Observability inequalities on measurable sets for the Stokes system and applications
.. OBSERVABILITY INEQUALITIES ON MEASURABLE SETS FOR THESTOKES SYSTEM AND APPLICATIONS
FELIPE W. CHAVES-SILVA , DIEGO A. SOUZA , AND
CAN ZHANGAbstract.
In this paper, we establish spectral inequalities on measurable sets of positiveLebesgue measure for the Stokes operator, as well as an observability inequalities on space-timemeasurable sets of positive measure for non-stationary Stokes system. Furthermore, we providetheir applications in the theory of shape optimization and time optimal control problems.
Keywords : spectral inequality, observability inequality, Stokes equations, shape optimizationproblems, time optimal control problem.
Mathematics Subject Classification (2010) : Introduction and main results
Let
T >
0, and let Ω ⊂ R N , N ≥
2, be a bounded connected open set with a smooth boundary ∂ Ω. We will use the notation Q = Ω × (0 , T ), Σ = ∂ Ω × (0 , T ), and we will denote by ν = ν ( x )the outward unit normal vector to Ω at x ∈ ∂ Ω. Throughout the paper spaces of R N -valuedfunctions, as well as their elements, are represented by boldface letters.The present paper deals with an observability inequality on measurable sets of positive mea-sure for the Stokes system (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) z t − ∆ z + ∇ q = in Q, div z = 0 in Q, z = on Σ , z ( · ,
0) = z in Ω . (1.1)System (1.1) is a linearization of the Navier-Stokes system for a homogeneous viscous incom-pressible fluid (with unit density and unit kinematic viscosity) subject to homogeneous Dirichletboundary conditions. Here, z is the R N -valued velocity field and q stands for the scalar pressure.Our motivation to obtain an observability inequality on measurable sets for the Stokes system(1.1) comes from the well-known fact that observability inequalities are equivalent to controlla-bility properties. In the case we are dealing with, this will be equivalent to the null controllabilityof system (1.1) with bounded controls acting on measurable sets with positive measure, and willhave important applications in shape optimization problems and in the study of the bang-bangproperty for time and norm optimal control problems for system (1.1) (see Section 3). Department of Mathematics, Federal University of Pernambuco, CEP 50740-545, Recife, PE, Brazil. E-mail: [email protected] . F. W. Chaves-Silva was supported ERC Project No. 320845: Semi Classical Analysis ofPartial Differential Equations, ERC-2012-ADG. .Department of Mathematics, Federal University of Pernambuco, CEP 50740-545, Recife, PE, Brazil. E-mail: [email protected] . D. A. Souza was supported by the ERC advanced grant 668998 (OCLOC) underthe EU’s H2020 research program.School of Mathematics and Statistics, Wuhan University, 430072 Wuhan, China; Sorbonne Universit´es,UPMC Univ Paris 06, CNRS UMR 7598, Laboratoire Jacques-Louis Lions, F-75005 Paris, France. E-mail: [email protected]. a r X i v : . [ m a t h . O C ] A ug FELIPE W. CHAVES-SILVA , DIEGO A. SOUZA , AND
CAN ZHANG
Observability inequalities for system (1.1) from a cylinder ω × (0 , T ), with ω ⊂ Ω being anon-empty open set, have been proved in different ways by several authors in the past few years.For instance, in [11], the observability inequality for the Stokes system is obtained by meansof global Carleman inequalities for parabolic equations with zero Dirichlet boundary conditions(see also [7] and [10]). Another proof is given in [12] by means of Carleman inequalities forparabolic equations with non-homogeneous Dirichlet boundary conditions applied to the systemsatisfied by the vorticity curl z . More recently, in [6], a new proof was established based on aspectral inequality for the eigenfunctions of the Stokes operator.Concerning observability inequalities over general measurable sets in space and time variables,as far as we know, the first result was obtained in [2] for the heat equation in a bounded andlocally star-shaped domain, and later extended in [8] and [9] to the case of parabolic systems withtime-independent analytic coefficients associated to possibly non self-adjoint elliptic operatorsand higher order parabolic evolutions with the analytic coefficients depending on space and timevariables, when the boundary of the bounded domain in which the equation evolves is globalanalytic. We also refer the interested reader to [1, 19, 24] for some earlier and closely relatedresults on this subject.For the Stokes system, the only result we know is the one in [25], which gives an observabilityinequality from a measurable subset with positive measure in the time variable. In there, theargument is mainly based on the theory of analytic semigroups. In this paper, we extended theresult in [25] to the case of observations from sets of positive measure in both time and spacevariables.Before presenting our main results, we first introduce the usual spaces in the context of fluidmechanics: V = { y ∈ H (Ω) N ; div y = 0 } , H = { y ∈ L (Ω) N ; div y = 0 , y · ν = 0 on ∂ Ω } . Throughout the paper, the following notation will be used: B R ( x ) denotes a ball in R N ofradius R > x ∈ R N ; | ω | is the Lebesgue measure of a subset ω ⊂ Ω and C ( · · · ) stands for a positive constant depending only on the parameters within the brackets,and it may vary from line to line in the context.Our first result is a L -observability inequality from measurable sets with positive measurefor system (1.1). Theorem 1.1.
Let B R ( x ) ⊂ Ω . For any measurable subset M ⊂ B R ( x ) × (0 , T ) with positivemeasure, there exists a positive constant C obs = C ( N, R, Ω , M , T ) such that the observabilityestimate (cid:107) z ( T, · ) (cid:107) H ≤ C obs (cid:90) M | z ( x , t ) | d x dt (1.2) holds for all z ∈ H . Remark 1.2.
When the observation set is M = B R ( x ) × (0 , T ) , one can see that the observ-ability constant C obs has the form Ce C/T with C = C ( N, Ω , R ) > . This is in accordance withthe very recent result [6, Theorem 1.1] . Remark 1.3.
The above technical assumption imposed on the measurable set M is just tosimplify the statement of the main result. Without loss of generality, for any measurable set M ⊂ Ω × (0 , T ) with positive measure, one can always assume that M ⊂ B R ( x ) × (0 , T ) with B R ( x ) ⊂ Ω BSERVABILITY ESTIMATE AND APPLICATIONS 3 for some
R > and x ∈ R N . Otherwise, one may choose a new measurable set (cid:101) M ⊂ M with | (cid:101) M | ≥ c | M | for some constant < c < . The method we shall use to prove Theorem 1.1 relies mainly on the telescoping series method[2] (which is in part inspired by [15] and [22]), the propagation of smallness for real-analyticfunctions on measurable sets [23] as well as an spectral inequality for Stokes system.Let { e j } j ≥ be the sequence of eigenfunctions of the Stokes system (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) − ∆ e j + ∇ p j = λ j e j in Ω , div e j = 0 in Ω , e j = on ∂ Ω , (1.3)with the sequence of eigenvalues { λ j } j ≥ satisfying0 < λ ≤ λ ≤ . . . and lim j →∞ λ j = + ∞ . The following inequality is proved in [6].
Theorem 1.4. [6, Theorem 3.1]
For any non-empty open subset O ⊂ Ω , there exists a constant C = C ( N, Ω , O ) > such that (cid:88) λ j ≤ Λ a j = (cid:90) Ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) λ j ≤ Λ a j e j ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d x ≤ Ce C √ Λ (cid:90) O (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) λ j ≤ Λ a j e j ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d x , (1.4) for any sequence of real numbers { a j } j ≥ ∈ (cid:96) and any positive number Λ . Spectral inequality (1.4) allow us to control the low frequencies of the Stokes system witha precise estimate on the cost of controllability with respect to the frequency length which,combined with the decay of solutions of (1.1), implies the null controllability of Stokes systemwith L -controls applied to arbitrarily small open sets.Our second main result is an extension of the spectral inequality (1.4) from open sets tomeasurable sets of positive measure. Theorem 1.5.
Let B R ( x ) ⊂ Ω and let ω ⊂ B R ( x ) be a measurable set with positive measure.Then, there exists a constant C = C ( N, R, Ω , | ω | ) > such that (cid:88) λ j ≤ Λ a j / ≤ Ce C √ Λ (cid:90) ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) (cid:88) λ j ≤ Λ a j e j ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d x , (1.5) for all Λ > and any sequence of real numbers { a j } j ≥ ∈ (cid:96) . Remark 1.6.
Inequality (1.5) leads to a null controllability result for the Stokes system with L ∞ -controls (see Theorem 3.5). Recall that (cid:96) (cid:44) (cid:40) { a j } j ≥ : + ∞ (cid:80) j =1 a j < + ∞ (cid:41) . FELIPE W. CHAVES-SILVA , DIEGO A. SOUZA , AND
CAN ZHANG
As we will see below, the proof of Theorem 1.5 strongly depends on quantitative estimates ofthe interior spatial analyticity for finite sums of eigenfunctions of the Stokes system (1.3). Asfar as we know, for the Navier-Stokes equations, this kind of interior analyticity has been firstanalyzed in [13] and [14], where the authors consider a nonlinear elliptic system satisfied by thevelocity z and the vorticity curl z and show the interior analyticity for the velocity z . However,since the boundary condition for the curl z is not prescribed, the analyticity up to the boundarycannot be achieved by this method.In this paper, in order to establish the spectral inequality (1.5), we adapted the arguments in[13] and [14], and [2, Theorem 5], to the low frequencies of the Stokes system.The paper is organized as follows. In Section 2, we shall present the proofs of Theorems1.1 and 1.5. Section 3 deals with some applications of main theorems for shape optimizationand time optimal control problems of Stokes system. Finally, in Appendix A, we prove somereal-analytic estimates for solutions of the Poisson equation.2. Spectral and Observability inequalities
Spectral inequality on measurable sets.
This section is devoted to the proof of Theo-rem 1.5. Compared to the proof of [2, Theorem 5] for the Laplace operator, we here encounterthe difficulty due to the pressure in the Stokes system. To circumvent that, we consider theequation satisfied by the curl of the low frequencies, which is an equation without pressurebut with no boundary conditions. This allow us recover and quantify the interior real-analyticestimates based on the curl operator.We begin with an estimate of the propagation of smallness for real-analytic functions onmeasurable sets with positive measure, which plays a core ingredient in the proof of Theorem 1.5.
Lemma 2.1.
Assume that f : B R ( x ) ⊂ R N −→ R N is real-analytic and verifies | ∂ αx f ( x ) | ≤ M | α | !( ρR ) | α | , for x ∈ B R ( x ) , α ∈ N N , for some M > and < ρ ≤ .For any measurable set ω ⊂ B R ( x ) with positive measure, there are positive constants C = C ( R, N, ρ, | ω | ) and θ = θ ( R, N, ρ, | ω | ) , with θ ∈ (0 , , such that (cid:107) f (cid:107) L ∞ ( B R ( x )) ≤ C (cid:18)(cid:90) ω | f ( x ) | d x (cid:19) θ M − θ . The above-mentioned local observability inequality for real-analytic functions was first estab-lished in [23]. The interested reader can also find a simpler proof of Lemma 2.1 in [1, Section3], and a more general extension in [8, Lemma 2].
Proof of Theorem 1.5 . For each real number Λ > { a j } j ≥ ∈ (cid:96) , wedefine u Λ ( x ) = (cid:88) λ j ≤ Λ a j e j ( x ) , x ∈ Ω , and v Λ ( x , s ) = (cid:88) λ j ≤ Λ a j e s √ λ j d e j ( x ) , ( x , s ) ∈ Ω × ( − , , BSERVABILITY ESTIMATE AND APPLICATIONS 5 where d denotes the curl operator. Because v Λ ( · ,
0) = d u Λ and div x u Λ = 0, we have that∆ x u Λ ( x ) = d ∗ v Λ ( x , , x ∈ Ω , (2.1)where d ∗ is the adjoint of d .Let us now obtain an estimate of the propagation of smallness for u Λ on measurable sets withpositive measure. According to Lemma 2.1, it is sufficient to quantify the analytic estimates ofhigher-order derivatives of u Λ .Since v Λ ( · , · ) satisfies − ∂ ss v Λ ( x , s ) − ∆ x v Λ ( x , s ) = 0 , ( x , s ) ∈ Ω × ( − , , we have that d ∗ v Λ verifies − ∂ ss d ∗ v Λ ( x , s ) − ∆ x d ∗ v Λ ( x , s ) = 0 , ( x , s ) ∈ Ω × ( − , f ≡ d ∗ v Λ is real-analytic in B R ( x ,
0) and thefollowing estimate holds (cid:107) ∂ α x ∂ βs d ∗ v Λ (cid:107) L ∞ ( B R ( x , ≤ C ( | α | + β )!( ρR ) | α | + β (cid:32) – (cid:90) B R ( x , | d ∗ v Λ ( x , s ) | d x ds (cid:33) / , ∀ α ∈ N N , β ≥ , where the positive constants ρ and C only depend on the dimension N .Taking β = 0 in the previous estimate, we readily obtain (cid:107) ∂ αx d ∗ v Λ ( · , (cid:107) L ∞ ( B R ( x )) ≤ C | α | !( ρR ) | α | (cid:32) – (cid:90) B R ( x , | d ∗ v Λ ( x , s ) | d x ds (cid:33) / , ∀ α ∈ N N . (2.2)To bound the right-hand side in (2.2), we set w Λ ( x , s ) = (cid:88) λ j ≤ Λ a j e s √ λ j e j ( x ) , ( x , s ) ∈ Ω × ( − , (cid:107) d ∗ v Λ (cid:107) L ( B R ( x , ≤ C (cid:107) w Λ (cid:107) L (( − , H (Ω)) ≤ C (cid:90) − (cid:107) Aw Λ ( · , s ) (cid:107) H ds, where we have used the fact that there exists C = C ( N, Ω) > C (cid:107) y (cid:107) H (Ω) ≤ (cid:107) Ay (cid:107) H ≤ C (cid:107) y (cid:107) H (Ω) , ∀ y ∈ D ( A ) , with A being the Stokes operator In fact, d is the differential which maps 1-forms into 2-forms. When a vector field w is identified with a 1-form,then dw can be identified with a N ( N − The Stokes operator A : D ( A ) −→ H is defined by A = − P ∆, with D ( A ) = (cid:8) y ∈ V : Ay ∈ H (cid:9) and P : L (Ω) = H ⊕ H ⊥ −→ H is the Leray projection. FELIPE W. CHAVES-SILVA , DIEGO A. SOUZA , AND
CAN ZHANG
Since { e j } j ≥ is an orthonormal basis of H , the last estimate yields (cid:107) d ∗ v Λ (cid:107) L ( B R ( x , ≤ Ce C √ Λ (cid:88) λ j ≤ Λ a j , (2.3)for some C > (cid:107) ∂ αx d ∗ v Λ ( · , (cid:107) L ∞ ( B R ( x )) ≤ C | α | !( ρR ) | α | e C √ Λ (cid:88) λ j ≤ Λ a j / , ∀ α ∈ N N , (2.4)where C = C ( N, Ω).Since u Λ solves the Poisson equation (2.1), we have that u Λ is real-analytic whenever theexterior force d ∗ v Λ ( · ,
0) is real-analytic. Now, thanks to (2.4), we can apply again Lemma A.1to obtain that (cid:107) ∂ αx u Λ (cid:107) L ∞ ( B R ( x )) ≤ ( R ˜ ρ ) −| α |− | α | ! (cid:107) u Λ (cid:107) L ( B R ( x )) + Ce C √ Λ (cid:88) λ j ≤ Λ a j / , ∀ α ∈ N N , for some constant ˜ ρ > (cid:107) u Λ (cid:107) L ( B R ( x )) ≤ (cid:107) u Λ (cid:107) H = (cid:88) λ j ≤ Λ a j , one can see that (cid:107) ∂ αx u Λ (cid:107) L ∞ ( B R ( x )) ≤ | α | !( ρR ) | α | e K √ Λ (cid:88) λ j ≤ Λ a j / , ∀ α ∈ N N , (2.5)where ρ and K are positive constants independent of Λ.Using (2.5) and Lemma 2.1, applied to the real-analytic function u Λ , we obtain the estimate (cid:107) u Λ (cid:107) L ∞ ( B R ( x )) ≤ C (cid:18)(cid:90) ω | u Λ ( x ) | d x (cid:19) θ e K √ Λ (cid:88) λ j ≤ Λ a j / − θ (2.6)for some constants C = C ( N, R, Ω , | ω | ) > θ = θ ( N, R, Ω , | ω | ) ∈ (0 , C = C (Ω , R, N ) such that (cid:88) λ j ≤ Λ a j / ≤ Ce C √ Λ (cid:107) u Λ (cid:107) L ∞ ( B R ( x )) . The above inequality and (2.6) then leads to (cid:88) λ j ≤ Λ a j / ≤ Ce C √ Λ (cid:18)(cid:90) ω | u Λ ( x ) | d x (cid:19) θ (cid:88) λ j ≤ Λ a j (1 − θ ) / , BSERVABILITY ESTIMATE AND APPLICATIONS 7 which give us the desired observability inequality (cid:88) λ j ≤ Λ a j / ≤ Ce C √ Λ (cid:90) ω | u Λ ( x ) | d x . (cid:3) Observability inequality on measurable sets in space-time variables.
This Sectionis devoted to the proof of Theorem 1.1.We begin with an interpolation estimate for the solutions of the Stokes system, which willbe estimate a consequence of the spectral inequality given in Theorem 1.5 and the exponentialdecay of solutions of the Stokes system, and can be seen as a quantitative estimate of the stronguniqueness of solutions. We refer the reader to [2, 8, 25] for closely related results concerningthe strong unique continuation property for general parabolic equations.
Proposition 2.2.
Let B R ( x ) ⊂ Ω and let ω ⊂ B R ( x ) be a measurable set with positivemeasure. Then, there exists C = C (Ω , | ω | ) > such that (cid:107) z ( · , t ) (cid:107) H ≤ (cid:16) Ce Ct − s (cid:107) z ( · , t ) (cid:107) L ( ω ) (cid:17) / (cid:107) z ( · , s ) (cid:107) / H , ∀ z ∈ H , where ≤ s < t ≤ T and z is the solution of (1.1) associated to z .Proof. It suffices to prove the estimate in the case s = 0.For any Λ >
0, we set H Λ (cid:44) span (cid:8) e j ; λ j ≤ Λ (cid:9) . Given z ∈ H , the solution z of (1.1) can be split into z = z Λ + z ⊥ Λ , where z Λ and z ⊥ Λ arethe solutions of (1.1) (together with some pressures) associated to z , Λ ∈ H Λ and z ⊥ , Λ ∈ H ⊥ Λ 4 , z = z , Λ + z ⊥ , Λ , respectively. Moreover, one has z Λ ( · , t ) ∈ H Λ and (cid:107) z ⊥ Λ ( · , t ) (cid:107) H ≤ e − Λ t (cid:107) z (cid:107) H , (2.7)for every t > t > (cid:107) z ( · , t ) (cid:107) H ≤ (cid:107) z Λ ( · , t ) (cid:107) H + (cid:107) z ⊥ Λ ( · , t ) (cid:107) H ≤ Ce C √ Λ (cid:107) z Λ ( · , t ) (cid:107) L ( ω ) + e − Λ t (cid:107) z (cid:107) H ≤ Ce C √ Λ (cid:16) (cid:107) z ( · , t ) (cid:107) L ( ω ) + (cid:107) z ⊥ Λ ( · , t ) (cid:107) H (cid:17) + e − Λ t (cid:107) z (cid:107) H ≤ Ce C √ Λ (cid:0) (cid:107) z ( · , t ) (cid:107) L ( ω ) + e − Λ t (cid:107) z (cid:107) H (cid:1) + e − Λ t (cid:107) z (cid:107) H ≤ C e C √ Λ − Λ2 t (cid:16) e Λ2 t (cid:107) z ( · , t ) (cid:107) L ( ω ) + e − Λ2 t (cid:107) z (cid:107) H (cid:17) ≤ C e C t (cid:107) z ( t ) (cid:107) / L ( ω ) (cid:107) z (cid:107) / H , where in the last inequality we have used that C √ Λ − t Λ2 ≤ C t , for any Λ > H ⊥ Λ = span (cid:8) e j ; λ j > Λ (cid:9) . FELIPE W. CHAVES-SILVA , DIEGO A. SOUZA , AND
CAN ZHANG and the following lemma:
Lemma 2.3 ([21]) . Let C , C be positive and M , M and M be nonnegative. Assume thereexist C > such that M ≤ C M and δ > such that M ≤ e − C δ M + e C δ M for every δ ≥ δ . Then, there exits C such that M ≤ C M C / ( C + C )1 M C / ( C + C )2 . (cid:3) For the proof of Theorem 1.1, we will use the following result concerning the property ofLebesgue density point for a measurable set in R . Lemma 2.4 ([19], Proposition 2 . . Let E be a measurable set in (0 , T ) with positive measureand let l be a density point of E . Then, for each µ > , there is l = l ( µ, E ) in ( l, T ) such thatthe sequence { l m } m ≥ defined as l m +1 = l + µ − m ( l − l ) , m = 1 , , . . . satisfies | E ∩ ( l m +1 , l m ) | ≥
13 ( l m − l m +1 ) , ∀ m ≥ . (2.8) Proof of Theorem 1.1 . For each t ∈ (0 , T ), let us define the slice M t = { x ∈ Ω : ( x, t ) ∈ M } and E = (cid:26) t ∈ (0 , T ); | M t | ≥ | M | T (cid:27) . From Fubini’s Theorem, it follows that M t ⊂ Ω is measurable for a.e. t ∈ (0 , T ), E is measurablein (0 , T ) and | E | ≥ | M | | B R ( x ) | and χ E ( t ) χ M t ( x ) ≤ χ M ( x , t ) , in Ω × (0 , T ) . For a.e. t ∈ E , we apply Proposition 2.2 to M t to find a constant C = C (Ω , R, | M | / ( T | B R ( x ) | ))such that (cid:107) z ( · , t ) (cid:107) H ≤ (cid:16) Ce Ct − s (cid:107) z ( · , t ) (cid:107) L ( M t ) (cid:17) / (cid:107) z ( · , s ) (cid:107) / H , (2.9)for 0 ≤ s < t .Let l be any density point in E . For µ > { l m } m ≥ thesequence associated to l and µ as in Lemma 2.4. For each m ≥
1, we set τ m = l m +1 + ( l m − l m +1 )6hence, | E ∩ ( τ m , l m ) | = | E ∩ ( l m +1 , l m ) | − | E ∩ ( l m +1 , τ m ) | ≥ ( l m − l m +1 )6 . Taking s = l m +1 in (2.9), we get (cid:107) z ( · , t ) (cid:107) H ≤ (cid:18) Ce Clm − lm +1 (cid:107) z ( · , t ) (cid:107) L ( M t ) (cid:19) / (cid:107) z ( · , l m +1 ) (cid:107) / H , for a.e. t ∈ E ∩ ( τ m , l m ) . (2.10) BSERVABILITY ESTIMATE AND APPLICATIONS 9
Integrating (2.10) with respect to t over E ∩ ( τ m , l m ), we obtain (cid:107) z ( · , l m ) (cid:107) H ≤ (cid:32) Ce Clm − lm +1 (cid:90) l m l m +1 χ E ( t ) (cid:107) z ( · , t ) (cid:107) L ( M t ) dt (cid:33) / (cid:107) z ( l m +1 ) (cid:107) / H , which implies that (cid:107) z ( · , l m ) (cid:107) H ≤ (cid:15) (cid:107) z ( · , l m +1 ) (cid:107) H + (cid:15) − Ce Clm − lm +1 (cid:90) l m l m +1 χ E ( t ) (cid:107) z ( · , t ) (cid:107) L ( M t ) , for any (cid:15) > (cid:15) = e − lm − lm +1) in the above inequality, we have e − C + 12 lm − lm +1 (cid:107) z ( · , l m ) (cid:107) H − e − C +1 lm − lm +1 (cid:107) z ( · , l m +1 ) (cid:107) H ≤ C (cid:90) l m l m +1 χ E ( t ) (cid:107) z ( · , t ) (cid:107) L ( M t ) dt. (2.11)Finally, choosing µ = C +1)2 C +1 , where C is any constant for which inequality (2.11) holds, wereadly obtain e − C + 12 lm − lm +1 (cid:107) z ( · , l m ) (cid:107) H − e − C + 12 lm +1 − lm +2 (cid:107) z ( · , l m +1) (cid:107) H ≤ C (cid:90) l m l m +1 χ E ( t ) (cid:107) z ( · , t ) (cid:107) L ( M t ) dt, ∀ m ≥ , (2.12)because µ ( l m +1 − l m +2 ) = l m − l m +1 , for all m ≥ . Finally, adding the telescoping series in (2.12) from m = 1 to + ∞ , we get the observabilityinequality (cid:107) z ( · , T ) (cid:107) H ≤ C (cid:90) M ∩ (Ω × [ l,l ]) | z ( x , t ) | d x dt, with some constant C = C ( N, R, Ω , M , T ) > (cid:3) Applications
Shape optimization problems.
As a direct and interesting application of Theorem 1.5,we analyze the following shape optimization problem formulated in [17].Let { β νj } j ∈ N be a sequence of independent real random variables on a probability space(X , F , P ) having mean equal to 0, variance equal to 1, and a super exponential decay (for in-stance, independent Gaussian or Bernoulli random variables, see [5, Assumption (3.1)] for moredetails). For every ν ∈ X, the solution of (1.1) corresponding to the initial datum z ν = (cid:88) j ≥ β νj a j e j , with { a j } j ≥ ∈ (cid:96) , (3.1)is given by z ν ( · , t ) = (cid:88) j ≥ β νj a j e − tλ j e j . (3.2)Given L ∈ (0 , U L = (cid:110) χ ω ∈ L ∞ (Ω; { , } ) : ω ⊂ Ω is a measurable subset of measure | ω | = L | Ω | (cid:111) . FELIPE W. CHAVES-SILVA , DIEGO A. SOUZA , AND
CAN ZHANG
For each χ ω ∈ U L , we then define the randomized observability constant by C T,rand ( χ ω ) = inf || z ν ( T ) || =1 E (cid:90) T (cid:90) ω | z ν ( x, t ) | d x dt, Using (3.2), the properties of random variables β νj , and the change of variable b j = a j e − T λ j ,we deduce that C T,rand ( χ ω ) = inf (cid:80) ∞ j =1 | b j | =1 E (cid:90) T (cid:90) ω (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:88) j ≥ β νj b j e tλ j e j ( x ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) d x dt, where E is the expectation over the space X with respect to the probability measure P .From Fubini’s theorem and the independence of the random variables { β νj } j ∈ N , a simplecomputation gives C T,rand ( χ ω ) = inf j ≥ e T λ j − λ j (cid:90) ω | e j ( x ) | d x . We now consider the optimal design problem of maximizing the randomized observabilityconstant C T,rand ( χ ω ) over the set of admissible designs U L . In other words, we study theproblem ( P T ) : sup χ ω ∈ U L C T,rand ( χ ω ) = sup χ ω ∈ U L inf j ≥ e T λ j − λ j (cid:90) ω | e j ( x ) | d x . (3.3)The optimal shape design problem (3.3) models the best sensor shape and location problem forthe control of the Stokes system (1.1).We have the following result: Theorem 3.1.
The problem ( P T ) has a unique solution.Proof. We only have to check the following two conditions:i. If there exists E ⊂ Ω of positive Lebesgue measure, an integer m ∈ N ∗ , β , . . . , β m ∈ R + ,and C ≥ (cid:80) mj =1 β j | e j ( x ) | = C almost everywhere on E , then there musthold C = 0 and β = β = . . . = β m = 0.ii. For every a ∈ L ∞ (Ω; [0 , (cid:82) Ω a ( x ) d x = L | Ω | , one haslim inf j → + ∞ e T λ j − λ j (cid:90) Ω a ( x ) | e j ( x ) | d x > e T λ − λ . By the analyticity of the eigenfunctions of Stokes system with homogeneous Dirichlet bound-ary conditions, it is not difficult to show that the first condition holds.For the second condition, notice that there exists (cid:15) > E ⊂ Ω of positive measure suchthat a ≥ (cid:15)χ E and (cid:90) Ω a ( x ) | e j ( x ) | d x ≥ (cid:15) (cid:90) E | e j ( x ) | d x . From Theorem 1.5, we easily see thatlim inf j → + ∞ e T λ j − λ j (cid:90) Ω a ( x ) | e j ( x ) | d x = + ∞ . BSERVABILITY ESTIMATE AND APPLICATIONS 11
From [17, Theorem 1], it follows that problem ( P T ) has a unique solution. (cid:3) Remark 3.2.
The optimal set given by Theorem 3.1 is open and semi-analytic . This followsfrom the fact that the eigenfunctions of the Stokes system with homogeneous Dirichlet boundaryconditions are analytic. Remark 3.3.
A proof of Theorem 3.1 when Ω is the unit disk of R can be found in [17] .However, such proof relies on an explicity knowledge of the eigenfunctions of the Stokes operator,which of course can not be extended to general domains. Notice that to prove Theorem 3.1, inthe general case, the key point is to obtain an observability inequalities with observations overmeasurable sets of positive measure as in Theorem 1.5. Null controllability for Stokes system with bounded controls.
Let ω be a non-empty open subset of Ω and consider the following controlled Stokes system (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) u t − ∆ u + ∇ p = v χ ω in Q, div u = 0 in Q, u = on Σ , u ( · ,
0) = u in Ω . (3.4)It is well known that for any T > u ∈ H , and v ∈ L ( ω × (0 , T )), there exists exactly onesolution ( u , p ) to the Stokes equations (3.4) with u ∈ C ([0 , T ]; H ) ∩ L (0 , T ; V ) , p ∈ L (0 , T ; U ) , where U := (cid:26) ψ ∈ H (Ω); (cid:90) Ω ψ ( x ) d x = 0 (cid:27) . In the context of the Stokes system (3.4), for 1 ≤ p ≤ ∞ , the L p - null controllability problemat time T reads as follows: For any u ∈ H , find a control v ∈ L p ( ω × (0 , T )) such that the associatedsolution to (3.4) satisfies u ( x , T ) = 0 in Ω . (3.5)The following result is well-known. Theorem 3.4.
For any non-empty open subset ω of Ω and any T > , the Stokes system (3.4) is L -null controllable. For the proof, we refer the reader to [6, 10, 11].In practice it would be interesting to take the control steering the solution of the Stokes systemto rest to be in L ∞ ( ω × (0 , T )). Nevertheless, it is not clear how to construct L ∞ ( ω × (0 , T ))controls from L ( ω × (0 , T )) controls. Notice that for the case of the heat equation this is always Here, it is understood that the optimal set is unique up to the set of zero measure. A subset of a real analyticfinite-dimensional manifold is said to be semi-analytic if it can be written in terms of equalities and inequalitiesof real analytic functions. FELIPE W. CHAVES-SILVA , DIEGO A. SOUZA , AND
CAN ZHANG possible since one can use local regularity results (for more details, see [3]), which is no longerthe case for the Stokes system.From Theorem 1.1 we are able to deduce a null controllability for Stokes system with L ∞ -controls. More precisely, we have: Theorem 3.5.
For any non-empty open subset ω of Ω and any T > , the Stokes system (3.4) is L ∞ -null-controllable.Proof. The proof follows from the duality between observability and controllability and the L -observability inequality (1.2). (cid:3) The observability inequality stablished in Theorem 1.1 allow us to conclude stronger control-lability properties for the Stokes system (3.4). In fact it is possible to control the Stokes systemwith L ∞ -controls supported in any measurable set of positive measure: Theorem 3.6.
For any
T > and any measurable set of positive measure γ ⊂ Ω × [0 , T ] , theStokes system (3.4) is L ∞ -null controllable with control supported in γ . Time optimal control problem for the Stokes system.
Let | · | r : R N → [0 , ∞ ) bethe r -euclidean norm in R N , i.e., | x | r = (cid:40) ( | x | r + . . . + | x N | r ) r if r ∈ [1 , ∞ ) , max {| x | , . . . , | x N |} if r = ∞ , for every x ∈ R N .For r ∈ [1 , ∞ ] fixed and any M >
0, we consider the set of admissible controls U M,rad = { v ∈ L ∞ ( ω × [0 , ∞ )) ; | v ( x , t ) | r ≤ M a.e. in ω × [0 , ∞ ) } and for u ∈ H given, we define the set of reachable states starting from u : R ( u , U M,rad ) = (cid:110) u ( · , τ ) ; τ > u is the solution of (3.4) with v ∈ U M,rad (cid:111) . Thanks to Theorem 3.5, it follows that ∈ R ( u , U M,rad ), for any u ∈ H .In this section, we study the following time optimal control problem: given u ∈ H and u f ∈ R ( u , U M,rad ) , find v (cid:63)r ∈ U M,rad such that the correspondingsolution u (cid:63) of (3.4) satisfies u (cid:63) ( τ (cid:63)r ( u , u f )) = u f , (3.6) where τ (cid:63)r ( u , u f ) is the minimal time needed to steer the initial datum u to thetarget u f with controls in U M,rad , i.e. τ (cid:63)r ( u , u f ) = min v ∈ U M,rad { τ ; u ( · , τ ) = u f } . (3.7)We have the following result: Theorem 3.7.
Let
M > and r ∈ [1 , ∞ ] be given. For every u ∈ H and any u f ∈ R ( u , U M,rad ) ,the time optimal problem (3.7) has at least one solution. Moreover, any optimal control v (cid:63)r satisfies the bang-bang property: | v (cid:63)r ( x , t ) | r = M for a.e. ( x , t ) ∈ ω × [0 , τ (cid:63)r ( u , u f )] . BSERVABILITY ESTIMATE AND APPLICATIONS 13
Proof.
Since u f ∈ R ( u , U M,rad ), there exists a minimizing sequence ( τ n , v n ) n ≥ such that τ n −−−→ n →∞ τ (cid:63)r ( u , u f ) and ( v n ) n ≥ ⊂ U M,rad has the property that the associated solution u n to (3.4) satisfies u n ( · , τ n ) = u f for all n ≥
1. Also, because ( v n ) n ≥ ⊂ U M,rad , it follows that ( v n ) n ≥ convergesweakly- (cid:63) to some vector-function v (cid:63) ∈ U M,rad in L ∞ ( ω × (0 , τ (cid:63)r ( u , u f ))). Claim: v (cid:63) is a solution of the time optimal problem (3.6). Proof of the Claim.
We only have to show that u (cid:63) ( · , τ (cid:63)r ( u , u f )) = u f , where u (cid:63) is the solutionof (3.4) associated to v (cid:63) .To show this, let ¯ u be the solution of (3.4) with v ≡ and w = u (cid:63) − ¯ u , w n = u n − ¯ u solutionsof (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w t − ∆ w + ∇ π = v (cid:63) ω in Q, div w = 0 in Q, w = on Σ , w n (0) = in Ω , and (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) w n,t − ∆ w n + ∇ π n = v n ω in Q, div w n = 0 in Q, w n = on Σ , w n (0) = in Ω , respectively.Now, thanks to the continuity in time of ¯ u and that τ n −−−→ n →∞ τ (cid:63) ( u , u f ), it follows that¯ u ( · , τ n ) −−−→ n →∞ ¯ u ( · , τ (cid:63)r ( u , u f )) in H . Moreover, it is not difficult to see that (cid:104) w n ( τ n ) − w n ( τ (cid:63)r ( u , u f )) , ϕ (cid:105) → ∀ ϕ ∈ H , (cid:104) w n ( τ (cid:63)r ( u , u f )) , ϕ (cid:105) → (cid:104) w ( τ (cid:63)r ( u , u f )) , ϕ (cid:105) ∀ ϕ ∈ H and (cid:104) w n ( τ n ) , ϕ (cid:105) → (cid:104) w ( τ (cid:63)r ( u , u f )) , ϕ (cid:105) ∀ ϕ ∈ H . Since u f = ¯ u ( · , τ n ) + w n ( · , τ n ), we have that (cid:104) u f , ϕ (cid:105) = (cid:104) ¯ u ( · , τ n ) + w n ( · , τ n ) , ϕ (cid:105) for all ϕ ∈ H and (cid:104) u f , ϕ (cid:105) = (cid:104) ¯ u ( · , τ (cid:63)r ( u , u f )) + w ( · , τ (cid:63)r ( u , u f )) , ϕ (cid:105) = (cid:104) u (cid:63) ( · , τ (cid:63)r ( u , u f )) , ϕ (cid:105) , for all ϕ ∈ H . (cid:3) Now, let us show that any optimal control v (cid:63) ∈ U M,rad satisfies the bang-bang property. To dothis, we argue by contradiction.We consider u (cid:63) the corresponding state (with some pressure) to (3.4) and suppose that thereexist (cid:15) > γ ⊂ ω × (0 , τ (cid:63)r ( u , u f )) such that | v (cid:63) ( x , t ) | r < M − (cid:15) (( x , t ) ∈ γ ) . (3.8)Choosing δ > (cid:26) τ = τ (cid:63)r ( u , u f ) − δ > , the set Γ = { ( x , t ) ∈ ω × (0 , τ ) : ( x , t ) ∈ γ } has positive measure , and using the time continuity of u (cid:63) , there exists δ ∈ (0 , δ ) such that (cid:107) u − u (cid:63) ( · , δ ) (cid:107) H ≤ (cid:15)C obs ( τ , Γ) , (3.9) FELIPE W. CHAVES-SILVA , DIEGO A. SOUZA , AND
CAN ZHANG where C obs ( τ , Γ) is the observability constant given by Theorem 1.1 for the control domain Γat time τ .From Theorem 3.6, there exists a control v ∈ L ∞ ( ω × (0 , τ )) with supp v ⊂ Γ , the associated solution u satisfies u ( · ,
0) = u − u (cid:63) ( · , δ ) and u ( · , τ ) = , (cid:107) v (cid:107) L ∞ (Γ) ≤ C obs ( τ , Γ) (cid:107) u − u (cid:63) ( δ ) (cid:107) H . Thus, from (3.9) we have that (cid:107) v (cid:107) L ∞ ( ω × (0 ,τ )) ≤ (cid:15). Now, let (cid:98) v ∈ L ∞ ( ω × (0 , τ )) be defined by (cid:98) v ( x, t ) = v (cid:63) ( x , t + δ ) + v ( x , t ) ( t ∈ [0 , τ ]) . Noticing that τ + δ ≤ τ (cid:63)r ( u , u f ), using the fact that supp v ⊂ Γ and estimate (3.8), it followsthat (cid:98) v ∈ U M,rad .Finally, setting (cid:98) u ( x , t ) = u (cid:63) ( x , t + δ ) + u ( x , t ) and (cid:98) p ( x , t ) = p (cid:63) ( x , t + δ ) + p ( x , t ), we have that (cid:98) u ( · ,
0) = u , (cid:98) u ( τ (cid:63)r ( u , u f ) − δ ) = u f and that (cid:98) u t − ∆ (cid:98) u + ∇ (cid:98) p = (cid:98) v ω . Hence, (cid:98) v ∈ U M,rad is a control which steers u to u f at time τ (cid:63)r ( u , u f ) − δ . This contradictsthe definition of τ (cid:63)r ( u , u f ) and then the bang-bang property holds. (cid:3) About the uniqueness of the optimal control for problem (3.7), we have the following result:
Proposition 3.8.
Let
M > and r ∈ (1 , ∞ ) . For any u ∈ H and every u f ∈ R ( u , U M,rad ) ,the time optimal control problem (3.6) - (3.7) has a unique solution v (cid:63)r which satisfies a bang-bangproperty: | v (cid:63)r ( x , t ) | r = M for a.e. ( x , t ) ∈ ω × [0 , τ (cid:63)r ( u , u f )] .Proof. The existence of solution and the bang-bang property is a consequence of Theorem 3.7.We only have to prove the uniqueness of solution. Thus, let v and h be two time optimal controlsin U M,rad . Thanks to the linearity, w = ( v + h ) is also a time optimal control. From Theorem3.7, w also satisfies the bang-bang property. Therefore, we have that | v ( x , t ) | r = | h ( x , t ) | r = | w ( x , t ) | r = M , a.e. in ω × (0 , τ (cid:63)r ( u , u )). Now, if v ( x , t ) (cid:54) = h ( x , t ) in a measurable set of positivemeasure D ⊂ ω × (0 , τ (cid:63)r ( u , u )), then, thanks to the fact that any norm | · | r for r ∈ (1 , ∞ )is uniformly convex in R N , we have that | w ( x , t ) | r < M a.e. in D ⊂ ω × (0 , τ (cid:63)r ( u , u )). Thiscontradicts the bang-bang property for w . (cid:3) Appendix A. Real-analytic estimates for solutions to the Poisson equation
In this appendix we prove the following lemma which was used in the proof of Theorem 1.5.
Lemma A.1.
Assume that f is an real-analytic function in B R ( x ) verifying | ∂ α x f ( x ) | ≤ M | α | !( Rρ ) | α | for all x ∈ B R ( x ) and α ∈ N N , (A.1) with some positive constants M and ρ . Let u ∈ L ( B R ( x )) satisfying the Poisson equation − ∆ u = f in B R ( x ) . (A.2) BSERVABILITY ESTIMATE AND APPLICATIONS 15
Then, u is real-analytic in B R/ ( x ) and has the estimate (cid:107) ∂ α x u (cid:107) L ∞ ( B R/ ( x )) ≤ | α | !( R ˜ ρ ) | α | +1 (cid:0) (cid:107) u (cid:107) L ( B R ( x )) + M (cid:1) , for all α ∈ N N , (A.3) where ˜ ρ is a constant depending only on the dimension N and ρ . A proof of the lemma A.1 for f ≡ Proof.
By rescaling, it suffices to prove the estimate (A.3) when R = 1 and x = .Since f is real-analytic in B ( ), by the interior regularity for solutions of elliptic equations,we have that u is smooth in B ( ). Hence, we have that − ∆ ∂ α x u ( x ) = ∂ α x f ( x ) for all x ∈ B ( ) , for every α = ( α , . . . , α N ) ∈ N N .Multiplying the above equation by (1 − | x | ) | α | +1) ∂ α x u gives − (1 − | x | ) | α | +1) ∂ α x u ( x )∆ ∂ α x u ( x ) = (1 − | x | ) | α | +1) ∂ α x u ( x ) ∂ α x f ( x ) , ∀ x ∈ B ( ) , (A.4)and integration by parts gives (cid:90) (cid:90) B ( ) (1 − | x | ) | α | +1) |∇ ∂ α x u | d x = 4( | α | + 1) (cid:90) (cid:90) B ( ) (1 − | x | ) | α | +1 ( ∇ ∂ α x u · x ) ∂ α x u d x + (cid:90) (cid:90) B ( ) (1 − | x | ) | α | +1) ∂ α x u∂ α x f d x . Now, thanks to the Young’s inequality, we have the following estimate (cid:90) (cid:90) B ( ) (1 − | x | ) | α | +1) |∇ ∂ α x u | d x ≤ [16( | α | + 1) + 1] (cid:90) (cid:90) B ( ) (1 − | x | ) | α | | ∂ α x u | d x + (cid:90) (cid:90) B ( ) | ∂ α x f | d x . Since f satisfies (A.1), we get (cid:90) (cid:90) B ( ) (1 − | x | ) | α | +1) |∇ ∂ α x u | d x ≤ | α | + 1) (cid:90) (cid:90) B ( ) (1 − | x | ) | α | | ∂ α x u | d x + | B (0) | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) M | α | ! ρ | α | (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . Therefore, we obtain (cid:13)(cid:13)(cid:13) (1 − | x | ) | α | +1 ∇ ∂ α x u (cid:13)(cid:13)(cid:13) L ( B ( )) ≤ (cid:34) ( | α | + 1) (cid:13)(cid:13)(cid:13) (1 − | x | ) | α | ∂ α x u (cid:13)(cid:13)(cid:13) L ( B ( )) + M | α | ! ρ | α | (cid:35) , (A.5)for every α = ( α , . . . , α N ) ∈ N N . In particular, taking α = (0 , . . . , (cid:13)(cid:13) (1 − | x | ) ∇ u (cid:13)(cid:13) L ( B ( )) ≤ (cid:0) (cid:107) u (cid:107) L ( B ( )) + M (cid:1) . By induction, we have the inequality (cid:13)(cid:13) (1 − | x | ) | α | ∂ α x u (cid:13)(cid:13) L ( B ( )) ≤ ρ −| α |− | α | ! (cid:0) (cid:107) u (cid:107) L ( B ( )) + M (cid:1) , (A.6) FELIPE W. CHAVES-SILVA , DIEGO A. SOUZA , AND
CAN ZHANG for some constant 0 < ρ < min (cid:8) ρ , (cid:9) and every α = ( α , . . . , α N ) ∈ N N .It is not difficult to see that estimate (A.6) leads to (A.3). (cid:3) Acknowledgements . The authors would like to appreciate Prof. E. Tr´elat and Prof. G.Lebeau for the stimulating conversations during this work.
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