j . (4.6) Remark 4.1
The Laplace distribution can be thought of as two exponential distributions(with an additional location parameter) spliced together back-to-back. It governs the differenceof two independent identically distributed exponential random variables and can be regardedas the generalization of the exponential distribution on the whole real line. More details canbe found in [29]. ase 5 . For the Chi-squared distribution it holds: Z Ω ˆ y ( x, t, ω ; ˆ y ) d P ( ω ) = Z ∞ α k − e − α k Γ( k ) ∞ X j =1 ˆ y ,j e − iλ j αt e j dα = ∞ X j =1 iλ j t ) k ˆ y ,j e − iλ j t e j . (4.7) Remark 4.2
A Chi-squared distributed random variable is the sum of the squares of k in-dependent standard normally distributed random variables. It is one of the most widely usedprobability distributions in inferential statistics. We refer the readers to [25] for more details. Case 6 . For the Cauchy distribution we have Z Ω ˆ y ( x, t, ω ; ˆ y ) d P ( ω ) = Z ∞−∞ π (1 + α ) ∞ X j =1 ˆ y ,j e − iλ j αt e j dα = ∞ X j =1 ˆ y ,j e − λ j t e j . (4.8) Remark 4.3
The Cauchy distribution is associated with many processes, including reso-nance energy distribution, impact and natural spectral and quadratic stark line broadening.It also has important connections with other random variables. For example, when γ and γ are two independent standard normally distributed random variables, then the ratio γ /γ has the standard Cauchy distribution. More details can be found in [25]. Remark 4.4
From (4.8) , we know that when α is a random variable with Cauchy distri-bution, then the average of the solution to the random Schr¨odinger equation (4.2) becomesa solution of the heat equation. This is another example that after averaging, one enjoyenhanced regularity properties. According to the above results, there are essentially two different dynamics for the aver-ages, depending on whether the time-exponentials entering in the Fourier expansion are realor imaginary. In cases 3 and 6, the average has a heat-like behavior. However, in cases 1, 2,4 and 5, the average has a Schr¨odinger-like behavior.
We first recall the following assumptions on G and E :( A1 ) Let G ⊂ G be a nonempty open subset.( A2 ) Let E ⊂ [0 , T ] be a Lebesgue measurable set with positive measure. Theorem 4.1
Let ( A1 ) and ( A2 ) hold. If α is a random variable with normal distributionor Cauchy distribution, then the system (4.1) is null controllable in average with control u ( · ) ∈ L (0 , T ; L ( G )) . Further, there is a constant C > such that | u | L (0 ,T ; L ( G )) ≤ C | y | L ( G ) . (4.9)18 emark 4.5 Note that, in the present case, the random Schr¨odinger equations is null con-trollable in average without any assumption on the support G of the control, other than beingof positive measure. This is in contrast with the well known results on the null controllabilityof Schr¨odinger equations, where G is assumed, for instance, to fulfill the classical GeometricControl Condition (GCC)(see [32, 33] for example) or other geometric restrictions associatedto multiplier techniques or Carleman inequalities(see [30, 42] for example). In the presentcase, these restrictions on G are not needed since the averages behave in a parabolic fashion. To prove Theorem 4.1, we introduce the adjoint system of (4.1) as follows: z t + iα ∆ z = 0 in G × (0 , T ) ,z = 0 on ∂G × (0 , T ) ,z ( T ) = z in G. (4.10)By Theorem A.2, we only need to prove the following result. Theorem 4.2
Under the assumptions of Theorem 4.1 the system (4.10) is null observablein average if α is a random variable with normal or Cauchy distribution. Indeed, we have the following stronger observability estimates.
Proposition 4.1
Let ( A1 ) and ( A2 ) hold. Assume that α ( · ) is a random variable withnormal distribution or Cauchy distribution. Then there exists a constant C > such thatfor any z ∈ L ( G ) , it holds that (cid:12)(cid:12)(cid:12) Z Ω z (0 , ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) L ( G ) ≤ C Z E (cid:12)(cid:12)(cid:12) Z Ω z ( x, t, ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) L ( G ) dt. (4.11)The proof of Proposition 4.1 is very similar to the one for Theorem 3.2. We omit it here. Remark 4.6 If α ( · ) is a random variable with Cauchy distribution, then we have that Z Ω z ( x, t, ω ; z ) d P ( ω ) = Z ∞−∞ π (1 + α ) ∞ X j =1 z ,j e − iλ j α ( T − t ) e j dα = ∞ X j =1 z ,j e − λ j ( T − t ) e j . (4.12) In this case, Proposition 4.1 is an immediate corollary of the observability estimate for heatequations.This is an example of a system that is null but not exactly controllable in average.
In this subsection we consider the cases where the averages behave as Schr¨odinger-like sem-groups. We assume that E = [0 , T ] so that the control is active in any time instant withinthe time interval [0 , T ]. 19et us first consider the following Schr¨odinger equation ϕ t + κi ∆ ϕ = 0 in G × (0 , T ] ,ϕ = 0 on ∂G × (0 , T ) ,ϕ (0) = ϕ in G, (4.13)where κ ∈ R \ { } and ϕ ∈ L ( G ).We make the following assumption in this subsection on G :( A3 ) Whatever T > k = 0 are, there is a constant C > ϕ ∈ L ( G ), the solution ϕ ( · , · ) to (4.13) satisfies | ϕ | L ( G ) ≤ C Z T Z G | ϕ | dxdt. (4.14)We refer the readers to [3, 24, 32, 47] for the study of (4.14) under different conditions for G . In those articles one can find various sufficient conditions on the subset G so that theobservability inequality above holds, depending of the techniques of proof employed (multi-pliers, Carleman inequalities, Microlocal analysis). In particular, this observability inequalityfor the Schr¨odinger equation holds as soon as it is satisfied for the wave equation in sometime horizon. Thus, in particular, it holds under the classical Geometric Control Condition(GCC) guaranteeing, roughly, that all rays of geometric optics enter the observation set G in some uniform time.We have the following observability result for the system (4.13). Theorem 4.3
The following results hold: • Let α ( · ) be a uniformly distributed random variable on an interval [ a, b ] . Then, thesystem (4.1) is exactly controllable in average with V = H = L ( G ) and U = H − ( G ) . • Let α ( · ) be an exponentially distributed random variable. Then, the system (4.1) isexactly controllable in average with H = L ( G ) , V = H ( G ) ∩ H ( G ) and U = L ( G ) . • Let α ( · ) be a random variable with Laplace distribution. Then, the system (4.1) isexactly controllable in average with H = L ( G ) , V = H ( G ) ∩ H ( G ) and U = L ( G ) . • Let α ( · ) be a random variable with standard Chi-squared distribution. Then, the system (4.1) is exactly controllable in average with H = L ( G ) , V = H k ( G ) ∩ H ( G ) and U = L ( G ) .Proof of the first conclusion in Theorem 4.3 : We divide the proof into two steps. Step 1 . In this step, we show that the average of solutions can be represented by thedifference of the solutions of two Schr¨odinger equations. This was already noticed in [58].We present the argument here for the sake of completeness.We let z ∈ L ( G ) in (4.10) and ˜ z ( x, t ; z ) = R Ω z ( x, T − t, ω ; z ) d P ( ω ). Assume that z = P ∞ j =1 z ,j e j . Similar to (4.3), we have that˜ z ( x, t ; z ) = 1 b − a ∞ X j =1 iλ j t z ,j (cid:0) e − iλ j at − e − iλ j bt (cid:1) e j . (4.15)20et z a ( · , · ) = ∞ X j =1 z ,j iλ j e − iλ j at , z b ( · , · ) = ∞ X j =1 z ,j iλ j e − iλ j bt . Clearly, z a ( · , · ) and z b ( · , · ) solve the following equations respectively: z a,t + ai ∆ z a = 0 in G × [0 , T ] ,z a = 0 on ∂G × [0 , T ] ,z a (0) = A − z in G, (4.16) z b,t + bi ∆ z b = 0 in G × [0 , T ] ,z b = 0 on ∂G × [0 , T ] ,z b (0) = A − z in G. (4.17)Further, t ˜ z ( · , · ) = ( z a − z b )( · , · ) . Step 2 . In this step, we establish the exactly averaged observability estimate. From(4.16) and (4.17), we have that( i∂ t + a ∆)( i∂ t + b ∆)( z a − z b )= ( i∂ t + a ∆)( i∂ t + b ∆) z a − ( i∂ t + a ∆)( i∂ t + b ∆) z b = ( i∂ t + b ∆)( i∂ t + a ∆) z a = 0 . Hence, we know that ( i∂ t + b ∆)( z a − z b ) solves iϕ t + a ∆ ϕ = 0 in G × (0 , T ] ,ϕ = 0 on ∂G × (0 , T ) ,ϕ (0) = ( b − a ) z in G. (4.18)By assumption ( A3 ), for any z ∈ L ( G ), it holds that | z | L ( G ) ≤ C Z T Z G | ϕ ( x, t ) | dxdt ≤ C Z T Z G | ( i∂ t + b ∆)( z a − z b )( x, t ) | dxdt ≤ C Z T Z G (cid:12)(cid:12) ( i∂ t + b ∆) (cid:2) t ˜ z ( x, t ; z ) (cid:3)(cid:12)(cid:12) dxdt ≤ C Z T (cid:12)(cid:12) t ˜ z ( x, t ; z ) (cid:12)(cid:12) H ( G ) dt ≤ C Z T | ˜ z ( x, t ; z ) | H ( G ) dt. (4.19)The proofs of the second to the fourth conclusion in Theorem 4.3 are very similar. Weonly give that for the second one. Proof of the second conclusion in Theorem 4.3 : Let z ∈ V ′ = [ H (Ω) ∩ H (Ω)] ′ in (4.10)and ˜ z ( x, t ; z ) = R Ω z ( x, T − t, ω ; z ) d P ( ω ). We only need to prove that | z | V ′ ≤ C Z T Z G | ˜ z ( x, t ; z ) | dxdt. (4.20)21e divide the proof into two steps. Step 1 . In this step, we prove that a “weak” version of the exact averaged observability,that is, there is a lower order term in the right hand side of the inequality.Assume that z = P ∞ j =1 z ,j e j ∈ V ′ . Then, z ( x, t, ω ; z ) = ∞ X j =1 z ,j e − iαλ j ( T − t ) e j . Similar to (4.4), we have that˜ z ( x, t ; z ) = Z Ω z ( x, T − t, ω ; z ) d P ( ω ) = ∞ X j =1 iλ j t + 1 z ,j e − iλ j t e j . (4.21)This implies that for any δ > | ˜ z ( · , · ; z ) | L ( δ,T ; L ( G )) ≤ C ( δ, T ) | z | V ′ . (4.22)Let v ( x, t ) = ∞ X j =1 iλ j t z ,j e − iλ j t e j . From assumption ( A3 ), for a fixed δ >
0, we have that | z | V ′ = (cid:12)(cid:12)(cid:12) ∞ X j =1 iλ j z ,j e − iλ j t e j (cid:12)(cid:12)(cid:12) L ( G ) ≤ C Z Tδ Z G (cid:12)(cid:12)(cid:12) ∞ X j =1 z ,j λ j e − iλ j t e j (cid:12)(cid:12)(cid:12) dxdt = C Z Tδ Z G | tv ( x, t ) | dxdt. (4.23)Therefore, | z | V ′ ≤ C Z Tδ Z G | tv ( x, t ) | dxdt ≤ C h Z Tδ Z G | t ˜ z ( x, t ; z ) | dxdt + Z Tδ Z G | tv ( x, t ) − t ˜ z ( x, t ; z ) | dxdt i ≤ C Z Tδ Z G | t ˜ z ( x, t ; z ) | dxdt + C Z Tδ Z G (cid:12)(cid:12)(cid:12) ∞ X j =1 iλ j ( iλ j t + 1) z ,j e − iλ j t e j (cid:12)(cid:12)(cid:12) dxdt ≤ C Z Tδ Z G | t ˜ z ( x, t ; z ) | dxdt + C Z Tδ Z G (cid:12)(cid:12)(cid:12) ∞ X j =1 iλ j ( iλ j t + 1) z ,j e − iλ j t e j (cid:12)(cid:12)(cid:12) dxdt ≤ C Z Tδ Z G | t ˜ z ( x, t ; z ) | dxdt + C Z Tδ Z G (cid:12)(cid:12)(cid:12) ∞ X j =1 λ j z ,j e − iλ j t e j (cid:12)(cid:12)(cid:12) dxdt ≤ C Z Tδ Z G | t ˜ z ( x, t ; z ) | dxdt + C | A − z | V ′ . (4.24)22 tep 2 . In this step, we get rid of the term | A − z | V ′ in the right hand side of (4.24) bya compactness–uniqueness argument. More precisely, we are going to prove that | z | V ′ ≤ C Z Tδ Z G | t ˜ z ( x, t ; z ) | dxdt. (4.25)If (4.25) is not true, then we can find a sequence { z n } ∞ n =1 ⊂ L ( G ) with | z n | H − ( G ) = 1 suchthat Z Tδ Z G | t ˜ z ( x, t ; z n ) | dxdt ≤ n . (4.26)Since { z n } ∞ n =1 is bounded in V ′ , we can find a subsequence { z n k } ∞ k =1 ⊂ { z n } ∞ n =1 such that z n k converges weakly to some z ∗ ∈ H − ( G ) (4.27)and A − z n k converges strongly to A − z ∗ in V ′ . (4.28)According to (4.26) and (4.24), we know that | A − z n k | V ′ ≥ C − n k . This, together with (4.28), implies that there is a positive constant
C > | A − z ∗ | V ′ ≥ C . (4.29)Thus, the limit z ∗ is non trivial.Further, from (4.21) and (4.22), we know that ˜ z ( · , · ; z n k ) converges weakly to ˜ z ( · , · ; z ∗ ) in L ( δ, T ; L ( G )). Hence, Z Tδ Z G | ˜ z ( x, t ; z ∗ ) | dxdt ≤ lim k →∞ Z Tδ Z G | ˜ z ( x, t ; z n k ) | dxdt ≤ lim k →∞ δn k = 0 . Therefore, we find that ˜ z ( · , · ; z ∗ ) = 0 in G × ( δ, T ) . (4.30)We would like to show that this leads to z ∗ ≡ E △ = (cid:8) z ∈ V ′ : the solution to (4.10) with the initial datum z fulfills˜ z ( · , · ; z ) = 0 in G × ( δ, T ) (cid:9) . Clearly, z ∗ given in (4.27) belongs to E . We want to prove that E = { } , which would be incontradiction with the fact that z ∗ is nonzero. Step 3.
To show the claim that E = { } and conclude the proof, we proceed in severalsteps. Step 3.1.
We first prove that
E ⊂ L ( G ).23o do this, given ε ∈ (0 , δ ) and any solution ˜ z , we introduce the discrete time-derivativeˆ z ε ( x, t ; z ) = ˜ z ( x, t + ε ; z ) − ˜ z ( x, t ; z ) ε for t ∈ [0 , T − δ ] . (4.31)Then we have thatˆ z ε ( x, t ; z )= 1 ε ∞ X j =1 iλ j ( t + ε ) + 1 z ,j e − iλ j ( t + ε ) e j − ε ∞ X j =1 iλ j t + 1 z ,j e − iλ j t e j = 1 ε ∞ X j =1 iλ j ( t + ε )+1 z ,j (cid:0) e − iλ j ( t + ε ) − e − iλ j t (cid:1) e j + 1 ε ∞ X j =1 h iλ j ( t + ε )+1 − iλ j t +1 i z ,j e − iλ j t e j = ∞ X j =1 iλ j ( t + ε )+1 z ,j e − iλ j ε − ε e − iλ j t e j − ∞ X j =1 iλ j [ iλ j ( t + ε )+1]( iλ j t + 1) z ,j e − iλ j t e j (4.32)and ˆ z ε ( x, z ) = ∞ X j =1 iλ j ε + 1 z ,j e − iλ j ε − ε e j − ∞ X j =1 iλ j iλ j ε + 1 z ,j e j . (4.33)Let v ε ( x, t ) = ∞ X j =1 iλ j ( t + ε ) z ,j e − iλ j ε − ε e − iλ j t e j − ∞ X j =1 iλ j ( t + ε ) z ,j e − iλ j t e j . (4.34)Then, again, by assumption ( A3 ), as in the proof of (4.23), we have (cid:12)(cid:12)(cid:12) ˜ z ε ( x, ε ; z ) − ˜ z ε ( x, z ) ε (cid:12)(cid:12)(cid:12) V ′ = (cid:12)(cid:12) ˆ z ε ( x, z ) (cid:12)(cid:12) V ′ ≤ C Z T − δδ Z G | ( t + ε ) v ε ( x, t ) | dxdt, where the constant C is independent of ε . Thus, we obtain that (cid:12)(cid:12)(cid:12) ˜ z ε ( x, ε ; z ) − ˜ z ε ( x, z ) ε (cid:12)(cid:12)(cid:12) V ′ ≤ C Z T − δδ Z G | ˆ z ε ( x, t ; z ) | dxdt + C Z T − δδ Z G | ˆ z ε ( x, t ; z ) − v ε ( x, t ) | dxdt ≤ C Z T − δδ Z G | ˆ z ε ( x, t ; z ) | dxdt + C Z T − δδ Z G | ˆ z ε ( x, t ; z ) − v ε ( x, t ) | dxdt. (4.35)Let us estimate the second term in the right hand side of (4.35). From (4.32) and (4.34), we24ave that Z T − δδ Z G | ˆ z ε ( x, t ; z ) − v ε ( x, t ) | dxdt ≤ Z T − δδ Z G (cid:12)(cid:12)(cid:12) ∞ X j =1 h iλ j ( t + ε ) + 1 − iλ j ( t + ε ) i z ,j e − iλ j ε − ε e − iλ j t e j (cid:12)(cid:12)(cid:12) dxdt +2 Z T − δδ Z G (cid:12)(cid:12)(cid:12) ∞ X j =1 h iλ j [ iλ j ( t + ε ) + 1]( iλ j t + 1) − iλ j ( t + ε ) i z ,j e − iλ j t e j (cid:12)(cid:12)(cid:12) dxdt ≤ C ∞ X j =1 λ j z ,j (cid:12)(cid:12)(cid:12) e − iλ j ε − ε (cid:12)(cid:12)(cid:12) + C ∞ X j =1 λ j z ,j . (4.36)If λ j < ε , then (cid:12)(cid:12)(cid:12) e − iλ j ε − ε (cid:12)(cid:12)(cid:12) = 1 ε (cid:12)(cid:12)(cid:12) ∞ X k =1 k ! ( − iλ j ε ) k (cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12) iλ j ∞ X k =1 k ! ( − iλ j ε ) k − (cid:12)(cid:12)(cid:12) ≤ λ j ∞ X k =1 k ! ≤ λ j . (4.37)If λ j > ε , then (cid:12)(cid:12)(cid:12) e − iλ j ε − ε (cid:12)(cid:12)(cid:12) = 1 ε (cid:12)(cid:12) e − iλ j ε − (cid:12)(cid:12) ≤ λ j (cid:12)(cid:12) e − iλ j ε − (cid:12)(cid:12) ≤ λ j . (4.38)According to (4.36), (4.37) and (4.38), we obtain that Z T − δδ Z G | ˆ z ε ( x, t ; z ) − v ε ( x, t ) | dxdt ≤ C ∞ X j =1 λ j z ,j . (4.39)As a result of (4.35) and (4.39), we have that (cid:12)(cid:12)(cid:12) ˜ z ε ( x, ε ; z ) − ˜ z ε ( x, z ) ε (cid:12)(cid:12)(cid:12) V ′ ≤ C Z T − δδ Z G | ˆ z ε ( x, t ; z ) | dxdt + C ∞ X j =1 λ j z ,j . (4.40)Since z ∈ E , we know that ˆ z ε ( x, t ; z ) = 0 in G × ( δ, T ) . This, together with (4.40), implies that for any ε ∈ (0 , δ ), (cid:12)(cid:12)(cid:12) ˜ z ε ( x, ε ; z ) − ˜ z ε ( x, z ) ε (cid:12)(cid:12)(cid:12) V ′ ≤ C ∞ X j =1 z ,j λ j ≤ C | z | V ′ . (4.41)Letting ε → | z | L ( G ) ≤ C (cid:12)(cid:12)(cid:12) ∞ X j =1 iλ j z ,j e j (cid:12)(cid:12)(cid:12) V ′ ≤ C ∞ X j =1 z ,j λ j ≤ C | z | V ′ . (4.42)Then, we have that E ⊂ L ( G ). 25 tep 3.2. The same estimate deduces that E is a finite-dimensional subspace of L ( G ),since | z | L ( G ) ≤ C | z | V ′ . (4.43) Step 3.3.
We now claim that A ∆ E ⊂ E .Utilizing (4.21) again, we have that˜ z ( · , · ; z ) ∈ C ([0 , T ]; L ( G )) for any z ∈ E . Since ˜ z ( · , · ; z ) = 0 in G × ( δ, T ) for any δ >
0, we see that z = 0 in G .Thanks to E ⊂ L ( G ), we get that A ∆ E ⊂ V ′ and ˜ z ( x, t ; A ∆ z ) = A ∆ ˜ z ( x, t ; z ) = 0 in G × ( δ, T ). Therefore, we obtain that A ∆ E ⊂ E . Step 3.4.
To conclude, assume that
E 6 = { } . Then, there would exist a non-trivialeigenfunction ψ ∈ E and an eigenvalue µ ∈ R such that − ∆ ψ = µψ in G,ψ = 0 on ∂G,ψ = 0 in G . However, by the classical unique continuation property for elliptic equations, ψ = 0 in G ,which contradicts that ψ is an eigenfunction.This concludes the proof of the fact that E = { } .As a consequence, we derive the desired estimate (4.20). Remark 4.7
We have utilized a compactness-uniqueness argument in the above proof, whichhas been extensively used in the proof of observability estimates (see [7] for example). Notehowever that, normally, this is done for solutions of PDE models. The averages underconsideration not being solutions of a specific PDE this argument needs to be carefully adaptedas we have done above.
In this subsection, for convenience, we assume that ∂G is C ∞ smooth, although mostcomments and results make sense with different weaker regularity assumptions.We have solved the internal averaged controllability problems for some particular classesof random heat and Schr¨odinger equations. The same could be done for boundary controlproblems.Let us consider the following heat equation with boundary control and random constantdiffusivity: y t − α ∆ y = 0 in G × (0 , T ] ,y = u on Γ × (0 , T ) ,y = 0 on ( ∂G \ Γ ) × (0 , T ) ,y (0) = y in G. (5.1)26ere Γ is an open subset of ∂G , α ( · ) is a random variable, u ∈ L (0 , T ; L (Γ )) and y ∈ L ( G ).We have the following result. Theorem 5.1
Let α ( · ) be a uniformly distributed or an exponentially distributed randomvariable. The system (5.1) is null controllable in average with control u ( · ) ∈ L (0 , T ; L (Γ )) .Further, there is a constant C > such that | u | L (0 ,T ; L (Γ )) ≤ C | y | L ( G ) . (5.2)To prove Theorem 5.1, we consider the adjoint system of (5.1) as follows: z t + α ∆ z = 0 in G × (0 , T ] ,z = 0 on ∂G × (0 , T ) ,z ( T ) = z in G. (5.3)One only need to prove the following result. Theorem 5.2
There exists a constant
C > such that for any z ∈ L ( G ) , and either α ( · ) is a uniformly distributed or an exponentially distributed random variable, it holds that (cid:12)(cid:12)(cid:12) Z Ω z ( · , , ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) L ( G ) ≤ C Z T Z Γ (cid:12)(cid:12)(cid:12) Z Ω ∂z ( x, t, ω ; z ) ∂ν d P ( ω ) (cid:12)(cid:12)(cid:12) d Γ dt. (5.4)The proof is very similar to the one for Theorem 3.2. We only give a sketch here. Let usassume that α ( · ) is an exponentially distributed random variable. From [34, Page 345], wehave the following result: X λ j ≤ r a j ≤ C e C √ r e t − t Z t t Z Γ (cid:12)(cid:12)(cid:12) X λ j ≤ r e √ λ j t a j ∂e j ∂ν (cid:12)(cid:12)(cid:12) d Γ dt for any 0 ≤ t < t ≤ T. (5.5)Let T k = (1 − k − ) T for k ∈ N and r k = 2 k +1) [ln(6 C ) + C ]. Similar to the proof of(3.25), we can obtain that T k +1 − T k C e C √ r k | ˜ z ( · , T k ; z ) | L ( G ) − C e C √ r k + 1 C e C √ r k ( T k +1 − T k ) e − Tk +1 − Tk r k | ˜ z ( · , T k +1 ; z ) | L ( G ) ≤ Z T k +1 T k Z Γ (cid:12)(cid:12)(cid:12) ∂ ˜ z ( x, t ; z ) ∂ν (cid:12)(cid:12)(cid:12) d Γ dt. (5.6)By summarizing the inequality (5.6) from k = 1 to k = ∞ , we obtain that T − T C e C √ r | z | L ( G ) + ∞ X k =1 f k | ˜ z ( · , T k +1 ; z ) | L ( G ) ≤ Z T Z Γ (cid:12)(cid:12)(cid:12) ∂ ˜ z ( x, t ; z ) ∂ν (cid:12)(cid:12)(cid:12) d Γ dt, (5.7)where f k = T k +2 − T k +1 C e C √ r k +1 − C e C √ r k + 1 C e C √ r k ( T k +1 − T k ) e − Tk +1 − Tk r k , k = 1 , , · · · r k = 2 k +1) [ln(6 C ) + C ], we have that f k ≥ k = 1 , , · · · This, together with (5.7), deduces that | z | L ( G ) ≤ C e C √ r T − T Z T Z Γ (cid:12)(cid:12)(cid:12) ∂ ˜ z ( x, t ; z ) ∂ν (cid:12)(cid:12)(cid:12) d Γ dt. (5.8)This completes the proof. Consider the following random Schr¨odinger equations: y t − αi ∆ y = 0 in G × (0 , T ] ,y = u on Γ × (0 , T ) ,y = 0 on ( ∂G \ Γ ) × (0 , T ) ,y (0) = y in G. (5.9)Here Γ is an open subset of ∂G , α ( · ) is a random variable, u ∈ L (0 , T ; L (Γ )) and y ∈ L ( G ). We have the following controllability results. Theorem 5.3
System (5.9) is null controllable in average if α is a random variable withnormal distribution or Cauchy distribution. Further, we assume the following condition holds:( A4 ) Whatever T > k = 0 are, there are constants C and C such that for any ϕ ∈ H ( G ), the solution ϕ ( · , · ) to (4.13) satisfies | ϕ | H ( G ) ≤ C Z T Z G (cid:12)(cid:12)(cid:12) ∂ϕ∂ν (cid:12)(cid:12)(cid:12) d Γ dt ≤ C | ϕ | H ( G ) . (5.10)We refer the readers to [33] for the conditions on Γ for which ( A4 ) holds. Theorem 5.4
Assume that ( A4 ) holds. We have the following results: • Let α ( · ) be a uniformly distributed random variable on an interval [ a, b ] . Then, thesystem (4.1) is exactly controllable in average with V = H = L ( G ) and U = H − (Γ ) . • Let α ( · ) be an exponentially distributed random variable. Then, the system (4.1) isexactly controllable in average with H = L ( G ) , V = H ( G ) and U = L (Γ ) . • Let α ( · ) be a random variable with Laplace distribution. Then, the system (4.1) isexactly controllable in average with H = L ( G ) , V = H ( G ) ∩ H ( G ) and U = L (Γ ) . • Let α ( · ) be a random variable with standard Chi-squared distribution. Then, the system (4.1) is exactly controllable in average with H = L ( G ) , V = H k − ( G ) ∩ H ( G ) and U = L (Γ ) .
28s usual, we introduce the adjoint system of (5.9) as follows: z t + iα ∆ z = 0 in G × (0 , T ) ,z = 0 on ∂G × (0 , T ) ,z ( T ) = z in G. (5.11)We can prove the following results. Theorem 5.5
Let α ( · ) be a standard normally distributed random variable. There exists aconstant C > such that for any z ∈ L ( G ) , it holds that (cid:12)(cid:12)(cid:12) ∞ X j =1 z ,j e − λ j T e j (cid:12)(cid:12)(cid:12) L ( G ) ≤ C Z T Z Γ (cid:12)(cid:12)(cid:12) Z Ω ∂z ( x, t, ω ; z ) ∂ν d P ( ω ) (cid:12)(cid:12)(cid:12) d Γ dt. (5.12) Theorem 5.6
Let α ( · ) be a random variable with standard Cauchy distribution. There existsa positive constant C such that for any z ∈ L ( G ) , it holds that (cid:12)(cid:12)(cid:12) ∞ X j =1 z ,j e − λ j T e j (cid:12)(cid:12)(cid:12) L ( G ) ≤ C Z T Z Γ (cid:12)(cid:12)(cid:12) Z Ω ∂z ( x, t, ω ; z ) ∂ν d P ( ω ) (cid:12)(cid:12)(cid:12) d Γ dt. (5.13)By means of (4.12), Theorem 5.6 is nothing but the boundary observability estimate forheat equations. The proof of Theorem 5.5 is very similar to the one for Theorem 5.2. Weomit it here.Further, the proof of Theorem 5.4 is also very analogous to the proofs for Theorems 4.3.We only give a sketch of the proof of the second conclusion in Theorem 5.4.Let z ∈ H − ( G ) in (4.10) and ˜ z ( x, t ; z ) = R Ω z ( x, T − t, ω ; z ) d P ( ω ). Then, we only needto prove that | z | H − ( G ) ≤ C Z T Z Γ (cid:12)(cid:12)(cid:12) ∂ ˜ z ( x, t ; z ) ∂ν (cid:12)(cid:12)(cid:12) d Γ dt. (5.14)Assume that z = P ∞ j =1 z ,j e j . We have˜ z ( x, t ; z ) = ∞ X j =1 iλ j t + 1 z ,j e − iλ j t e j . (5.15)Let v ( x, t ) = ∞ X j =1 iλ j t z ,j e − iλ j t e j . From ( A4 ), for a fixed δ ∈ (0 , T ), we have that | z | H − ( G ) = (cid:12)(cid:12)(cid:12) ∞ X j =1 iλ j z ,j e − iλ j t e j (cid:12)(cid:12)(cid:12) H ( G ) ≤ C Z Tδ Z Γ (cid:12)(cid:12)(cid:12) t ∂v ( x, t ) ∂ν (cid:12)(cid:12)(cid:12) d Γ dt. (5.16)29herefore, | z | H − ( G ) ≤ C Z Tδ Z Γ (cid:12)(cid:12)(cid:12) t ∂v ( x, t ) ∂ν (cid:12)(cid:12)(cid:12) d Γ dt ≤ C h Z Tδ Z Γ (cid:12)(cid:12)(cid:12) t ∂ ˜ z ( x, t ; z ) ∂ν (cid:12)(cid:12)(cid:12) d Γ dt + Z Tδ Z Γ (cid:12)(cid:12)(cid:12) t ∂v ( x, t ) ∂ν − t ∂ ˜ z ( x, t ; z ) ∂ν (cid:12)(cid:12)(cid:12) d Γ dt i ≤ C Z Tδ Z Γ (cid:12)(cid:12)(cid:12) t ∂ ˜ z ( x, t ; z ) ∂ν (cid:12)(cid:12)(cid:12) d Γ dt + C Z Tδ (cid:12)(cid:12)(cid:12) ∞ X j =1 iλ j ( iλ j t + 1) z ,j e − iλ j t e j (cid:12)(cid:12)(cid:12) H ( G ) dt ≤ C Z Tδ Z Γ (cid:12)(cid:12)(cid:12) t ∂ ˜ z ( x, t ; z ) ∂ν (cid:12)(cid:12)(cid:12) d Γ dt + C | z | H − ( G ) . (5.17)We claim that | z | H − ( G ) ≤ C Z T Z Γ (cid:12)(cid:12)(cid:12) t ∂ ˜ z ( x, t ; z ) ∂ν (cid:12)(cid:12)(cid:12) d Γ dt. (5.18)If (5.18) is not true, then we can find a sequence { z n } ∞ n =1 ⊂ L ( G ) with | z n | H − ( G ) = 1 suchthat Z T Z Γ (cid:12)(cid:12)(cid:12) t ∂ ˜ z ( x, t ; z ) ∂ν (cid:12)(cid:12)(cid:12) d Γ dt ≤ n . (5.19)Since { z n } ∞ n =1 is bounded in H − ( G ), we can find a subsequence { z n k } ∞ k =1 ⊂ { z n } ∞ n =1 suchthat z n k converges weakly to some z ∗ ∈ H − ( G ). From (5.19), we know that | z ∗ | H − ( G ) ≥ C (5.20)for a positive constant and ∂ ˜ z ( · , · ; z ∗ ) ∂ν = 0 on Γ × (0 , T ) . (5.21)Put e E △ = { z ∈ H − ( G ) : the solution to (5.11) with the final datum z fulfills (5.21) } . Analogous to the proof that E = { } , we can prove that e E = { } , which contradicts (5.20).Hence, we know that (5.18) holds. We have considered the averaged control problems of random heat and Schr¨odinger equa-tions for the internal control ( resp. boundary control) supported in G × E ( resp. Γ × E ),where G ⊂ G ( resp. Γ ⊂ ∂G ) is a nonempty open subset. By means of the method in [4],one can consider the case that the internal control ( resp. the boundary control) is supported30n a measurable subset D ⊂ G × (0 , T ) ( resp. G ⊂ ∂G × (0 , T )). For instance, we can considerthe following systems: y t − α ∆ y = χ D u in G × (0 , T ) ,y = 0 on ∂G × (0 , T ) ,y (0) = y in G, (5.22)and y t − iα ∆ y = χ D u in G × (0 , T ) ,y = 0 on ∂G × (0 , T ) ,y (0) = y in G. (5.23)Here y ∈ L ( G ), D ⊂ G × (0 , T ) is a Lebesgue measurable set with positive Lebesguemeasure and u ∈ L ∞ ( G × (0 , T )). One can combine the proof of Theorem 3.1 and Corollary1 in [4] to prove the following results. Theorem 5.7
Assume that, either α ( · ) is a uniformly or exponentially distributed randomvariable. Then, the system (5.22) is null controllable in average with control u ( · ) ∈ L ∞ ( G × (0 , T )) . Further, there is a constant C > such that | u | L ∞ ( G × (0 ,T )) ≤ C | y | L ( G ) . Theorem 5.8 If α is a random variable with normal distribution or Cauchy distribution,then the system (5.23) is null controllable in average with control u ( · ) ∈ L ∞ ( G × (0 , T )) .Further, there is a constant C > such that | u | L ∞ ( G × (0 ,T )) ≤ C | y | L ( G ) . Next, let us consider the following systems: y t − α ∆ y = 0 in G × (0 , T ] ,y = u on G ,y = 0 on [ ∂G × (0 , T )] \ G ,y (0) = y in G, (5.24)and y t − αi ∆ y = 0 in G × (0 , T ] ,y = u on G ,y = 0 on [ ∂G × (0 , T )] \ G ,y (0) = y in G. (5.25)Here G is a Lebesgue measurable subset of ∂G × (0 , T ) with positive Lebesgue measure, α ( · )is a random variable, u ∈ L ∞ ( ∂G × (0 , T )) and y ∈ L ( G ).One can combine the proof of Theorem 5.1 and Corollary 1 in [4] to prove the followingresults. 31 heorem 5.9 Assume that, either α ( · ) is a uniformly or exponentially distributed randomvariable. Then, the system (5.24) is null controllable in average with control u ( · ) ∈ L ∞ ( G × (0 , T )) . Further, there is a constant C > such that | u | L ∞ ( ∂G × (0 ,T )) ≤ C | y | L ( G ) . Theorem 5.10 If α is a random variable with normal distribution or Cauchy distribution,then the system (5.25) is null controllable in average with control u ( · ) ∈ L ∞ ( G × (0 , T )) .Further, there is a constant C > such that | u | L ∞ ( ∂G × (0 ,T )) ≤ C | y | L ( G ) . We have studied the averaged controllability problems for some random heat equationsand random Schr¨odinger equations. In the results proved so far we have obtained averagedcontrollability for parameter-depending equations that were controllable for each value ofthe parameter. Here, we give an example of model which is not null controllable when onefixes an ω but that gains null controllability by the averaging process.Consider the following equation: ( iy t + αA γ ∆ y = Bu in (0 , T ] ,y (0) = y . (5.26)Here γ ∈ ( , ), y ∈ L ( G ), u ∈ L (0 , T ; L ( G )) and Bu = χ G u .The adjoint system of (5.26) reads ( iz t − αA γ ∆ z = 0 in [0 , T ) ,z ( T ) = z , (5.27)where z ∈ L ( G ). Assume that z = P ∞ j =1 z ,j e j . If α ( · ) : Ω → R is a standard normallydistributed random variable, then we know that Z Ω z ( x, t, ω ; z ) d P ( ω ) = 1 √ π Z ∞−∞ e − α ∞ X j =1 z ,j e − iλ γj α ( T − t ) e j dα = ∞ X j =1 z ,j e − λ γj ( T − t ) e j . (5.28)Similar to the proof of Theorem 4.1, we can establish the following result. Theorem 5.11
Let E ⊂ [0 , T ] be a measurable set with positive Lebesgue measure m ( E ) .There exists a constant C > such that for any z ∈ L ( G ) , it holds that (cid:12)(cid:12)(cid:12) ∞ X j =1 z ,j e − λ γj T e j (cid:12)(cid:12)(cid:12) L ( G ) ≤ C Z E (cid:12)(cid:12)(cid:12) Z Ω z ( x, t, ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) L ( G ) dt. (5.29)32heorem 5.11 implies that the system (5.26) is null controllable in average. However, itis not null controllable for any fixed α ∈ R even for d = 1. For example, let α = 1 and G = (0 , z ( x, t ) = √ ∞ X j =1 z ,j e − i ( jπ ) γ t sin( jπx ) . Since lim j →∞ (cid:8) [( j + 1) π )] γ − ( jπ ) γ (cid:9) = 0 , we know that the following inequality | z ( · , T ) | L (0 , = ∞ X j =1 z ,j ≤ C Z T Z G | z ( x, t ) | dxdt does not hold for any C > We can also consider the approximate averaged controllability problem of some moregeneral random heat equations. We make the following assumptions on the coefficients a jk : G × Ω → R n × n ( j, k = 1 , , · · · , n ): (H1) a jk ( · , ω ) : G → R is analytic, P -a.s., and a jk = a kj . (H2) For a.e. ω ∈ Ω , there is a constant C ( ω ) > such that for any multi-index η = ( η , · · · , η n ) ∈ ( N ∪ { } ) n , (cid:12)(cid:12)(cid:12) ∂ η a jk ( x, ω ) ∂x η (cid:12)(cid:12)(cid:12) ≤ C ( ω ) | η | ! R | η | , for any j, k = 1 , · · · , n, where R is a positive constant larger than max x ∈ G | x | , and C ( · ) satisfies that Z Ω C ( ω ) d P ( ω ) < ∞ . (H3) There exists a constant s > such that n X j,k =1 a jk ( ω, t, x ) ξ j ξ k ≥ s | ξ | , ∀ ( ω, x, ξ ) ≡ ( ω, x, ξ , · · · , ξ n ) ∈ Ω × G × R n . (5.30)Consider the following random heat equation: y t − n X j,k =1 (cid:0) a jk y x j (cid:1) x k = χ G u in G × (0 , T ) ,y = 0 on ∂G × (0 , T ) ,y (0) = y in G. (5.31)Here the initial datum y ∈ L ( G ).We have the following result. 33 heorem 5.12 Under the assumptions (H1) – (H3) above system (5.31) is approximatelycontrollable in average in any time T > and from any open non-empty subset G of G .Proof : By Theorem A.3, we only need to prove that the adjoint system of (5.31) satisfiesthe averaged unique continuation property. Its adjoint system reads z t + n X j,k =1 (cid:0) a jk z x j (cid:1) x k = 0 in G × (0 , T ) ,z = 0 on ∂G × (0 , T ) ,z ( T ) = z in G, (5.32)where the final datum z ∈ L ( G ). From (H1) and (H3) , we know that for any t ∈ [0 , T )and a.e. ω ∈ Ω, the solution z ( · , t, ω ; z ) is analytic in G (see [18, 28] for example). Further,for any ball B r ⊂ G with radius r , there is a constant C > η ∈ ( N ∪ { } ) n , (cid:12)(cid:12)(cid:12) ∂ η z ( · , t, ω ; z ) ∂x η (cid:12)(cid:12)(cid:12) ≤ CC ( ω ) | η | ! r | η | in B r . From (H2) , we have that (cid:12)(cid:12)(cid:12) ∂ η R Ω z ( · , t, ω ) d P ( ω ) ∂x η (cid:12)(cid:12)(cid:12) ≤ C | η | ! r | η | . Then, we know that R Ω z ( · , t, ω ) d P ( ω ) is analytic in B r . Hence, it is analytic in G . Since R Ω z ( · , · , ω ) d P ( ω ) = 0 in G × (0 , T ), we get that it vanishes everywhere in G × (0 , T ). Notingthat it is continuous in L ( G ) with respect to t , we conclude that z = 0 in G , which impliesthat (5.32) satisfies the averaged unique continuation property. One can also consider the internal and boundary averaged controllability problems forrandom heat and random Schr¨odinger equations with random initial data. Let us firstconsider the following random heat equation: y t − α ∆ y = χ G × E u in G × (0 , T ) ,y = 0 on ∂G × (0 , T ) ,y (0 , ω ) = y ( ω ) in G, (5.33)Here y ( · ) ∈ L (Ω; L ( G )), and G and E are subsets of G and [0 , T ] respectively, where thecontrols are being applied. The constant diffusivity α : Ω → R + is assumed to be a randomvariable.According to Remark A.1, we know that to prove the averaged null controllability of(5.33), we only need to establish the following observability estimate: (cid:16) Z Ω (cid:12)(cid:12)(cid:12) z (0 , ω ; z ) (cid:12)(cid:12)(cid:12) L ( G ) d P ( ω ) (cid:17) ≤ C Z E (cid:12)(cid:12)(cid:12) Z Ω z ( t, ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) L ( G ) dt, (5.34)where z solves (3.7) and C is independent of z .34f α ( · ) is the exponentially distributed random variable, then Z Ω (cid:12)(cid:12)(cid:12) z (0 , ω ; z ) (cid:12)(cid:12)(cid:12) L ( G ) d P ( ω ) = Z ∞ e − ( α − ∞ X j =1 z ,j e − λ j αT dα = ∞ X j =1 λ j T + 1 z ,j e − λ j T . (5.35)From (5.35), similar to the proof of the inequality (3.8), one can obtain (5.34). The samething can be done if α ( · ) is a uniformly distributed random variable on [ a, b ] for 0 < a < b . Proposition 5.1
Let ( A1 ) and ( A2 ) hold. Assume that, either α ( · ) is a uniformly orexponentially distributed random variable. Then, the system (5.33) is null controllable inaverage with control u ( · ) ∈ L (0 , T ; L ( G )) . Further, there is a constant C > such that | u | L (0 ,T ; L ( G )) ≤ C | y | L (Ω; L ( G )) . (5.36)Thanks to Remark A.1, we know that in order to show that (5.33) is approximatelycontrollable in average, one just needs to prove that the solution to (3.7) satisfies the averagedunique continuation property, which is obtained in the proof of Theorem 3.3. Hence, we havethe following result: Proposition 5.2
Let ( A1 ) and ( A2 ) hold. System (5.33) is approximately controllable inaverage, provided that α ( · ) is a uniformly distributed or an exponentially distributed randomvariable. Next, we consider the averaged controllability problem for the following randomSchr¨odinger equation: y t − iα ∆ y = χ G × E u in G × (0 , T ) ,y = 0 on ∂G × (0 , T ) ,y (0) = y in G. (5.37)Here y ( · ) ∈ L (Ω; V ), and G and E are subsets of G and [0 , T ] respectively, where thecontrols are being applied. α : Ω → R is assumed to be a random variable.In virtue of Remark A.1, we know that to get the averaged exact controllability of thesystem (5.37), one only need to prove that the solution to the equation (4.10) is exactlyaveraged observable. As a result of these facts, we know that the conclusions in Theorem4.3 also hold for the system (5.37). More precisely, we have the following results: Proposition 5.3
The following results hold: • Let α ( · ) be a uniformly distributed random variable on an interval [ a, b ] . Then, the sys-tem (5.37) is exactly controllable in average with V = H = L ( G ) and U = H − ( G ) . • Let α ( · ) be an exponentially distributed random variable. Then, the system (5.37) isexactly controllable in average with H = L ( G ) , V = H ( G ) ∩ H ( G ) and U = L ( G ) . Let α ( · ) be a random variable with Laplace distribution. Then, the system (5.37) isexactly controllable in average with H = L ( G ) , V = H ( G ) ∩ H ( G ) and U = L ( G ) . • Let α ( · ) be a random variable with standard Chi-squared distribution. Then, the system (5.37) is exactly controllable in average with H = L ( G ) , V = H k ( G ) ∩ H ( G ) and U = L ( G ) . Further, let us consider the averaged null controllability problem for the system (5.37).Due to Remark A.1, we only need to prove the following observability estimate: (cid:16) Z Ω (cid:12)(cid:12)(cid:12) z (0 , ω ; z ) (cid:12)(cid:12)(cid:12) L ( G ) d P ( ω ) (cid:17) ≤ C Z E (cid:12)(cid:12)(cid:12) Z Ω z ( x, t, ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) L ( G ) dt. (5.38)When α ( · ) is a normally distributed random variable, we have that Z Ω (cid:12)(cid:12)(cid:12) z (0 , ω ; z ) (cid:12)(cid:12)(cid:12) L ( G ) d P ( ω ) = 1 √ π Z ∞−∞ e − α ∞ X j =1 z ,j e − iλ j αT dα = ∞ X j =1 z ,j e − λ j t . (5.39)When α ( · ) is a random variable with the Cauchy distribution, we have that Z Ω (cid:12)(cid:12)(cid:12) z (0 , ω ; z ) (cid:12)(cid:12)(cid:12) L ( G ) d P ( ω ) = Z ∞−∞ π (1 + α ) ∞ X j =1 z ,j e − iλ j αT dα = ∞ X j =1 z ,j e − λ j T . (5.40)Thanks to (5.39) and (5.39), and similar to the proof of Theorem 3.2, one can showthat the inequality (5.38) holds when α ( · ) is a random variable with normal or Cauchydistribution. Therefore, we have the following result: Proposition 5.4
Under the assumptions of Theorem 4.1 the system (5.37) is null control-lable in average if α is a random variable with normal or Cauchy distribution. (2.1)If a system is null( resp. exactly) controllable in average, then we can drive the average ofthe state to rest ( resp. a given destination). It is then natural to analyse whether one cancontrol the higher order moments of the state and, in particular, the covariance Cor( y ) ofthe state y , which is defined as follows:Cor( y ) = Z Ω (cid:16) y ( T, ω ) − Z Ω y ( T, ω ) d P ( ω ) (cid:17) ⊗ (cid:16) y ( T, ω ) − Z Ω y ( T, ω ) d P ( ω ) (cid:17) d P ( ω ) , where ⊗ denotes the tensor product of two elements in H .36or( y ) is an element in H ⊗ H , which is also a Hilbert space. With the goal of reinforcingthe averaged controllability property studied in this paper, a first thought could be to lookfor the controllability of the covariance. But this is not suitable since the variance measureshow far the random variable is spread out. In other words, loosely speaking, the smallerthe | Cor( y ( T )) | H ⊗ H is, the closer the state y ( T, · ) is concentrated to an element in H and,therefore, closer to the simultaneous controllability property. A more natural goal is to lookfor the null controllability of the covariance. However, from the definition of Cor( · ), it istrivial to see that Cor( y ( T )) = 0 if and only if y ( T, · ) = R Ω y ( T, ω ) d P ( ω ) with probabilityone, i.e., y ( T, ω ) is independent of ω with probability one. Nevertheless, this is impossibleif we only use parameter independent control (see Remark 2.4). Therefore, since we wantto make | Cor( y ( T )) | V ⊗ V as small as possible, it is natural to study the following optimalcontrol problem: Problem (OP2) : Minimize | Cor( y ( T )) | V ⊗ V for u ( · ) ∈ U null [0 , T ] △ = n u ∈ L (0 , T ; U ) : the average of solutions to (2.1) correspondsto u satisfies that Z Ω y ( T, ω ; y ) dω = 0 o . Problem (OP2) is an optimal control problem with a terminal constraint. Following [36, 53],one can derive a Pontryagin type Maximum Principle for it. However, this is beyond thescope of this paper.
There are many interesting and important (at least for us) problems in this topic. We presentsome of them here briefly: • Averaged controllability problems for general random heat and Schr¨odingerequations.
We have only studied the averaged controllability problems for some very special classesof random heat and Schr¨odinger equations, for which the averaged dynamics could becomputed explicitly. It would be interesting to investigate some more general classesof parameter dependent systems. For example, is it (3.1) null controllable in averageif we take α to be a random variable with Chi-squared distribution?Furthermore, the method we use to study (3.1) and (4.1) depends on the fact that theeigenfunctions of − αA ∆ are independent of ω . Thus, more general random heat andSchr¨odinger equations (as, for instance, (5.31)) where the principal part of the PDEdepends on ω cannot be treated in this way.The method of proof employed to show the approximate averaged controllability of(5.31) is based on the use of the space-time analyticity properties of solutions (5.32)to derive unique continuation properties, but it does not provide any quantitativeinformation. 37 The relationship between averaged controllability properties and the ran-dom variable .We have shown that different random variable α ( · ) may lead to different controllabilityproperty of the system (4.1). It is interesting to give a description of the relationshipbetween the random variable α ( · ) and the controllability property of the system (4.1).For instance, for what kind of random variables, the system (4.1) is null controllablein average? Is there a random variable such that the system (4.1) is neither exactlycontrollable in average nor null controllable in average? • The null averaged controllability problem for random fractional heat equa-tions .We have proven that the random fractional Schr¨odinger equations are null control-lable in average for γ ∈ ( , ). It is more interesting to study the same problem forrandom fractional heat equations, which describe the anomalous diffusion process. Inparticularly, consider the following system: ( y t + αA γ ∆ y = Bu in (0 , T ] ,y (0) = y . (5.41)Here γ > y ∈ L ( G ), u ∈ L (0 , T ; L ( G )) and Bu = χ G u .It is clear that when γ ≤ and α ( · ) is a uniformly or exponentially distributed randomvariables, the system (5.41) is not null controllable in average. However, is it possibleto find some random variable α ( · ) such that the system (5.41) is null controllable inaverage for some γ ∈ (0 , ]? • Averaged controllability problems for general random evolution partial dif-ferential equations .The method used in this paper can only be applied to the case of equations in whichthe random operator is α ∆, which is very restrictive.Several methods have been developed to solve the controllability problems for deter-ministic partial differential equations. For the heat equation, the existing methodsinclude Carleman estimates ([23]) and the moment method ([5]). For Schr¨odingerequations, Carleman estimates can also be applied ([30]), together with the momentmethod ([5]), multipliers ([42]), microlocal analysis ([32]), etc.It would be interesting to generalize these method to deal with the averaged control-lability problems for more general random heat and Schr¨odinger equations. The paper[2] contains an interesting survey on the controllability of parabolic systems. It wouldbe interesting to explore possible applications to averaged control. • Averaged controllability problems for nonlinear random evolution equa-tions . 38e have studied the averaged controllability problems for linear random evolutionequations. The same problem can be considered for nonlinear random evolution equa-tions. A possible method to handle the nonlinear problem is to follow what people dofor the classical controllability problems, that is, combining the controllability resultand some fixed point theorem or inverse mapping theorem. However, as the averageof the solution of linear transport equation with respect to velocity helps people studythe nonlinear transport equations, we expect that one can get better result than theones obtained by applying the method mentioned above directly. For example, canone prove that a random Schr¨odinger equation with a cubic term is exactly or nullcontrollable in average? • Numerical approximation of averaged controls.
The numerical approximation of control problems is studied extensively in the litera-ture. We refer the readers to [15, 20, 55] and the rich references therein for this topic.This question also arises in the context of averaged control. This can be done basedon the variational characterization of the average controls given in this paper.
A Appendix
A.1 Reduction of averaged controllability to averaged observabil-ity
We have the following results.
Theorem A.1
System (2.1) is exactly controllable in average in E with the control cost C > if and only if the adjoint system (2.8) is exactly observable in average in E .Proof of Theorem A.1 : The “if” part. Let us define a linear subspace X ⊂ L ( E ; U ) as X △ = n χ E ( · ) Z Ω B ∗ ( ω ) z ( · , ω ; z ) d P ( ω ) : z ∈ V ′ o . and a linear functional F on X as F (cid:16) χ E ( · ) Z Ω B ∗ ( ω ) z ( · , ω ; z ) d P ( ω ) (cid:17) = h y , z i V,V ′ − D y , Z Ω z (0 , ω ; z ) d P ( ω ) E V,V ′ . From (2.9), we know that F (cid:16) χ E ( · ) Z Ω B ∗ ( ω ) z ( · , ω ; z ) P ( ω ) (cid:17) ≤ C (cid:0) | y | V + | y | V (cid:1)(cid:16) | z | V ′ + (cid:12)(cid:12)(cid:12) Z Ω z (0 , ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) V ′ (cid:17) ≤ C (cid:0) | y | V + | y | V (cid:1)(cid:12)(cid:12)(cid:12) χ E ( · ) Z Ω B ∗ ( ω ) z ( · , ω ; z ) P ( ω ) (cid:12)(cid:12)(cid:12) L (0 ,T ; U ) . Hence, F is a bounded linear functional on X with norm | F | L ( X , R ) ≤ C ( | y | V + | y | V ). Then,it can be extended to a bounded linear functional on L ( E ; U ) with the same norm. We still39enote by F the extension if there is no confusion. Then, by Riesz Representation Theorem,there is a u ( · ) ∈ L ( E ; U ) such that for any v ( · ) ∈ L ( E ; U ), F ( v ) = h v, u i L ( E ; U ) and | u | L ( E ; U ) = | F | L ( X , R ) ≤ C ( | y | V + | y | V ) . From the definition of F ( · ), we know that for any z ∈ V ′ , h y , z i V,V ′ − D y , Z Ω z (0 , ω ; z ) d P ( ω ) E V,V ′ = Z T χ E ( t ) D u ( t ) , Z Ω B ∗ ( ω ) z ( t, ω ; z ) d P ( ω ) E U dt. (A.1)We claim that u ( · ) is the control we need. Indeed, taking the dual product of V ′ and V of z = z ( t, ω ; z ) with (2.1) and integrating the product with respect to t in (0 , T ) and ω in Ω,we obtain that Z T χ E ( t ) D u ( t ) , Z Ω B ∗ ( ω ) z ( t, ω ; z ) d P ( ω ) E U dt = Z T χ E ( t ) Z Ω h u ( t ) , B ∗ ( ω ) z ( t, ω ; z ) i U d P ( ω ) dt = Z T χ E ( t ) Z Ω h B ( ω ) u ( t ) , z ( t, ω ; z ) i V,V ′ d P ( ω ) dt = Z Ω h y ( T, ω ; y ) , z i V,V ′ d P ( ω ) − Z Ω h y , z (0 , ω ; z ) i V,V ′ d P ( ω )= D Z Ω y ( T, ω ; y ) d P ( ω ) , z E V,V ′ − D y , Z Ω z (0 , ω ; z ) d P ( ω ) E V,V ′ . (A.2)From (A.1) and (A.2), we conclude that D Z Ω y ( T, ω ; y ) d P ( ω ) , z E V,V ′ = h y , z i V,V ′ , ∀ z ∈ V ′ , which deduces that R Ω y ( T, ω ; y ) d P ( ω ) = y .The “only if” part. Let z ∈ V ′ . We choose y , y ∈ V ′ which satisfy that | y | V ′ = | y | V ′ ≤ , − D y , Z Ω z (0 , ω ; z ) d P ( ω ) E V,V ′ = (cid:12)(cid:12)(cid:12) Z Ω z (0 , ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) V ′ , h y , z i V,V ′ = | z | V ′ . Let u ( · ) be the control such that | u | L ( E ; U ) ≤ C ( | y | V + | y | V ) ≤ C (A.3)and Z Ω y ( T, ω ; y ) d P ( ω ) = y . | z | V ′ + (cid:12)(cid:12)(cid:12) Z Ω z (0 , ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) V ′ = Z T χ E ( t ) D u ( t ) , Z Ω B ∗ ( ω ) z ( t, ω ; z ) d P ( ω ) E U dt. Thus, we find that | z | V ′ + (cid:12)(cid:12)(cid:12) Z Ω z (0 , ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) V ′ ≤ | u | L ( E ; U ) (cid:12)(cid:12)(cid:12) Z Ω B ∗ ( ω ) z ( · , ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) L ( E ; U ) . This, together with (A.3), implies that | z | V ′ ≤ C (cid:12)(cid:12)(cid:12) Z Ω B ∗ ( ω ) z ( · , ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) L ( E ; U ) . Theorem A.2
System (2.1) is null controllable in average in E with control the cost C > if and only the adjoint system (2.8) is null observable in average. The proof of Theorem A.2 is very similar to the one for Theorem A.1. We omit it here.
Theorem A.3
System (2.1) is approximately controllable in average in E if and only if theadjoint system (2.8) satisfies the averaged unique continuation property in E .Proof of Theorem A.3 : The “if” part. Since the system (2.1) is linear, we may assume that y = 0. Then, we only need to prove the following set A T △ = n Z Ω y ( T, ω ; 0) d P ( ω ) : y solves (2.1) with some control u ( · ) o is dense in V . We do this by contradiction argument. If A T was not dense in V , then wecan find a ϕ ∈ V ′ with | ϕ | V ′ = 1 such that h ψ, ϕ i V,V ′ = 0 , ∀ ψ ∈ A T . On the other hand, similar to (A.2), we have that Z T χ E ( t ) D u ( t ) , Z Ω B ∗ ( ω ) z ( t, ω ; z ) d P ( ω ) E U dt = D Z Ω y ( T, ω ; 0) d P ( ω ) , z E V,V ′ . (A.4)Let z = ϕ in (A.4). We have that Z T χ E ( t ) D u ( t ) , Z Ω B ∗ ( ω ) z ( t, ω ; ϕ ) d P ( ω ) E U dt = 0 , ∀ u ( · ) ∈ L (0 , T ; U ) . Hence, we find that χ E ( · ) Z Ω B ∗ ( ω ) z ( · , ω ; ϕ ) d P ( ω ) = 0 in L (0 , T ; U ) , which implies that ϕ = 0 and leads to a contradiction.41he “only if” part. We utilize the contradiction argument again. We assume that thereis a z ∈ V ′ with | z | V ′ = 1, such that χ E ( · ) Z Ω B ∗ ( ω ) z ( · , ω ; z ) d P ( ω ) = 0 in L (0 , T ; U ) . This, together with (A.4), implies that
D Z Ω y ( T, ω ; 0) d P ( ω ) , z E V,V ′ = 0 , ∀ u ( · ) ∈ L ( E ; U ) . (A.5)On the other hand, since (2.1) is approximately controllable in average, we can find a u ( · ) ∈ L ( E ; U ) such that (cid:12)(cid:12)(cid:12) Z Ω y ( T, ω ; 0) d P ( ω ) − z (cid:12)(cid:12)(cid:12) V,V ′ < . It is clear that for this R Ω y ( T, ω ; 0) d P ( ω ), D Z Ω y ( T, ω ; 0) d P ( ω ) , z E V,V ′ > , which contradicts (A.5). Remark A.1
One can also consider the case where y ∈ L (Ω; V ) , i.e. when the datum tobe controlled depends on the parameter ω as well.The proof of Theorem A.1 applies replacing the terms h y , R Ω z (0 , ω ; z ) d P ( ω ) i V,V ′ and | y | V by the terms R Ω h y ( ω ) , z (0 , ω ; z ) i V,V ′ d P ( ω ) and | y ( · ) | L (Ω; V ) , respectively. The samecan be said about Theorem A.3.The situation is different and much more delicate in the context of averaged null con-trollability. Note that, when considering initial data to be controlled depending on ω , onerequires an observability estimate of the form Z Ω (cid:12)(cid:12)(cid:12) z (0 , ω ; z ) (cid:12)(cid:12)(cid:12) V ′ d P ( ω ) ≤ C Z T χ E ( t ) (cid:12)(cid:12)(cid:12) Z Ω B ( ω ) ∗ z ( t, ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) U dt, (A.6) which, in principle, is much stronger than the one we have in which, we get observabilityestimates on (cid:12)(cid:12)(cid:12) Z Ω z (0 , ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) V ′ . This difficulty does not arise in the context of exact averaged controllability where, werecover the norm of z , and this yields also estimates on z (0 , ω ; z ) for all ω and in particularon R Ω (cid:12)(cid:12)(cid:12) z (0 , ω ; z ) (cid:12)(cid:12)(cid:12) V ′ d P ( ω ) .But the property of null averaged controllability with initial data independent of ω doesnot seem to suffice to derive the same property with initial data that depend on ω . This isan interesting issue for further work. .2 Variational characterization of the controls of minimal norm We have shown the existence of the exactly averaged control( resp. null averaged control,approximately averaged control), provided that the adjoint system is exactly averaged ob-servable( resp. null averaged observable, satisfying averaged unique continuation property).These results allow concluding whether a system is controllable in an averaged sense. In thissection, we give variational characterizations of the controls. Such kind of results not onlyderive characterizations of the controls but also serve for computational purposes.
Theorem A.4
If the system (2.8) is exactly averaged observable, then the exact averagedcontrol for the system (2.1) of minimal L (0 , T ; U ) -norm is given by u ( t ) = χ E ( t ) Z Ω B ∗ ( ω ) z (0 , ω ; ˆ z ) P ( ω ) , (A.7) where ˆ z ∈ V ′ minimizes the functional J ( z ) = 12 Z T χ E ( t ) (cid:12)(cid:12)(cid:12) Z Ω B ∗ ( ω ) z ( t, ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) U dt − h y , z i V,V ′ + D y , Z Ω z (0 , ω ; z ) d P ( ω ) E V,V ′ . (A.8) Remark A.2
The control given by (A.7) is an average of functions of the form B ( · ) ∗ ˆ z ( t, · ) .For each sample point ω , B ( ω ) ∗ ˆ z ( t, ω ) can be chosen to be a control. However, generallyspeaking, it does not steer the initial datum y to the final one y .Proof of Theorem A.4 : Define a functional J ( · ) : V ′ → R as follows: J ( z ) = 12 Z T χ E ( t ) (cid:12)(cid:12)(cid:12) Z Ω B ∗ ( ω ) z ( t, ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) U dt − h y , z i V,V ′ + D y , Z Ω z (0 , ω ; z ) d P ( ω ) E V,V ′ . (A.9)It is easy to see that the functional J : V ′ → R is continuous and convex. From (2.9), wehave that J ( · ) is coercive. Then, we know that J ( · ) has a unique minimizer ˆ z . Let ˆ z ( · , · ) bethe corresponding solution of the adjoint system. By computing the first variation of J ( · ),it can be seen that h y , z i V,V ′ = Z T χ E ( t ) D Z Ω B ∗ ( ω ) z ( t, ω ; ˆ z ) dω, Z Ω B ∗ ( ω ) z ( t, ω ; z ) d P ( ω ) E U dt + D y , Z Ω z (0 , ω ; z ) d P ( ω ) E V,V ′ , ∀ z ∈ V ′ . (A.10)From (A.10), we know that if we choose the control as (A.7), then Z Ω y ( T, ω ; y ) d P ( ω ) = y . u ( · ) given by (A.7) is the control with minimal L (0 , T ; U )-norm,which drives the mathematical expectation of the solution of the system (2.1) from y to y .Let us choose z = ˆ z in (A.10). We have that h y , ˆ z i V,V ′ − D y , Z Ω z (0 , ω ; ˆ z ) d P ( ω ) E V,V ′ = Z T χ E ( t ) (cid:12)(cid:12)(cid:12) Z Ω B ∗ ( ω ) z ( t, ω ; ˆ z ) d P ( ω ) (cid:12)(cid:12)(cid:12) U dt. (A.11)Let ˜ u ( · ) be another control which steers the mathematical expectation of the solution to thesystem (2.1) from y to y . Then we obtain that h y , ˆ z i V,V ′ = Z T χ E ( t ) D ˜ u ( t ) , Z Ω B ∗ ( ω ) z ( t, ω ; z ) d P ( ω ) E U dt + D y , Z Ω z (0 , ω ; ˆ z ) d P ( ω ) E V,V ′ , ∀ z ∈ H. (A.12)From (A.11) and (A.12), we get that Z T χ E ( t ) (cid:12)(cid:12)(cid:12) Z Ω B ∗ ( ω ) z (0 , ω ; ˆ z ) d P ( ω ) (cid:12)(cid:12)(cid:12) U dt = Z T χ E ( t ) D ˜ u ( t ) , Z Ω B ∗ ( ω ) z (0 , ω ; ˆ z ) d P ( ω ) E U dt ≤ (cid:12)(cid:12)(cid:12) χ E Z Ω B ∗ ( ω ) z (0 , ω ; ˆ z ) d P ( ω ) (cid:12)(cid:12)(cid:12) L (0 ,T ; U ) | χ E ˜ u | L (0 ,T ; U ) , which implies that (cid:12)(cid:12)(cid:12) χ E Z Ω B ∗ ( ω ) z (0 , ω ; ˆ z ) d P ( ω ) (cid:12)(cid:12)(cid:12) L (0 ,T ; U ) ≤ | χ E ˜ u | L (0 ,T ; U ) . Remark A.3
The control is the solution of the adjoint system with the final datum whichis the minimizer of the quadratic, convex and coercive functional J ( · ) in V ′ . One can usenumerical methods such as implementing gradient like iterative algorithms to compute it (see[15] for example). However, one will meet the same difficulty as employing this method tocompute the control for the exact control problems of PDEs. Similar to Theorem A.4, we can prove the following result.
Theorem A.5
If the system (2.8) is null averaged observable, then the null averaged controlof minimal L (0 , T ; U ) -norm is given by u ( t ) = χ E ( t ) Z Ω B ∗ ( ω ) z (0 , ω ; ˆ z ) d P ( ω ) , (A.13) where z ∈ V ′ minimizes the functional J ( z ) = 12 Z T χ E ( t ) (cid:12)(cid:12)(cid:12) Z Ω B ∗ ( ω ) z ( t, ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) U dt + D y , Z Ω z (0 , ω ; z ) d P ( ω ) E V,V ′ . (A.14)44 heorem A.6 Suppose that the system (2.8) satisfies the averaged unique continuationproperty. Then for any ε > , the approximately averaged control is given by u ( t ) = χ E ( t ) Z Ω B ∗ ( ω )ˆ z ( t, ω ) d P ( ω ) . (A.15) Here ˆ z is the solution to the adjoint system (2.8) corresponding to the datum z ∈ V ′ whichminimizes the functional J ε ( z ) = 12 Z T χ E ( t ) (cid:12)(cid:12)(cid:12) Z Ω B ∗ ( ω ) z ( t, ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) U dt − h y , z i V,V ′ + D y , Z Ω z (0 , ω ; z ) d P ( ω ) E V,V ′ + ε | z | V ′ . (A.16)Borrowing some idea in [54], we can prove the following result stronger than TheoremA.6. Theorem A.7
Suppose that the system (2.8) satisfies the averaged unique continuationproperty. For any ε > and any finite dimensional space X ⊂ V , the solution to (2.1) with the control u ( t ) = χ E ( t ) Z Ω B ∗ ( ω )ˆ z ( t, ω ) d P ( ω ) (A.17) satisfies that (cid:12)(cid:12)(cid:12) y − Z Ω y ( T, ω ; y ) d P ( ω ) (cid:12)(cid:12)(cid:12) V < ε, Π X y = Π X Z Ω y ( T, ω ; y ) d P ( ω ) . (A.18) Here Π X denotes the orthogonal projection from V to X and ˆ z ( · ) is the solution to the adjointsystem (2.8) corresponding to the final datum ˆ z ∈ V ′ which minimizes the functional J ε ( z ) = 12 Z T χ E ( t ) (cid:12)(cid:12)(cid:12) Z Ω B ∗ ( ω ) z ( t, ω ; z ) d P ( ω ) (cid:12)(cid:12)(cid:12) U dt − h y , z i V,V ′ + D y , Z Ω z (0 , ω ; z ) d P ( ω ) E V,V ′ + ε | ( I − Π ∗ X ) z | V ′ . (A.19) Proof : Clearly, J ε ( · ) is continuous and convex. We only need to show that it is coercive.For this, we prove that J ε ( z ) → ∞ as | z | V ′ → ∞ . (A.20)Let { z j } ∞ j =1 ⊂ V ′ be a sequence such that | z j | V ′ → ∞ as j → ∞ . Put ˇ z j = | z j | − V ′ z j for j ∈ N . Then z j | z j | V ′ = 12 | z j | V ′ Z T χ E ( t ) (cid:12)(cid:12)(cid:12) Z Ω B ( ω ) ∗ z ( t, ω ; ˇ z j ) d P ( ω ) (cid:12)(cid:12)(cid:12) U dt − h y , ˇ z j i V,V ′ + D y , Z Ω z (0 , ω ; ˇ z j ) d P ( ω ) E V,V ′ + ε | ( I − Π ∗ X )ˇ z j | V ′ . (A.21)If lim j →∞ Z T χ E ( t ) (cid:12)(cid:12)(cid:12) Z Ω B ( ω ) ∗ z ( t, ω ; ˇ z j ) d P ( ω ) (cid:12)(cid:12)(cid:12) U dt > , J ε ( z j ) → ∞ as j → ∞ . Hence, we only need to consider the case thatlim j →∞ Z T χ E ( t ) (cid:12)(cid:12)(cid:12) Z Ω B ( ω ) ∗ z ( t, ω ; ˇ z j ) d P ( ω ) (cid:12)(cid:12)(cid:12) U dt = 0 . Since { ˇ z j } ∞ j =1 is bounded in V ′ , we can find a subsequence of it, which is still denoted by { ˇ z j } ∞ j =1 if there is no confusion, such that ˇ z j converges to some ˇ z in V ′ weakly. Thus, wehave that χ E ( · ) R Ω B ( ω ) ∗ z ( · , ω ; ˇ z j ) d P ( ω ) converges to χ E ( · ) R Ω B ( ω ) ∗ z ( · , ω ; ˇ z ) d P ( ω ) weakly in L (0 , T ; U ). Then, we have that Z T χ E ( t ) (cid:12)(cid:12)(cid:12) Z Ω B ( ω ) ∗ z ( t, ω ; ˇ z ) d P ( ω ) (cid:12)(cid:12)(cid:12) U dt ≤ lim j →∞ Z T χ E ( · ) (cid:12)(cid:12)(cid:12) Z Ω B ( ω ) ∗ z ( t, ω ; ˇ z ) d P ( ω ) (cid:12)(cid:12)(cid:12) U dt = 0 , which implies that χ E R Ω B ( ω ) ∗ z ( t, ω ; ˇ z ) d P ( ω ) = 0 in L (0 , T ; U ). Thus, we know thatˇ z = 0. Since X is finite dimensional, we have thatlim j →∞ | ( I − Π ∗ X )ˇ z j | V ′ = 1 . Therefore,lim j →∞ J ε ( z j ) | z j | V ′ ≥ lim j →∞ (cid:16) − h y , ˇ z j i V,V ′ + D y , Z Ω z (0 , ω ; ˇ z j ) d P ( ω ) E V,V ′ + ε (cid:17) = ε, which implies that J ε ( z j ) → ∞ as | z j | V ′ → ∞ . By this, we get the coercivity of J ε ( · ). Hence,we know that there is a minimizer ˆ z of J ε ( · ).For any δ > z ∈ V ′ , we have that0 ≤ h (cid:2) J ε (ˆ z + δz ) − J ε (ˆ z ) (cid:3) , which implies that − ε | ( I − Π ∗ X ) z | V ′ ≤ Z T χ E ( t ) D Z Ω B ( ω ) ∗ z ( t, ω ; ˆ z ) d P ( ω ) , Z Ω B ( ω ) ∗ z ( t, ω ; z ) d P ( ω ) E U dt −h y , z i V,V ′ + D y , Z Ω z (0 , ω ; z ) d P ( ω ) E V,V ′ . Similarly, we have that ε | ( I − Π ∗ X ) z | V ′ ≥ Z T χ E ( t ) D Z Ω B ( ω ) ∗ z ( t, ω ; ˆ z ) d P ( ω ) , Z Ω B ( ω ) ∗ z ( t, ω ; z ) d P ( ω ) E U dt −h y , z i V,V ′ + D y , Z Ω z (0 , ω ; z ) d P ( ω ) E V,V ′ . z ∈ V ′ , ε | ( I − Π ∗ X ) z | V ′ ≥ (cid:12)(cid:12)(cid:12) Z T χ E ( t ) D Z Ω B ( ω ) ∗ z ( t, ω ; ˆ z ) d P ( ω ) , Z Ω B ( ω ) ∗ z ( t, ω ; z ) d P ( ω ) E U dt −h y , z i V,V ′ + D y , Z Ω z (0 , ω ; z ) d P ( ω ) E V,V ′ (cid:12)(cid:12)(cid:12) . This, together with (A.2), implies that for any z ∈ V ′ , (cid:12)(cid:12)(cid:12)D z , y − Z Ω y ( T, ω ; y ) E V,V ′ (cid:12)(cid:12)(cid:12) ≤ ε | ( I − Π ∗ X ) z | V ′ . Hence, we get (A.18).
Acknowledgement
This work is supported by the Advanced Grants NUMERIWAVES/FP7-246775 of the Eu-ropean Research Council Executive Agency, FA9550-14-1-0214 of the EOARD-AFOSR,PI2010-04 and the BERC 2014-2017 program of the Basque Government, the MTM2011-29306-C02-00 and SEV-2013-0323 Grants of the MINECO a Humboldt Research Award(University of Erlangen-N¨urnberg), and the NSF of China under grant 11101070.The authors thank the CIMI - Toulouse for the hospitality and support during the prepa-ration of this work in the context of the Excellence Chair in “PDE, Control and Numerics”.The author acknowledges Martin Lazar for helpful comments.
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