aa r X i v : . [ m a t h . O C ] A p r THE TURNPIKE PROPERTY IN SEMILINEAR CONTROL
Dario Pighin
Departamento de Matem´aticas, Universidad Aut´onoma de Madrid28049 Madrid, SpainChair of Computational Mathematics, Fundaci´on DeustoUniversity of Deusto, 48007, Bilbao, Basque Country, Spain
Abstract.
An exponential turnpike property for a semilinear control problemis proved. The state-target is assumed to be small, whereas the initial datumcan be arbitrary.Turnpike results have also been obtained for large targets, requiring thatthe control acts everywhere. In this case, we prove the convergence of theinfimum of the averaged time-evolution functional towards the steady one.Numerical simulations have been performed.
Introduction
In this manuscript, the long time behaviour of semilinear optimal control prob-lems as the time-horizon tends to infinity is analyzed. Our results are global,meaning that we do not require smallness of the initial datum for the governingstate equation.In [33], A. Porretta and E. Zuazua studied turnpike property for control problemsgoverned by a semilinear heat equation, with dissipative nonlinearity. In particular,[33, Theorem 1] yields the existence of a solution to the optimality system fulfillingthe turnpike property, under smallness conditions on the initial datum and thetarget. Our first goal is to(1) prove that in fact the (exponential) turnpike property is satisfied by theoptimal control and state;(2) remove the smallness assumption on the initial datum.We keep the smallness assumption on the target. This leads to the smallness anduniqueness of the steady optima (see [33, subsection 3.2]), whence existence anduniqueness of the turnpike follows. We also treat the case of large targets, underthe added assumption that control acts everywhere. In this case, we prove a weakturnpike result, which stipulates that the averaged infimum of the time-evolutionfunctional converges towards the steady one. We also provide an L bound of thetime derivative of optimal states, uniformly in the time horizon.Generally speaking, in turnpike theory a time-evolution optimal control problemis considered together with its steady version. The “Turnpike Property” is verified Mathematics Subject Classification.
Primary: 49N99; Secondary: 35K91.
Key words and phrases.
Optimal control problems, long time behavior, the turnpike property,semilinear parabolic equations.This project has received funding from the European Research Council (ERC) under the Eu-ropean Unions Horizon 2020 research and innovation programme (grant agreement No 694126-DYCON).We acknowledge professor Enrique Zuazua for his helpful remarks on the manuscript. τ T − τ Tyy O ty steady optimumquasi-optimumoptimum
Figure 1. quasi-optimal turnpike strategiesif the time-evolution optima remain close to the steady optima up to some thininitial and final boundary layers.An extensive literature is available on the topic. A pioneer on the topic has beenJohn von Neumann [43]. In econometrics the topic has been widely investigatedby several scholars including P. Samuelson and L.W. McKenzie [13, 38, 27, 28, 29,9, 21]. Long time behaviour of optimal control problems have been studied by P.Kokotovic and collaborators [44, 2], by R.T. Rockafellar [37] and by A. Rapaportand P. Cartigny [35, 36]. A.J. Zaslavski wrote a book [47] on the topic. A turnpike-like asymptotic simplification have been obtained in the context of optimal designof the diffusivity matrix for the heat equation [1]. In the papers [12, 18, 17, 39], theconcept of (measure) turnpike is related to the dissipativity of the control problem.Recent papers on long time behaviour of Mean Field games [7, 8, 31] motivatednew research on the topic. A special attention have been paid in providing anexponential estimate, as in the work [32] by A. Porretta and E. Zuazua, wherelinear quadratic control problems were considered. These results have later beenextended in [41, 33, 46, 40, 20, 19] to control problems governed by a nonlinear stateequation and applied to optimal control of the Lotka-Volterra system [23]. Recentlyturnpike property have been studied around nonsteady trajectories [40, 15]. Theturnpike property is intimately related to asymptotic behaviour of the Hamilton-Jacobi equation [24].Note that for a general optimal control problem, even in absence of a turnpikeresult, we can construct quasi-optimal turnpike strategies (see [22, Remark 7]) asin fig. 1:(1) in a short time interval [0 , τ ] drive the state from the initial configuration y to the turnpike y ;(2) in a long time arc [ τ, T − τ ], remain on y ;(3) in a short final arc [ T − τ, T ], use to control to match the required terminalcondition at time t = T .In general, the corresponding control and state are not optimal, being not smooth.However, they are easy to construct. HE TURNPIKE PROPERTY IN SEMILINEAR CONTROL 3
The proof of turnpike results is harder than the above construction. In fact, toprove turnpike results, one has to ensure that there is not another time-evolvingstrategy which is significantly better than the above one. In case the turnpikeproperty is verified, the above strategy is quasi-optimal.
Statement of the main results.
We consider the semilinear optimal controlproblem: min u ∈ L ((0 ,T ) × ω ) J T ( u ) = 12 Z T Z ω | u | dxdt + β Z T Z ω | y − z | dxdt, (1)where: y t − ∆ y + f ( y ) = uχ ω in (0 , T ) × Ω y = 0 on (0 , T ) × ∂ Ω y (0 , x ) = y ( x ) in Ω . (2)As usual, Ω is a regular bounded open subset of R n , with n = 1 , ,
3. The nonlin-earity f is C nondecreasing, with f (0) = 0. The action of the control is localizedby multiplication by χ ω , characteristic function of the open subregion ω ⊆ Ω. Thetarget z is assumed to be in L ∞ ( ω ). Since that the nonlinearity is nondecreasing,the semilinear problem (2) is well-posed [4, chapter 5] . Namely, given an initialdatum y ∈ L (Ω) and a control u ∈ L ((0 , T ) × ω ), there exists a unique solution y ∈ C ([0 , T ]; L (Ω)) ∩ L (0 , T ; H (Ω)) .ω ⊆ Ω is an open subset and β ≥ β increases, thedistance between the optimal state and the target decreases.By the direct method in the calculus of variations [10, 42], there exists a globalminimizer of (1). As we shall see, uniqueness can be guaranteed, provided that theinitial datum and the target are small enough in the uniform norm.Taking the Gˆateaux differential of the functional (1) and imposing the Fermatstationary condition, we realize that any optimal control reads as u T = − q T χ ω ,where (cid:0) y T , q T (cid:1) solves y Tt − ∆ y T + f ( y T ) = − q T χ ω in (0 , T ) × Ω y T = 0 on (0 , T ) × ∂ Ω y T (0 , x ) = y ( x ) in Ω − q Tt − ∆ q T + f ′ ( y T ) q T = β ( y T − z ) χ ω in (0 , T ) × Ω q T = 0 on (0 , T ) × ∂ Ω q T ( T, x ) = 0 in Ω . (3)In order to study the turnpike, we need to study the steady version of (2)-(1):min u s ∈ L (Ω) J s ( u s ) = 12 Z ω | u s | dx + β Z ω | y s − z | dx, (4)where: ( − ∆ y s + f ( y s ) = u s χ ω in Ω y s = 0 on ∂ Ω . (5) THE TURNPIKE PROPERTY IN SEMILINEAR CONTROL
Under the same assumptions required for the problem (2)-(1), for any given control u s ∈ L (Ω), there exists a unique state y s ∈ H (Ω) ∩ H (Ω) solution to (5) (seee.g. [5]).By adapting the techniques of [10], we have the existence of a global minimizer u for (4). The corresponding optimal state is denoted by y . If the target is sufficientlysmall in the uniform norm, the optimal control is unique (see [33, subsection 3.2]).Furthermore any optimal control u = − qχ ω , where the pair ( y, q ) satisfies thesteady optimality system − ∆ y + f ( y ) = − qχ ω in Ω y = 0 on ∂ Ω − ∆ q + f ′ ( y ) q = β ( y − z ) χ ω in Ω q = 0 on ∂ Ω . (6)The analysis in [33, section 3] leads to the following local result. Theorem 0.1 (Porretta-Zuazua) . Consider the control problem (5) - (4) . Thereexists δ > such that if the initial datum and the target fulfill the smallnesscondition k y k L ∞ (Ω) ≤ δ and k z k L ∞ ( ω ) ≤ δ, there exists a solution (cid:0) y T , q T (cid:1) to the Optimality System y Tt − ∆ y T + f ( y T ) = − q T χ ω in (0 , T ) × Ω y T = 0 on (0 , T ) × ∂ Ω y T (0 , x ) = y ( x ) in Ω − q Tt − ∆ q T + f ′ ( y T ) q T = β ( y T − z ) χ ω in (0 , T ) × Ω q T = 0 on (0 , T ) × ∂ Ω q T ( T, x ) = 0 in Ω satisfying for any t ∈ [0 , T ] k q T ( t ) − q k L ∞ (Ω) + k y T ( t ) − y k L ∞ (Ω) ≤ K [exp ( − µt ) + exp ( − µ ( T − t ))] , where K and µ are T -independent. We observe that the turnpike property is satisfied by one solution to the optimal-ity system. Since our problem may be not convex, we cannot directly assert thatsuch solution of the optimality system is the unique minimizer (optimal control)for (5)-(4).
Large initial data and small targets.
We start by keeping the running target small,but allowing the initial datum for (2) to be large.
Theorem 0.2.
Consider the control problem (2) - (1) . Let u T be a minimizer of (1) . There exists ρ > such that for every initial datum y ∈ L ∞ (Ω) and target z verifying k z k L ∞ ( ω ) ≤ ρ, (7) we have k u T ( t ) − u k L ∞ (Ω) + k y T ( t ) − y k L ∞ (Ω) ≤ K [exp ( − µt ) + exp ( − µ ( T − t ))] , ∀ t ∈ [0 , T ] , (8) the constants K and µ > being independent of the time horizon T . HE TURNPIKE PROPERTY IN SEMILINEAR CONTROL 5 t s Tyy O δty steady optimumoptimum
Figure 2. global-local argumentNote that ρ is smaller than the smallness parameter δ in Theorem 0.1.The main ingredients our proofs require are:(1) prove a L ∞ bound of the norm of the optimal control, uniform in the timehorizon T > small data and small targets . Note that,in Theorem 0.1, the authors prove the existence of a solution to the opti-mality system enjoying the turnpike property. In this preliminary step, for small data and small targets , we prove that any optimal control verifies theturnpike property (Lemma 1.2 in section 1.1);(3) for small targets and any data , proof of the smallness of k y T ( t ) k L ∞ (Ω) intime t large (section 1.2). This is done by estimating the critical time t s needed to approach the turnpike;(4) conclude concatenating the two former steps (section 1.2).Theorem 0.2 ensures that the conclusion of Theorem 0.1 holds for the optimalpair.Let us outline the proof of 3 (fig. 2), the existence of τ upper bound for theminimal time needed to approach the turnpike t s .Suppose, by contradiction, that the critical time t s to approach the turnpike is verylarge. Accordingly, the time-evolution optimal strategy obeys the following plan:(1) stay away from the turnpike for long time;(2) move close to the turnpike;(3) enjoy a final time-evolution performance, cheaper than the steady one.Then, in phase 1, with respect to the steady performance, an extra cost is generated,which should be regained in phase 3. At this point, we realize that this is preventedby validity of the local turnpike property. Indeed, once the time-evolution optimaapproach the turnpike at some time t s , the optimal pair satisfies the turnpikeproperty for larger times t ≥ t s . Hence, for t ≥ t s , the time-evolution performancecannot be significantly cheaper than the steady one. Accordingly, we cannot regainthe extra-cost generated in phase 1, so obtaining a contradiction. THE TURNPIKE PROPERTY IN SEMILINEAR CONTROL
Control acting everywhere: convergence of averages for arbitrary targets.
In sec-tion 2 we deal with large targets, supposing the control acts everywhere (i.e. ω = Ω).We prove that averages converge. Furthermore, we obtain an L bound for the timederivative of optimal states. The bound is uniform independent of the time horizon T , meaning that, if T is large, the time derivative of the optimal state is small formost of the time. Theorem 0.3.
Take an arbitrary initial datum y ∈ L ∞ (Ω) and an arbitrarytarget z ∈ L ∞ ( ω ) . Consider the time-evolution control problem (2) - (1) and itssteady version (5) - (4) . Assume ω = Ω . Then, averages converge T inf L ((0 ,T ) × ω ) J T −→ T → + ∞ inf L (Ω) J s . (9) Suppose in addition y ∈ L ∞ (Ω) ∩ H (Ω) . Let u T be an optimal control for (2) - (1) and let y T be the corresponding state, solution to (2) , with control u T and initialdatum y . Then, the L norm of the time derivative of the optimal state is boundeduniformly in T (cid:13)(cid:13) y Tt (cid:13)(cid:13) L ((0 ,T ) × Ω) ≤ K, (10) the constant K being T -independent. The proof of Theorem 0.3, available in section 2, is based on the following rep-resentation formula for the time-evolving functional (Lemma 2.2): J T ( u ) = Z T J s (cid:0) − ∆ y ( t, · ) + f ( y ( t, · )) (cid:1) dt + 12 Z T Z Ω | y t ( t, x ) | dxdt + 12 Z Ω h k∇ y ( T, x ) k + 2 F ( y ( T, x )) − k∇ y ( x ) k − F ( y ( x )) i dx, (11)where for a.e. t ∈ (0 , T ), J s (cid:0) − ∆ y ( t, · ) + f ( y ( t, · )) (cid:1) denotes the evaluation of thesteady functional J s at control u s ( · ) := − ∆ y ( t, · ) + f ( y ( t, · )) and y is the stateassociated to control u solving y t − ∆ y + f ( y ) = u in (0 , T ) × Ω y = 0 on (0 , T ) × ∂ Ω y (0 , x ) = y ( x ) in Ω . (12)Note that the above formula is valid for initial data y ∈ L ∞ (Ω) ∩ H (Ω). However,by the regularizing effect of (12) and the properties of the control problem, one canreduce to the case of smooth initial data.By means of (11), the functional J T can be seen as the sum of three terms:(1) R T J s (cid:0) − ∆ y ( t, · ) + f ( y ( t, · )) (cid:1) dt , which stands for the “steady” cost at a.e.time t ∈ (0 , T ) integrated over (0 , T );(2) R T R Ω | y t ( t, x ) | dxdt , which penalizes the time derivative of the func-tional;(3) R Ω h k∇ y ( T, x ) k + 2 F ( y ( T, x )) − k∇ y ( x ) k − F ( y ( x )) i dx , which de-pends on the terminal values of the state. HE TURNPIKE PROPERTY IN SEMILINEAR CONTROL 7
Choose now an optimal control u T for (2)-(1) and plug it in (11). By Lemma 1.1,the term R Ω h k∇ y ( T, x ) k + 2 F ( y ( T, x )) − k∇ y ( x ) k − F ( y ( x )) i dx can be estimated uni-formly in the time horizon. At the optimal control, the term R T R Ω | y t ( t, x ) | dxdt has to be small and the “steady” cost R T J s (cid:0) − ∆ y ( t, · ) + f ( y ( t, · )) (cid:1) dt is the dominant addendum. This is the basic ideaof our approach to prove turnpike results for large targets.The rest of the manuscript is organized as follows. In section 1 we prove The-orem 0.2. In section 2, we prove Theorem 0.3. In section 3 we perform somenumerical simulations. The appendix is mainly devoted to the proof of the uniformbound of the optima (Lemma 1.1) and a PDE result needed for Lemma 2.2.1. Proof of Theorem 0.2
Preliminary Lemmas.
As announced, we firstly exhibit an upper bound ofthe norms of the optima in terms of the data. Note that the Lemma below yieldsan uniform bound for large targets as well.
Lemma 1.1.
Consider the control problem (2) - (1) . Let R > , y ∈ L ∞ (Ω) and z ∈ L ∞ ( ω ) , satisfying k y k L ∞ (Ω) ≤ R and k z k L ∞ ( ω ) ≤ R . Let u T be an optimalcontrol for (2) - (1) . Then, u T and y T are bounded and k u T k L ∞ ((0 ,T ) × ω ) + (cid:13)(cid:13) y T (cid:13)(cid:13) L ∞ ((0 ,T ) × Ω) ≤ K (cid:2) k y k L ∞ (Ω) + k z k L ∞ ( ω ) (cid:3) , (13) where the constant K is independent of the time horizon T , but it depends on R . The proof is postponed to the Appendix.The second ingredient for the proof of Theorem 0.2 is the following Lemma.
Lemma 1.2.
Consider the control problem (2) - (1) . Let y ∈ L ∞ (Ω) and z ∈ L ∞ ( ω ) . There exists δ > such that, if k z k L ∞ ( ω ) ≤ δ and k y k L ∞ (Ω) ≤ δ, (14) the functional (1) admits a unique global minimizer u T . Furthermore, for every ε > there exists δ ε > such that, if k z k L ∞ ( ω ) ≤ δ ε and k y k L ∞ (Ω) ≤ δ ε , (15) the functional (1) admits a unique global minimizer u T and k u T ( t ) − u k L ∞ (Ω) + k y T ( t ) − y k L ∞ (Ω) ≤ ε [exp ( − µt ) + exp ( − µ ( T − t ))] , ∀ t ∈ [0 , T ] , (16)( u, y ) being the optimal pair for (4) . The constants δ ε and µ > are independentof the time horizon and µ is given by (cid:13)(cid:13)(cid:13) E ( t ) − b E (cid:13)(cid:13)(cid:13) L (L (Ω) , L (Ω)) ≤ C exp ( − µt ) , k exp ( − tM ) k L (L (Ω) , L (Ω)) ≤ exp ( − µt ) , M := − ∆ + f ′ ( y ) + b Eχ ω . (17) where E and b E denote respectively the differential and algebraic Riccati operators(see [33, equation (22)] ) and ∆ : H (Ω) −→ H − (Ω) is the Dirichlet laplacian. THE TURNPIKE PROPERTY IN SEMILINEAR CONTROL
Proof of Lemma 1.2.
We introduce the critical ball B := n u ∈ L ∞ ((0 , T ) × ω ) (cid:12)(cid:12)(cid:12) k u k L ∞ ((0 ,T ) × ω ) ≤ K (cid:2) k y k L ∞ (Ω) + k z k L ∞ ( ω ) (cid:3)o , (18)where K is the constant appearing in (13). Step 1
Strict convexity in B for small data By [11, section 5] or [10], the second order Gˆateaux differential of J reads as h d J T ( u ) w, w i = Z T Z ω w dxdt + Z T Z ω | ψ w | dxdt − Z T Z Ω f ′′ ( y ) q | ψ w | dxdt, where y solves (2) with control u and initial datum y , ψ w solves the linearizedproblem ( ψ w ) t − ∆ ψ w + f ′ ( y ) ψ w = wχ ω in (0 , T ) × Ω ψ w = 0 on (0 , T ) × ∂ Ω ψ w (0 , x ) = 0 in Ω (19)and − q t − ∆ q + f ′ ( y ) q = ( y − z ) χ ω in (0 , T ) × Ω q = 0 on (0 , T ) × ∂ Ω q ( T, x ) = 0 in Ω . (20)Since f ′ ( y ) ≥ k ψ w k L ((0 ,T ) × Ω) ≤ K k w k L ((0 ,T ) × ω ) . Let u ∈ B . By applying a comparison argument to (2) and (20), k y k L ∞ ((0 ,T ) × Ω) + k q k L ∞ ((0 ,T ) × Ω) ≤ K (cid:2) k y k L ∞ (Ω) + k z k L ∞ ( ω ) (cid:3) . Hence, h d J T ( u ) w, w i ≥ Z T Z ω | ψ w | dxdt + (cid:8) − K (cid:2) k y k L ∞ (Ω) + k z k L ∞ ( ω ) (cid:3)(cid:9) Z T Z ω | w | dxdt, If k y k L ∞ (Ω) and k z k L ∞ ( ω ) are small enough, we have h d J T ( u ) w, w i ≥ Z T Z ω | w | dxdt, whence the strict convexity of J in the critical ball B . Now, by (13) and (18), if k y k L ∞ (Ω) and k z k L ∞ ( ω ) are small enough, any optimal control u T belongs to B .Then, there exists a unique solution to the optimality system, with control in thecritical ball B and such control coincides with u T the unique global minimizer of(1). Step 2
Conclusion
Let ε >
0. By following the fixed-point argument developed in the proof of [33,Theorem 1 subsection 3.1] and in [33, subsection 3.2], we can find δ ε > k z k L ∞ ( ω ) ≤ δ ε and k y k L ∞ (Ω) ≤ δ ε , there exists a solution (cid:0) y T , q T (cid:1) to the optimality system such that k u T k L ∞ ((0 ,T ) × ω ) < ε and k u T ( t ) − u k L ∞ (Ω) + k y T ( t ) − y k L ∞ (Ω) ≤ K [exp ( − µt ) + exp ( − µ ( T − t ))] , ∀ t ∈ [0 , T ] . HE TURNPIKE PROPERTY IN SEMILINEAR CONTROL 9
By Step 1, if ε is small enough, u T := − q T χ ω is a strict global minimizer for J T .Then, being strict, it is the unique one. This finishes the proof. (cid:3) In the following Lemma, we compare the value of the time evolution functional(1) at a control u , with the value of the steady functional (4) at control u , supposingthat u and u satisfy a turnpike-like estimate. Lemma 1.3.
Consider the time-evolution control problem (2) - (1) and its steadyversion (5) - (4) . Fix y ∈ L (Ω) an initial datum and z ∈ L ( ω ) a target. Let u ∈ L ∞ (Ω) be a control and let y be the corresponding solution to (5) . Let u ∈ L ∞ ((0 , T ) × ω ) be a control and y the solution to (2) , with control u . Assume k u ( t ) − u k L ∞ (Ω) + k y ( t ) − y k L ∞ (Ω) ≤ K [exp ( − µt ) + exp ( − µ ( T − t ))] , ∀ t ∈ [0 , T ] , (21) with K = K (Ω , β, y ) and µ = µ (Ω , β ) . Then, | J T ( u ) − T J s ( u ) | ≤ C (cid:2) k u k L ∞ (Ω) + k z k L ∞ ( ω ) (cid:3) , (22) the constant C depending only on the above constant K and µ .Proof of Lemma 1.3. We estimate | J T ( u ) − T J s ( u ) | = (cid:12)(cid:12)(cid:12)(cid:12) k u k ((0 ,T ) × ω ) + β k y − z k ((0 ,T ) × ω ) − T (cid:20) k u k (Ω) + β k y − z k ( ω ) (cid:21)(cid:12)(cid:12)(cid:12)(cid:12) = (cid:12)(cid:12)(cid:12)(cid:12) k u − u k ((0 ,T ) × ω ) + β k y − y k ((0 ,T ) × ω ) + Z T Z ω ( u − u ) udxdt + β Z T Z ω ( y − y )( y − z ) dxdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ C (cid:2) k u k L ∞ ( ω ) + k z k L ∞ ( ω ) (cid:3) (Z T h k u − u k ∞ ( ω ) + k u − u k L ∞ ( ω ) i dt + Z T h k y − y k ∞ ( ω ) + k y − y k L ∞ ( ω ) i dt ) ≤ C (cid:2) k u k L ∞ ( ω ) + k z k L ∞ ( ω ) (cid:3) , where the last inequality follows from (21). (cid:3) The following Lemma (fig. 3) plays a key role in the proof of Theorem 0.2.Let u T be an optimal control for (2)-(1). Let y T be the corresponding optimalstate. For any ε >
0, let δ ε be given by (15). Set t s := inf (cid:8) t ∈ [0 , T ] | k y T ( t ) k L ∞ (Ω) ≤ δ ε (cid:9) , where we use the convention inf( ∅ ) = T . Lemma 1.4 (Global attactor property) . Consider the control problem (2) - (1) . Let y ∈ L ∞ (Ω) and z ∈ L ∞ ( ω ) . Let u T be an optimal control for (2) - (1) and let y T be the corresponding optimal state. For any ε > . there exist ρ ε = ρ ε (Ω , β, ε ) and τ ε = τ ε (Ω , β, y , ε ) , such that if k z k L ∞ ( ω ) ≤ ρ ε and T ≥ τ ε , k y T ( t s ) k L ∞ (Ω) ≤ δ ε , with the upper bound t s ≤ τ ε (23) t s Tyy O δty steady optimumoptimum
Figure 3. global-local argument employed in the proof of Lemma 1.4 and k u T ( t ) − u k L ∞ (Ω) + k y T ( t ) − y k L ∞ (Ω) ≤ ε [exp ( − µ ( t − t s )) + exp ( − µ ( T − ( t − t s )))] , ∀ t ∈ [ t s , T ] . (24) The constant µ is given by (17) and δ ε is given by (15) .Proof of Lemma 1.4. Throughout the proof, constant K = K (Ω , β ) is chosen assmall as needed, whereas constant K = K (Ω , β, y ) is chosen as large as needed. Step 1
Estimate of the L ∞ norm of steady optimal controls In this step, we follow the arguments of [33, subsection 3.2]. Let u ∈ L (Ω) be anoptimal control for (5)-(4). By definition of minimizer (optimal control),12 k u k ( ω ) ≤ J s ( u ) ≤ J s (0) = β k z k ( ω ) ≤ βµ leb ( ω )2 k z k ∞ ( ω ) . Now, any optimal control is of the form u = − qχ ω , where the pair ( y, q ) satisfiesthe optimality system (6). Since n = 1 , ,
3, by elliptic regularity (see, e.g. [14,Theorem 4 subsection 6.3.2]) and Sobolev embeddings (see e.g. [14, Theorem 6subsection 5.6.3]), q ∈ C (Ω) and k q k L ∞ (Ω) ≤ K k z k L ∞ ( ω ) , where K = K (Ω).This yields u ∈ C ( ω ) and k u k L ∞ ( ω ) ≤ K k z k L ∞ ( ω ) . (25) Step 2
There exist ρ ε = ρ ε (Ω , β, ε ) and τ ε = τ ε (Ω , β, y , ε ) , such that if k z k L ∞ ( ω ) ≤ ρ ε , then the critical time satisfies t s ≤ τ ε Let u be an optimal control for the steady problem. Then, by definition of minimizer(optimal control), J T (cid:0) u T (cid:1) ≤ J T ( u ) (26)and, by Lemma 1.3, J T ( u ) ≤ T inf L (Ω) J s + K . (27) HE TURNPIKE PROPERTY IN SEMILINEAR CONTROL 11
Now, we split the integrals in J T into [0 , t s ] and ( t s , T ] J T (cid:0) u T (cid:1) = 12 Z t s Z ω | u T | dt + β Z t s Z ω | y T − z | dxdt + 12 Z Tt s Z ω | u T | dt + β Z Tt s Z ω | y T − z | dxdt. (28)Set: c y ( t, x ) := f ( y T ( t, x )) y T ( t, x ) y T ( t, x ) = 0 f ′ (0) y T ( t, x ) = 0 . Since f is nondecreasing and f (0) = 0, we have c y ≥
0. Then, Lemma A.1 (withpotential c y and source term h := u T χ ω ) yields12 Z t s Z ω | u T | dt + β Z t s Z ω | y T − z | dxdt ≥ K Z t s k y T ( t ) k ∞ (Ω) dt − K . Furthermore, by definition of t s , for any t ∈ [0 , t s ], k y T ( t ) k L ∞ (Ω) ≥ δ ε . Then,12 Z t s Z ω | u T | dt + β Z t s Z ω | y T − z | dxdt ≥ K t s δ ε − K . (29)Once again, by definition of t s , k y T ( t s ) k L ∞ (Ω) = δ ε and k z k L ∞ ( ω ) ≤ δ ε , where δ ε is given by (15). Therefore, by Lemma 1.2, the turnpike estimate (16) issatisfied in [ t s , T ]. Lemma 1.3 applied in [ t s , T ] gives12 Z Tt s Z ω | u T | dt + β Z Tt s Z ω | y T − z | dxdt ≥ ( T − t s ) inf L (Ω) J s − K (cid:2) k u k L ∞ (Ω) + k z k L ∞ ( ω ) (cid:3) ≥ ( T − t s ) inf L (Ω) J s − K , (30)where the last inequality is due to (25) and k z k L ∞ ( ω ) ≤ δ ε .At this point, by section 1.1, (29) and section 1.1 J T (cid:0) u T (cid:1) ≥ K t s δ ε + ( T − t s ) inf L (Ω) J s − K . (31)Therefore, by (31), (26) and (27) K t s δ ε + ( T − t s ) inf L (Ω) J s − K ≤ T inf L (Ω) J s + K , whence t s (cid:20) K δ ε − inf L (Ω) J s (cid:21) ≤ K . (32)Now, by (25), there exists ρ ε = ρ ε (Ω , β, ε ) ≤ δ ε such that, if the target k z k L ∞ ( ω ) ≤ ρ ε , then inf L (Ω) J s ≤ K δ ε . This, together with (32), yields t s K δ ε ≤ K , whence t s ≤ K δ ε . Set τ ε := K δ ε + 1 . This finishes this step.
Step 3
Conclusion
By Step 2, for any T ≥ τ ε , there exists t s ≤ τ ε such that k y T ( t s ) k L ∞ (Ω) ≤ δ ε , (33)where δ ε is given by (16). Now, by Bellman’s Principle of Optimality, u T ↾ ( t s ,T ) isoptimal for (2)-(1), with initial datum y T ( t s ) and target z . We took ρ ε ≤ δ ε , Then,we also have k z k L ∞ ( ω ) ≤ ρ ε ≤ δ ε . (34)Then, we can apply Lemma 1.2, getting (24). This completes the proof. (cid:3) Proof of Theorem 0.2.
We now prove Theorem 0.2.
Proof of Theorem 0.2.
By Lemma 1.4, there exists ρ ε (Ω , β, ε ) > k z k L ∞ ( ω ) ≤ ρ ε , (35)any optimal control satisfies the turnpike estimate k u T ( t ) − u k L ∞ (Ω) + k y T ( t ) − y k L ∞ (Ω) ≤ ε [exp ( − µ ( t − t s )) + exp ( − µ ( T − ( t − t s )))] , ∀ t ∈ [ t s , T ] . (36)Set K := exp( µτ ) K (cid:2) k y k L ∞ (Ω) + δ (cid:3) , with µ > K is given by (13). Note that K = K (Ω , β, y ) and, in particular, it is independent of the time horizon. By theabove definition, for every T > t ∈ [0 , τ ε ] ∩ [0 , T ] k u T ( t ) − u k L ∞ (Ω) + k y T ( t ) − y k L ∞ (Ω) ≤ K exp( − µτ ) ≤ K exp( − µt ) . (37)On the other hand, for t ≥ t s , (36) holds. Then, (8) follows. (cid:3) Control acting everywhere: convergence of averages
In this section, we suppose that the control acts everywhere, namely ω = Ω inthe state equation (2). Our purpose is to prove Theorem 0.3, valid for any dataand targets.In the following Lemma, we observe that, even in the more general case ω ( Ω,we have an estimate from above of the infimum of the time-evolution functional interms of the steady functional. This is the easier task obtained by plugging thesteady optimal control in the time-evolution functional. The complicated task isto estimate from below the infimum of the time-evolution functional, in terms ofthe steady functional. Indeed, the lower bound indicates that the time-evolutionstrategies cannot perform significantly better than the steady one and this is ingeneral the hardest task in the proof of turnpike results. The key idea is indicatedin Lemma 2.2.
Lemma 2.1.
Consider the time-evolution control problem (2) - (1) and its steadyversion (5) - (4) . Arbitrarily fix y ∈ L ∞ (Ω) an initial datum and z ∈ L ∞ ( ω ) atarget. We have inf L ((0 ,T ) × ω ) J T ≤ T inf L (Ω) J s + K, (38) HE TURNPIKE PROPERTY IN SEMILINEAR CONTROL 13 the constant K being independent of T > . The proof is available in appendix C.The main idea for the proof of Theorem 0.3 is in the following Lemma, wherean alternative representation formula for the time-evolution functional is obtained.
Lemma 2.2.
Consider the functional introduced in (1) - (2) and its steady version (5) - (4) . Set F ( y ) := R y f ( ξ ) dξ . Assume ω = Ω . Suppose the initial datum y ∈ L ∞ (Ω) ∩ H (Ω) . Then, for any control u ∈ L ((0 , T ) × ω ) , we can rewrite thefunctional as J T ( u ) = Z T J s (cid:0) − ∆ y ( t, · ) + f ( y ( t, · )) (cid:1) dt + 12 Z T Z Ω | y t ( t, x ) | dxdt + 12 Z Ω h k∇ y ( T, x ) k + 2 F ( y ( T, x )) − k∇ y ( x ) k − F ( y ( x )) i dx, (39) where, for a.e. t ∈ (0 , T ) , J s (cid:0) − ∆ y ( t, · ) + f ( y ( t, · )) (cid:1) denotes the evaluation ofthe steady functional J s at control u s ( · ) := − ∆ y ( t, · ) + f ( y ( t, · )) and y is the stateassociated to control u solution to y t − ∆ y + f ( y ) = u in (0 , T ) × Ω y = 0 on (0 , T ) × ∂ Ω y (0 , x ) = y ( x ) in Ω . (40)In (39), the term R T R Ω | y t ( t, x ) | dxdt emerges. This means that the time deriv-ative of optimal states has to be small, whence the time-evolving optimal strategiesfor (1)-(2) are in fact close to the steady ones.The proof of Lemma 2.2 is based on the following PDE result, which basicallyasserts that the squared right hand side of the equation ( y t − ∆ y + f ( y ) = h in (0 , T ) × Ω y = 0 on (0 , T ) × ∂ Ωcan be written as k h k ((0 ,T ) × Ω) = k y t k ((0 ,T ) × Ω) + k− ∆ y + f ( y ) k ((0 ,T ) × Ω) + remainder , (41)where the remainder depends on the value of the solution at times t = 0 and t = T . Lemma 2.3.
Let Ω be a bounded open set of R n , n ∈ { , , } , with C ∞ boundary.Let f ∈ C ( R ; R ) be nondecreasing, with f (0) = 0 . Set F ( y ) := R y f ( ξ ) dξ . Let y ∈ L ∞ (Ω) ∩ H (Ω) be an initial datum and let h ∈ L ∞ ((0 , T ) × Ω) be a sourceterm. Let y be the solution to y t − ∆ y + f ( y ) = h in (0 , T ) × Ω y = 0 on (0 , T ) × ∂ Ω y (0 , x ) = y ( x ) in Ω . (42) Then, the following identity holds Z T Z Ω | h | dxdt = Z T Z Ω h | y t | + |− ∆ y + f ( y ) | i dxdt (43)+ Z Ω h k∇ y ( T, x ) k + 2 F ( y ( T, x )) − k∇ y ( x ) k − F ( y ( x )) i dx. Proof of lemma 2.3.
We start by proving our assertion for C ∞ -smooth data, withcompact support. By (42), we have Z T Z Ω | h | dxdt = Z T Z Ω | y t − ∆ y + f ( y ) | dxdt = Z T Z Ω h | y t | + |− ∆ y + f ( y ) | i dxdt + 2 Z T Z Ω y t [ − ∆ y + f ( y )] dxdt. (44)We now concentrate on the terms 2 R T R Ω y t [ − ∆ y ] dxdt and 2 R T R Ω y t f ( y ) dxdt .Integrating by parts in space, we get2 Z T Z Ω y t [ − ∆ y ] dxdt = Z T Z Ω ∂ ∇ y∂t · ∇ ydxdt = Z Ω h k∇ y ( T, x ) k − k∇ y ( x ) k i dx. (45)By using the chain rule and the definition F ( y ) := R y f ( ξ ) dξ , we have Z T Z Ω y t f ( y ) dxdt = Z T Z Ω ∂∂t [ F ( y )] dxdt = Z Ω [ F ( y ( T, x )) − F ( y ( x ))] dx. (46)By (44), (45) and (46), we get (43).The conclusion for general data follows from a density argument based on parabolicregularity (see [26, Theorem 7.32 page 182], [25, Theorem 9.1 page 341] or [45,Theorem 9.2.5 page 275]). (cid:3) We proceed now with the proof of Lemma 2.2.
Proof of Lemma 2.2.
For any control u ∈ L ((0 , T ) × ω ), by Lemma 2.3 applied to(40), we have12 Z T Z ω | u | dxdt = Z T Z Ω h | y t | + |− ∆ y + f ( y ) | i dxdt + Z Ω h k∇ y ( T, x ) k + 2 F ( y ( T, x )) − k∇ y ( x ) k − F ( y ( x )) i dx. whence J T ( u ) = 12 Z T Z Ω |− ∆ y + f ( y ) | dxdt + β Z T Z ω | y − z | dxdt + 12 Z T Z Ω | y t | dxdt + 12 Z Ω h k∇ y ( T, x ) k + 2 F ( y ( T, x )) − k∇ y ( x ) k − F ( y ( x )) i dx. HE TURNPIKE PROPERTY IN SEMILINEAR CONTROL 15
By the above equality and the definition of J s (5)-(4), formula (39) follows. (cid:3) The last Lemma needed to prove Theorem 0.3 is the following one.
Lemma 2.4.
Consider the time-evolution control problem (2) - (1) and its steadyversion (5) - (4) . Assume ω = Ω . Arbitrarily fix y ∈ L ∞ (Ω) ∩ H (Ω) an initialdatum and z ∈ L ∞ ( ω ) a target. Let u T be an optimal control for (2) - (1) and let y T be the corresponding state, solution to (2) , with control u T and initial datum y .Then,(1) there exists a T -independent constant K such that (cid:12)(cid:12)(cid:12)(cid:12) inf L ((0 ,T ) × ω ) J T − T inf L (Ω) J s (cid:12)(cid:12)(cid:12)(cid:12) ≤ K ; (47) (2) the L norm of the time derivative of the optimal state is bounded uniformlyin T (cid:13)(cid:13) y Tt (cid:13)(cid:13) L ((0 ,T ) × Ω) ≤ K, (48) with K independent of T > .Proof of Lemma 2.4. Step 1 Proof of inf L ((0 ,T ) × ω ) J T = J T (cid:0) u T (cid:1) ≥ T inf L (Ω) J s + 12 Z T Z Ω (cid:12)(cid:12) y Tt ( t, x ) (cid:12)(cid:12) dxdt − Z Ω h k∇ y ( x ) k + 2 F ( y ( x )) i dx. We start observing that, since the nonlinearity f is nondecreasing and f (0) = 0,the primitive F is nonnegative F ( y ) ≥ , ∀ y ∈ R . (49)Let u T be an optimal control for (2)-(1) and let y T be the corresponding state,solution to (2), with control u T and initial datum y . By Lemma 2.2 and (49), wehave J T (cid:0) u T (cid:1) = Z T J s (cid:0) − ∆ y T ( t, · ) + f (cid:0) y T ( t, · ) (cid:1) (cid:1) dt + 12 Z T Z Ω (cid:12)(cid:12) y Tt ( t, x ) (cid:12)(cid:12) dxdt + 12 Z Ω h(cid:13)(cid:13) ∇ y T ( T, x ) (cid:13)(cid:13) + 2 F (cid:0) y T ( T, x ) (cid:1) − k∇ y ( x ) k − F ( y ( x )) i dx ≥ Z T J s (cid:0) − ∆ y T ( t, · ) + f (cid:0) y T ( t, · ) (cid:1) (cid:1) dt + 12 Z T Z Ω (cid:12)(cid:12) y Tt ( t, x ) (cid:12)(cid:12) dxdt − Z Ω h k∇ y ( x ) k + 2 F ( y ( x )) i dx. (50)Now, for a.e. t ∈ (0 , T ), by definition of infimum J s (cid:0) − ∆ y T ( t, · ) + f (cid:0) y T ( t, · ) (cid:1) (cid:1) ≥ inf L (Ω) J s . The above inequality and (50) yield J T (cid:0) u T (cid:1) ≥ Z T J s (cid:0) − ∆ y T ( t, · ) + f (cid:0) y T ( t, · ) (cid:1) (cid:1) dt + 12 Z T Z Ω (cid:12)(cid:12) y Tt ( t, x ) (cid:12)(cid:12) dxdt − Z Ω h k∇ y ( x ) k + 2 F ( y ( x )) i dx ≥ Z T (cid:20) inf L (Ω) J s (cid:21) dt + 12 Z T Z Ω (cid:12)(cid:12) y Tt ( t, x ) (cid:12)(cid:12) dxdt − Z Ω h k∇ y ( x ) k + 2 F ( y ( x )) i dx = T inf L (Ω) J s + 12 Z T Z Ω (cid:12)(cid:12) y Tt ( t, x ) (cid:12)(cid:12) dxdt − Z Ω h k∇ y ( x ) k + 2 F ( y ( x )) i dx, whenceinf L ((0 ,T ) × ω ) J T = J T (cid:0) u T (cid:1) ≥ T inf L (Ω) J s + 12 Z T Z Ω (cid:12)(cid:12) y Tt ( t, x ) (cid:12)(cid:12) dxdt − Z Ω h k∇ y ( x ) k + 2 F ( y ( x )) i dx. (51) Step 2
Conclusion
On the one hand, by Lemma 2.1, we haveinf L ((0 ,T ) × ω ) J T − T inf L (Ω) J s ≤ K, (52)the constant K being independent of T >
0. On the other hand, by (51), we getinf L ((0 ,T ) × ω ) J T − T inf L (Ω) J s ≥ − K. (53)By (52) and (53), inequality (47) follows.It remains to prove (48). By (51) and Lemma 2.1, we have T inf L (Ω) J s + 12 Z T Z Ω (cid:12)(cid:12) y Tt ( t, x ) (cid:12)(cid:12) dxdt − K ≤ inf L ((0 ,T ) × ω ) J T ≤ T inf L (Ω) J s + K, whence 12 Z T Z Ω (cid:12)(cid:12) y Tt ( t, x ) (cid:12)(cid:12) dxdt ≤ K, as required. (cid:3) We are now ready to prove Theorem 0.3.
Proof of Theorem 0.3.
Estimate (10) follows directly from Lemma 2.4 (2.).It remains to prove the convergence of the averages. By the regularizing effect ofthe state equation (2) and Lemma 1.1, we can reduce to the case of initial datum y ∈ L ∞ (Ω) ∩ H (Ω). By Lemma 2.4, we have (cid:12)(cid:12)(cid:12)(cid:12) inf L ((0 ,T ) × ω ) J T − T inf L (Ω) J s (cid:12)(cid:12)(cid:12)(cid:12) ≤ K. (54)Then, (cid:12)(cid:12)(cid:12)(cid:12) T inf L ((0 ,T ) × ω ) J T − inf L (Ω) J s (cid:12)(cid:12)(cid:12)(cid:12) = 1 T (cid:12)(cid:12)(cid:12)(cid:12) inf L ((0 ,T ) × ω ) J T − T inf L (Ω) J s (cid:12)(cid:12)(cid:12)(cid:12) ≤ KT −→ T → + ∞ , as required. (cid:3) HE TURNPIKE PROPERTY IN SEMILINEAR CONTROL 17 Numerical simulations
This section is devoted to a numerical illustration of Theorem 0.2. Our goal isto check that turnpike property is fulfilled for small target, regardless of the size ofthe initial datum.We deal with the optimal control problemmin u ∈ L ((0 ,T ) × (0 , )) J T ( u ) = 12 Z T Z | u | dxdt + β Z T Z | y − z | dxdt, where: y t − y xx + y = uχ (0 , ) ( t, x ) ∈ (0 , T ) × (0 , y ( t,
0) = y ( t,
1) = 0 t ∈ (0 , T ) y (0 , x ) = y ( x ) x ∈ (0 , . We choose as initial datum y ≡
10 and as target z ≡ Y i +1 − Y i ∆ t − ∆ Y i +1 + Y i = U i χ (0 , ) i = 0 , . . . , N t − Y = y , where Y i and U i denote resp. a time discretization of the state and the control.The optimal control is determined by a Gradient Descent method, with constantstepsize. The optimal state is depicted in fig. 4. t Figure 4. graph of the function t −→ k y T ( t ) k L ∞ (Ω) (in blue) and k y k L ∞ (Ω) (in red), where y T denotes an optimal state, whereas y stands for an optimal steady state. Conclusions and open problems
In this manuscript we have obtained some global turnpike results for an optimalcontrol problem governed by a nonlinear state equation. For any data and smalltargets, we have shown that the exponential turnpike property holds (Theorem 0.2).For arbitrary targets, we have proved the convergence of averages (Theorem 0.3),under the added assumption of controlling everywhere. One of the main toolsemployed for our analysis is an L ∞ bound of the norm of the optima, uniform inthe time horizon (Lemma 1.1). Numerical simulation have been performed, whichconfirms the theoretical results.We present now an interesting open problem in the field.In Theorem 0.3 we have proved the convergence of averages for large targets,in the context of control everywhere. An interesting challenge is to prove theexponential turnpike property, even if the control is local (namely ω ( Ω). Thechallenge is to prove the following conjecture.
Conjecture 4.1.
Consider the control problem (2)-(1). Take any initial datum y ∈ L ∞ (Ω) and any target z ∈ L ∞ ( ω ). Let u T be a minimizer of (1). Thereexists an optimal pair ( u, y ) for (5)-(4) such that k u T ( t ) − u k L ∞ (Ω) + k y T ( t ) − y k L ∞ (Ω) ≤ K [exp ( − µt ) + exp ( − µ ( T − t ))] , ∀ t ∈ [0 , T ] , (55)the constants K and µ > T .In [30] special large targets z are constructed, such that the optimal control forthe steady problem (5)-(4) is not unique. For those targets, a question arises: ifthe turnpike property is satisfied, which minimizer for (5)-(4) attracts the optimalsolutions to (2)-(1)?Note that, in the context of internal control, the counterexample to uniqueness in[30] is valid in case of local control ω ( Ω.Generally speaking a further investigation is required for the linearized optimalitysystem determined in [33, subsection 3.1]. We introduce the problem. As in (3),consider the optimality system for (2)-(1) y Tt − ∆ y T + f ( y T ) = − q T χ ω in (0 , T ) × Ω y T = 0 on (0 , T ) × ∂ Ω y T (0 , x ) = y ( x ) in Ω − q Tt − ∆ q T + f ′ ( y T ) q T = β ( y T − z ) χ ω in (0 , T ) × Ω q T = 0 on (0 , T ) × ∂ Ω q T ( T, x ) = 0 in Ω . (56)Pick any optimal pair ( u, y ) for (5)-(4). By the first order optimality conditions,the steady optimal control reads as u = − qχ ω , with − ∆ y + f ( y ) = − qχ ω in Ω y = 0 on ∂ Ω − ∆ q + f ′ ( y ) q = β ( y − z ) χ ω in Ω q = 0 on ∂ Ω . (57) HE TURNPIKE PROPERTY IN SEMILINEAR CONTROL 19
As in [33], we introduce the perturbation variables η T := y T − y and ϕ T := q T − q (58)and we write down the linearized optimality system around ( u, y ) η Tt − ∆ η T + f ′ ( y ) η T = − ϕ T χ ω in (0 , T ) × Ω η T = 0 on (0 , T ) × ∂ Ω η T (0 , x ) = y ( x ) − y ( x ) in Ω − ϕ Tt − ∆ ϕ T + f ′ ( y ) ϕ T = ( βχ ω − f ′′ ( y ) q ) η T in (0 , T ) × Ω ϕ T = 0 on (0 , T ) × ∂ Ω ϕ T ( T, x ) = − q ( x ) in Ω . (59)As pointed out in [33, Theorem 1 in subsection 3.1], a key point is to check thevalidity of the turnpike property for the linearized optimality system (59). This iscomplicated because of the term βχ ω − f ′′ ( y ) q , whose sign is unknown for generallarge targets. Furthermore, in case of nonuniqueness of steady optimum, it wouldbe interesting to compute the spectrum of the linearized system around any steadyoptima to check if among them one is a better attractor. Appendix A. Parabolic regularity results
One of the key tool to carry on the proof of Lemma 1.1 is the following regularityresult.
Lemma A.1.
Let Ω be a bounded open set of R n , n ∈ { , , } , with C boundary.Let c ∈ L ∞ ((0 , T ) × Ω) be nonnegative. Let y ∈ L ∞ (Ω) be an initial datum andlet h ∈ L ∞ ((0 , T ) × Ω) be a source term. Let y be the solution to y t − ∆ y + cy = h in (0 , T ) × Ω y = 0 on (0 , T ) × ∂ Ω y (0 , x ) = y ( x ) in Ω . Then, y ∈ L ((0 , T ); L ∞ (Ω)) and we have k y k L ((0 ,T );L ∞ (Ω)) ≤ K (cid:2) k y k L ∞ (Ω) + k h k L ((0 ,T ) × Ω) (cid:3) , (60) where K is independent of the potential c ≥ , the time horizon T and the initialdatum y .Proof of Lemma A.1. Step 1 Comparison
Let ψ be the solution to: ψ t − ∆ ψ = | h | in (0 , T ) × Ω ψ = 0 on (0 , T ) × ∂ Ω ψ (0 , x ) = | y | . in Ω (61)Since c ≥
0, a.e. in (0 , T ) × Ω, by a comparison argument, for each t ∈ [0 , T ]: | y ( t, x ) | ≤ ψ ( t, x ) , a.e. x ∈ Ω . (62)Now, since y and h are bounded, again by comparison principle applied to (61), ψ is bounded. Hence, by (62), y is bounded as well and Z T k y ( t ) k ∞ (Ω) dt ≤ Z T k ψ ( t ) k ∞ (Ω) dt. (63) Then, to conclude it suffices to show k ψ k L (0 ,T ;L ∞ (Ω)) ≤ K (cid:2) k y k L ∞ (Ω) + k h k L ((0 ,T ) × Ω) (cid:3) , the constant K being independent of T . Step 2
Splitting
Split ψ = ξ + χ , where ξ solves: ξ t − ∆ ξ = 0 in (0 , T ) × Ω ξ = 0 on (0 , T ) × ∂ Ω ξ (0 , x ) = | y | in Ω (64)while χ satisfies: χ t − ∆ χ = | h | in (0 , T ) × Ω χ = 0 on (0 , T ) × ∂ Ω χ (0 , x ) = 0 in Ω . (65)First of all, we prove an estimate like (60) for ξ . We start by employing maximumprinciple (see [34]) to (63), getting k ξ k L ∞ ((0 ,T ) × Ω) ≤ k y k L ∞ (Ω) . (66)Now, if T ≥
1, by the regularizing effect and the exponential stability of the heatequation, for any t ∈ [1 , T ], we have k ξ ( t ) k L ∞ (Ω) ≤ K k ξ ( t − k L (Ω) ≤ K exp ( − λ ( t − k y k L (Ω) , (67)the constant K depending only on the domain Ω. Then, by (66) and (67), for any T >
0, for every t ∈ [0 , T ], k ξ ( t ) k L ∞ (Ω) ≤ K min { , exp ( − λ ( t − } k y k L ∞ (Ω) , (68)with K = K (Ω).Now, we focus on (65). By parabolic regularity (see e.g. [14, Theorem 5 subsec-tion 7.1.3]), χ ∈ L (0 , T ; H (Ω)), with χ t ∈ L ((0 , T ) × Ω). Then, by multiplying(65) by − ∆ χ and integrating over [0 , T ] × Ω, we obtain12 k∇ χ ( T ) k (Ω) + Z T Z Ω | ∆ χ | dxdt ≤ k h k L ((0 ,T ) × Ω) k ∆ χ k L ((0 ,T ) × Ω) . By Young’s Inequality, Z T Z Ω | ∆ χ | dxdt ≤ k h k ((0 ,T ) × Ω) + 12 k ∆ χ k ((0 ,T ) × Ω) , which leads to Z T Z Ω | ∆ χ | dxdt ≤ k h k ((0 ,T ) × Ω) . Now, by [14, Theorem 6 subsection 5.6.3] and [14, Theorem 4 subsection 6.3.2], Z T k χ k ∞ (Ω) dt ≤ K Z T k χ k (Ω) dt ≤ K Z T Z Ω | ∆ χ | dxdt ≤ K k h k ((0 ,T ) × Ω) . (69)Finally, by (63), (68) and (69), Z T k y k ∞ (Ω) dt ≤ Z T k ξ k ∞ (Ω) dt +2 K Z T k χ k ∞ (Ω) dt ≤ K h k y k ∞ (Ω) + k h k ((0 ,T ) × Ω) i , as required. (cid:3) HE TURNPIKE PROPERTY IN SEMILINEAR CONTROL 21
The following regularity result is employed in the proof of Lemma 1.1.
Lemma A.2.
Let Ω ⊂ R n be a bounded open set, with ∂ Ω ∈ C ∞ . Let c ∈ L ∞ ((0 , T ) × Ω) be nonnegative. Let y ∈ L ∞ (Ω) an initial datum and h ∈ L ∞ ((0 , T ) × Ω) a source term. Let T ∈ (0 , T ) and set N := ⌊ T /T ⌋ . Let y be the solution to y t − ∆ y + cy = h in (0 , T ) × Ω y = 0 on (0 , T ) × ∂ Ω y (0 , x ) = y ( x ) in Ω . Then, y ∈ L ∞ ((0 , T ) × Ω) and we have k y k L ∞ ((0 ,T ) × Ω) ≤ K (cid:20) k y k L ∞ (Ω) + max i =1 ,...,N k h k L ((( i − T ,iT );L ∞ (Ω)) + k h k L ( NT ,T ;L ∞ (Ω)) (cid:21) , (70) where K is independent of the potential c ≥ and the time horizon T .Proof of Lemma A.2. Step 1 Comparison argument
Let ψ be the solution to: ψ t − ∆ ψ = | h | in (0 , T ) × Ω ψ = 0 on (0 , T ) × ∂ Ω ψ (0 , x ) = | y | . in Ω (71)Since c ≥
0, a.e. in (0 , T ) × Ω, by a comparison argument, for each t ∈ [0 , T ]: | y ( t, x ) | ≤ ψ ( t, x ) , a.e. x ∈ Ω . (72)Now, since y and h are bounded, again by comparison principle applied to (71), ψ is bounded. Hence, by (72), y is bounded as well and k y k L ∞ ((0 ,T ) × Ω) ≤ k ψ k L ∞ ((0 ,T ) × Ω) . (73)Then, to conclude it suffices to show k ψ k L ∞ ((0 ,T ) × Ω) ≤ K (cid:20) k y k L ∞ (Ω) + max i =1 ,...,N k h k L ((( i − T ,iT );L ∞ (Ω)) + k h k L ( NT ,T ;L ∞ (Ω)) (cid:21) , the constant K being independent of T . Step 2
Conclusion
Let { S ( t ) } t ∈ R + be the heat semigroup on Ω, with zero Dirichlet boundary condi-tions. Fix ε ∈ (0 , T ). By the regularizing effect of the heat equation (see, e.g. [6,Theorem 10.1, section 10.1]), for any t ≥ ε , k S ( t ) y k L ∞ (Ω) ≤ K exp( − µ ( t − ε )) k y k L (Ω) ≤ K exp( − µ ( t − ε )) k y k L ∞ (Ω) . For t ∈ [0 , ε ], by comparison principle, we have k S ( t ) y k L ∞ (Ω) ≤ K k y k L ∞ (Ω) ≤ K exp( − µ ( t − ε )) k y k L ∞ (Ω) , being exp( − µ ( t − ε )) ≥
1. Hence, for any t ≥ k S ( t ) y k L ∞ (Ω) ≤ K exp( − µ ( t − ε )) k y k L ∞ (Ω) . (74)Then, by the Duhamel formula, for any t ∈ [0 , T ], we have ψ ( t ) = S ( t ) ( | y | ) + Z t S ( t − s ) ( | h ( s ) | ) ds. (75) Now, by (74), for any t ≥ k S ( t ) ( | y | ) k L ∞ (Ω) ≤ K exp ( − µ ( t − ε )) k y k L ∞ (Ω) . (76)Besides, by applying (74) to the integral term η ( t ) := R t S ( t − s ) ( | h ( s ) | ) ds in (75),we obtain k η ( t ) k L ∞ ≤ Z t k S ( t − s ) ( | h ( s ) | ) k L ∞ ds ≤ K Z t exp( − µ ( t − s − ε )) k h ( s ) k L ∞ ds ≤ K ⌊ tT ⌋ X i =1 exp( − µ ( t − ε − iT )) Z iT ( i − T exp( − µ ( iT − s )) k h ( s ) k L ∞ ds + K Z t ( ⌊ tT ⌋− ) T exp( − µ ( t − s − ε )) k h ( s ) k L ∞ ds ≤ K ⌊ tT ⌋ X i =1 exp( − µ ( t − ε − iT )) "Z iT ( i − T exp( − µ ( iT − s )) ds "Z iT ( i − T k h ( s ) k ∞ ds + K "Z t ( ⌊ tT ⌋− ) T exp( − µ ( t − s − ε )) ds "Z t ( ⌊ tT ⌋− ) T k h ( s ) k ∞ ds ≤ K ⌊ tT ⌋ X i =1 exp( − µ ( t − ε − iT )) k h k L (( i − T ,iT ;L ∞ (Ω)) + k h k L ( ⌊ tT ⌋ ,t ;L ∞ (Ω)) i ≤ K " + ∞ X i =1 exp( − µ ( t − ε − iT )) max i =1 ,...,N k h k L ((( i − T ,iT );L ∞ (Ω)) + k h k L ( ⌊ tT ⌋ ,t ;L ∞ (Ω)) i ≤ K h k h k L (( i − T ,iT ;L ∞ (Ω)) + k h k L ( NT ,T ;L ∞ (Ω)) i . (77)Then, by (76) and appendix A, for each t ∈ [0 , T ] k ψ ( t ) k L ∞ (Ω) ≤ K exp ( − µ ( t − ε )) h k y k L ∞ (Ω) + k h k L (( i − T ,iT ;L ∞ (Ω)) + k h k L ( NT ,T ;L ∞ (Ω)) i as desired. (cid:3) Remark A.3.
Lemma A.1 can be applied to a bounded solution y to (2). Indeed,set c y ( t, x ) := f ( y ( t, x )) y ( t, x ) y ( t, x ) = 0 f ′ (0) y ( t, x ) = 0 . Since f is increasing and f (0) = 0, we have c y ≥
0. Hence, we are in position toapply Lemma A.1, with potential c y . HE TURNPIKE PROPERTY IN SEMILINEAR CONTROL 23 t t Ty O ty T Figure 5.
The idea of the proof of Lemma 1.1 is to use control-lability for (2) to show that optima for (2)-(1) cannot oscillate toomuch. Indeed, consider a the time interval [ t , t ]. By controlla-bility, we can link y T ( t , · ) and y T ( t , · ) by a controlled trajectory(in blue). By optimality, the optimum (in black) is bounded bythe constructed trajectory. Appendix B. Uniform bounds of the optima
As pointed out in [33, subsection 3.2], the norms of optimal controls and statescan be estimated in terms of the initial datum for (2) and the running target in anaveraged sense, using the inequality J T (cid:0) u T (cid:1) ≤ J T (0) , (78)where u T is any optimal control for the time-evolution problem. We have to ensurethat the bounds actually holds for any time, i.e. we need to show that optimalcontrols and states do not oscillate too much.The proof of Lemma 1.1 follows the scheme: • divide the interval [0 , T ] into subintervals of T -independent length; • estimate the magnitude of the optima in each subinterval by using control-lability (fig. 5).In order to carry out the proof of Lemma 1.1, we need some preliminary lemmas.We start by stating some results on the controllability of a dissipative semilinearheat equation.B.1. Controllability of dissipative semilinear heat equation.Lemma B.1.
Let y ∈ L ∞ (Ω) be an initial datum. Let ˆ y ∈ L ∞ ((0 , + ∞ ) × Ω) bea target trajectory, solution to (2) , with control ˆ u ∈ L ∞ ((0 , T ) × ω ) . Let R > .Suppose k y k L ∞ (Ω) ≤ R and k ˆ y k L ∞ ((0 , + ∞ ) × Ω) ≤ R . Then, there exists T R = T R (Ω , f, ω, R ) , such that for any T ≥ T R there exists u ∈ L ∞ ((0 , T ) × ω ) such thatthe solution y to the controlled equation (2) , with initial datum y and control u ,verifies the final condition y ( T, x ) = ˆ y ( T, x ) in Ω (79) and k u − ˆ u k L ∞ ((0 ,T ) × ω ) ≤ K k y − ˆ y (0) k L ∞ (Ω) , (80) where the constant K depends only on Ω , f , ω and R . The proof of the above lemma is classical (see, e.g. [16, 3]).In order to prove Lemma 1.1, we introduce an optimal control problem, withspecified terminal states. Let t < t . Let ˆ y be a target trajectory, boundedsolution to (2) in ( t , t ), i.e. ˆ y t − ∆ˆ y + f (ˆ y ) = ˆ uχ ω in ( t , t ) × Ωˆ y = 0 on ( t , t ) × ∂ Ωˆ y ( t , x ) = ˆ y ( x ) in Ω , (81)with initial datum ˆ y ∈ L ∞ (Ω) and control ˆ u ∈ L ∞ (( t , t ) × ω ).For any control u ∈ L (( t , t ) × ω ), the corresponding state y is the solution to: y t − ∆ y + f ( y ) = uχ ω in ( t , t ) × Ω y = 0 on ( t , t ) × ∂ Ω y ( t , x ) = ˆ y ( t , x ) in Ω . (82)We introduce the set of admissible controls U ad := (cid:8) u ∈ L (( t , t ) × ω ) | y ( t , · ) = ˆ y ( t , · ) (cid:9) . By definition, ˆ u ∈ U ad . Hence, U ad = ∅ . We consider the optimal control problemmin u ∈ U ad J t ,t ( u ) = 12 Z t t Z ω | u | dxdt + β Z t t Z ω | y − z | dxdt, (83)with running target z ∈ L ∞ ( ω ). By the direct methods in the calculus of variations,the functional J t ,t admits a global minimizer in the set of admissible controls U ad .We now bound the minimal value of the functional (83), showing that the mag-nitude of the control ˆ u in the time interval [ t , t − T R ] can be neglected whenestimating the cost of controllability. Namely, what matters is the norm of ˆ u in thefinal time interval [ t − T R , t ]. Lemma B.2.
Consider the optimal control problem (82) - (83) , with t − t ≥ T R .Then, min U ad J t ,t ≤ K h k ˆ y ( t , · ) k ∞ (Ω) + ( t − t ) k z k ∞ ( ω ) + k ˆ u k ∞ (( t − T R ,t ) × ω ) + k ˆ y ( t − T R , · ) k ∞ (Ω) i , (84) the constant K being independent of the time horizon t − t ≥ T R .Proof of Lemma B.2. Step 1 A quasi-optimal control
To get the desired bound, we introduce a quasi-optimal control u for (82)-(83),linking ˆ y ( t , · ) and y T ( t , · ). The control strategy is the following(1) employ null control for time t ∈ [0 , t − T R ];(2) match the final condition by control w , for t ∈ [ t − T R , t ]. HE TURNPIKE PROPERTY IN SEMILINEAR CONTROL 25
Let us denote by y the solution to the semilinear problem with null control y t − ∆ y + f (cid:0) y (cid:1) = 0 in ( t , t ) × Ω y = 0 on ( t , t ) × ∂ Ω y ( t , x ) = ˆ y ( t , x ) in Ω . (85)By Lemma B.1, there exists w ∈ L ∞ (( t − T R , t ) × ω ), steering (82) from y ( t , · )to ˆ y ( t , · ) in the time interval ( t − T R , t ), with estimate k w − ˆ u k L ∞ (( t − T R ,t ) × ω ) ≤ K (cid:13)(cid:13) y ( t − T R ) − ˆ y ( t − T R ) (cid:13)(cid:13) L ∞ (Ω) , (86)Then, set u := ( , t − T R ) w in ( t − T R , t ) . (87)By (86), we can bound the norm of the control, k u k L ∞ (( t ,t ) × ω ) ≤ K h(cid:13)(cid:13) y ( t − T R ) − ˆ y ( t − T R ) (cid:13)(cid:13) L ∞ (Ω) + k ˆ u k L ∞ (( t − T R ,t ) × ω ) i . (88) Step 2
Conclusion
Consider the control u introduced in (87) and let y be the solution to (82), withinitial datum y and control u . Then, we havemin U ad J t ,t ≤ J t ,t ( u )= 12 Z t t Z ω | u | dxdt + β Z t t Z ω | y − z | dxdt = 12 Z t t − T R Z ω | w | dxdt + β Z t t Z ω | y − z | dxdt ≤ Z t t − T R Z ω | w | dxdt + β Z t t Z ω | y | dxdt + β Z t t Z ω | z | dxdt ≤ Z t t − T R Z ω | w | dxdt + β Z t t Z ω | y | dxdt + K ( t − t ) k z k ∞ ( ω ) ≤ K h k w k ∞ (( t − T R ,t ) × ω ) + ( t − t ) k z k ∞ ( ω ) + β Z t − T R t (cid:13)(cid:13) y ( t, · ) (cid:13)(cid:13) (Ω) dt + k y k (( t − T R ,t ) × Ω) ≤ K h(cid:13)(cid:13) y ( t − T R , · ) − ˆ y ( t − T R , · ) (cid:13)(cid:13) ∞ (Ω) + k ˆ u k ∞ (( t − T R ,t ) × ω ) (89)+( t − t ) k z k ∞ ( ω ) + k ˆ y ( t , · ) k ∞ (Ω) i ≤ K h k ˆ y ( t , · ) k ∞ (Ω) + ( t − t ) k z k ∞ ( ω ) (90)+ k ˆ u k ∞ (( t − T R ,t ) × ω ) + k ˆ y ( t − T R , · ) k ∞ (Ω) i , (91)where in (89) and in (90) we have employed the dissipativity of (85). This concludesthe proof. (cid:3) B.2.
A mean value result for integrals.
In the following Lemma we estimatethe value of a function at some point, with the value of its integral.
Lemma B.3.
Let h ∈ L ( c, d ) ∩ C ( c, d ) , with −∞ < c < d < + ∞ . Assume h ≥ a.e. in ( c, d ) . Then,(1) there exists t c ∈ (cid:0) c, c + d − c (cid:1) , such that h ( t c ) ≤ d − c Z dc hdt ; (2) there exists t d ∈ (cid:0) d − d − c , d (cid:1) , such that h ( t d ) ≤ d − c Z dc hdt. Proof of Lemma B.3.
By contradiction, for any t ∈ (cid:0) c, c + d − c (cid:1) , h ( t ) > d − c R dc hds .Then, we have Z dc hdt ≥ Z c + d − c c hdt > Z c + d − c c " d − c Z dc hds dt = Z dc hds, so obtaining a contradiction. The proof of (2.) is similar. (cid:3) B.3.
Proof of Lemma 1.1.
We are now in position to prove Lemma 1.1.
Proof of Lemma 1.1. Step 1
Estimates on subintervals
Let T R be given by Lemma B.1.The case T ≤ T R can be addressed by employing the inequality J T (cid:0) u T (cid:1) ≤ J T (0) and bootstrapping in the optimality system (3), as in [33, subsection 3.2].We address now the case T > T R .Set N T := ⌊ T T R ⌋ . Arbitrarily fix θ >
0, a degree of freedom, to be made preciselater. Consider the indexes i ∈ { , . . . , N T } , such that Z i T R ( i − T R h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i . (92)Set I T := (cid:26) i ∈ { , . . . , N T } (cid:12)(cid:12)(cid:12)(cid:12) the estimate (92) is not verified (cid:27) . (93)On the one hand, for any i ∈ { , . . . , N T } \ I T , by definition of I T Z i T R ( i − T R h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i . On the other hand, for every i ∈ I T , we seek to prove the existence of a constant K θ = K θ (Ω , f, R, θ ), possibly larger than θ , such that Z i T R ( i − T R h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ K θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i . (94)We start by considering the union of time intervals, where (92) is not verified W T := [ i ∈ I T [( i − T R , i T R ] . HE TURNPIKE PROPERTY IN SEMILINEAR CONTROL 27
The above set is made of a finite union of disjoint closed intervals, namely thereexists a natural M and { ( a j , b j ) } j =1 ,...,M , such that b j < a j +1 , j = 1 , . . . , M − W T = [ i ∈ I T [( i − T R , i T R ] = [ j =1 ,...,M [ a j , b j ] . For any j = 1 , . . . , M , set C j := { i ∈ I T | [( i − T R , i T R ] ⊆ [ a j , b j ] } . (95)We are going to prove (94), studying the optima in a neighbourhood of [ a j , b j ], for j = 1 , . . . , M . Three different cases may occur: • Case 1. a = 0 and b < T R N T , namely the left end of the interval [ a , b ]coincides with t = 0, while the right end is far from t = T ; • Case 2. a j > b j < T R N T , i.e. the left end of the interval [ a j , b j ] isfar from t = 0 and the right end is far from t = T ; • Case 3. a j > b j = 3 T R N T , i.e. the left end of the interval [ a j , b j ] isfar from t = 0, while the right end is close to t = T . Case 1. a = 0 and b < T R N T .Since b < T R N T , we have [ b , b + 3 T R N T ] ⊆ [0 , T ] \ W T . Hence, by (93), Z b +3 T R b h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i . Set c := b , d := b + 3 T R and h ( t ) := k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) . By LemmaB.3, there exist t c and t d , b < t c < b + T R and b + 2 T R < t d < b + 3 T R , (96)such that k q T ( t c ) k ∞ (Ω) + k y T ( t c ) k ∞ (Ω) ≤ T R Z b +3 T R b h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ θT R h k y k ∞ (Ω) + k z k ∞ ( ω ) i and k q T ( t d ) k ∞ (Ω) + k y T ( t d ) k ∞ (Ω) ≤ T R Z b +3 T R b h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ θT R h k y k ∞ (Ω) + k z k ∞ ( ω ) i . Parabolic regularity in the optimality system (3) in the interval [ t c , t d ] gives (cid:13)(cid:13) y T (cid:13)(cid:13) ∞ (( t c ,t d ) × Ω) + (cid:13)(cid:13) q T (cid:13)(cid:13) ∞ (( t c ,t d ) × Ω) ≤ K n k q T ( t d ) k ∞ (Ω) + k y T ( t c ) k ∞ (Ω) + k z k ∞ ( ω ) + Z b +3 T R b h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ) ≤ K θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i . (97) where the constant K θ is independent of the time horizon T , but it depends on θ .At this point, we want to apply Lemma B.2. To this purpose, we set up a controlproblem like (82)-(83) with specified final stateˆ y := y T t := 0 t := t d . By (84) and (97),min U ad J t ,t ≤ K h k y k ∞ (Ω) + t d k z k ∞ ( ω ) + k u T k ∞ (( t d − T R ,t d ) × ω ) + k y T ( t d − T R ) k ∞ (Ω) i ≤ K θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i + γt d k z k ∞ ( ω ) , (98)where K θ = K θ (Ω , f, R, θ ) and γ = γ (Ω , f, R ). In our case the target trajectory for(82)-(83) is the state y T associated to an optimal control u T for (2)-(1). Then, bydefinition of (82)-(83), J t ,t (cid:0) u T (cid:1) ≤ J t ,t ( u ) , ∀ u ∈ U ad . Hence, by (98), J t ,t (cid:0) u T (cid:1) ≤ min U ad J t ,t ≤ K θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i + γt d k z k ∞ ( ω ) . (99)By definition of I T (93) and C (95), we have Z b h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≥ X i ∈ C θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i = θb T R h k y k ∞ (Ω) + k z k ∞ ( ω ) i > θ ( t d − T R )3 T R h k y k ∞ (Ω) + k z k ∞ ( ω ) i , where in the last inequality we have used (96), which yields b > t d − T R . By theabove inequality, Lemma A.1, (97) and (99), θ ( t d − T R )6 T R h k y k ∞ (Ω) + k z k ∞ ( ω ) i + 12 Z b h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ Z b h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ K h J t ,t (cid:0) u T (cid:1) + k y k ∞ (Ω) + k q T ( t d ) k ∞ (Ω) i ≤ K θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i + γt d k z k ∞ ( ω ) , HE TURNPIKE PROPERTY IN SEMILINEAR CONTROL 29 whence Z b h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ K θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i +2 (cid:18) γt d − θ ( t d − T R )6 T R (cid:19) k z k ∞ ( ω ) ≤ K θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i +2 t d (cid:18) γ − θ T R (cid:19) k z k ∞ ( ω ) . If θ is large enough, we have γ − θ T R <
0. Hence, choosing θ large enough, weobtain the estimate Z b h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ K θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i . Case 2. a j > b j < T R N T .Since a j > b j < T R N T , we have Z a j a j − T R h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i (100)and Z b j +3 T R b j h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i . (101)In Case 2, we apply Lemma B.3: • in the interval [ a j − T R , a j ]; • in the interval [ b j , b j + 3 T R ].We start by applying Lemma B.3 in [ a j − T R , a j ]. To this end, set c := a j − T R , d := a j and h ( t ) := k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) . By Lemma B.3, there exist t a,c and t a,d , a j − T R < t a,c < a j − T R and a j − T R < t a,d < a j , (102)such that k q T ( t a,c ) k ∞ (Ω) + k y T ( t a,c ) k ∞ (Ω) ≤ T R Z a j a j − T R h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ θT R h k y k ∞ (Ω) + k z k ∞ ( ω ) i and k q T ( t a,d ) k ∞ (Ω) + k y T ( t a,d ) k ∞ (Ω) ≤ T R Z a j a j − T R h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ θT R h k y k ∞ (Ω) + k z k ∞ ( ω ) i . By parabolic regularity in the optimality system (3) in the interval [ t a,c , t a,d ], wehave (cid:13)(cid:13) y T (cid:13)(cid:13) ∞ (( t a,c ,t a,d ) × Ω) + (cid:13)(cid:13) q T (cid:13)(cid:13) ∞ (( t a,c ,t a,d ) × Ω) ≤ K n k q T ( t a,d ) k ∞ (Ω) + k y T ( t a,c ) k ∞ (Ω) + k z k ∞ ( ω ) + Z a j a j − T R h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ) ≤ K θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i . (103)where the constant K θ is independent of the time horizon T , but it depends on θ .We apply Lemma B.3 in [ b j , b j + 3 T R ]. To this extent, set c := b j , d := b j + 3 T R and h ( t ) := k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) . By Lemma B.3, there exist t b,c and t b,d ,0 < t b,c < b j + T R and b j + 2 T R < t b,d < b j + 3 T R , (104)such that k q T ( t b,c ) k ∞ (Ω) + k y T ( t b,c ) k ∞ (Ω) ≤ T R Z b j +3 T R b j h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ θT R h k y k ∞ (Ω) + k z k ∞ ( ω ) i and k q T ( t b,d ) k ∞ (Ω) + k y T ( t b,d ) k ∞ (Ω) ≤ T R Z b j +3 T R b j h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ θT R h k y k ∞ (Ω) + k z k ∞ ( ω ) i . By parabolic regularity in the optimality system (3) in the interval [ t b,c , t b,d ], wehave (cid:13)(cid:13) y T (cid:13)(cid:13) ∞ (( t b,c ,t b,d ) × Ω) + (cid:13)(cid:13) q T (cid:13)(cid:13) ∞ (( t b,c ,t b,d ) × Ω) ≤ K n k q T ( t b,d ) k ∞ (Ω) + k y T ( t b,c ) k ∞ (Ω) + k z k ∞ ( ω ) + Z b j +3 T R b j k q T ( t ) k ∞ (Ω) dt + Z b j +3 T R b j k y T ( t ) k ∞ (Ω) dt ) ≤ K θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i . (105)where the constant K θ is independent of the time horizon T , but it depends on θ .At this point, we want to apply Lemma B.2. To this purpose, we set up a controlproblem like (82)-(83) with specified final stateˆ y := y T t := t a,c t := t b,d . HE TURNPIKE PROPERTY IN SEMILINEAR CONTROL 31
By (84), (103) and (105),min U ad J t ,t ≤ K h k y T ( t a,c ) k ∞ (Ω) + ( t b,d − t a,c ) k z k ∞ ( ω ) , + k u T k ∞ (( t b,d − T R ,t b,d ) × ω ) + k y T ( t b,d − T R ) k ∞ (Ω) i ≤ K θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i + γ ( t b,d − t a,c ) k z k ∞ ( ω ) , (106)where K θ = K θ (Ω , f, R, θ ) and γ = γ (Ω , f, R ). In our case the target trajectory for(82)-(83) is the state y T associated to an optimal control u T for (2)-(1). Then, bydefinition of (82)-(83), J t ,t (cid:0) u T (cid:1) ≤ J t ,t ( u ) , ∀ u ∈ U ad . Hence, by (106), J t ,t (cid:0) u T (cid:1) ≤ min U ad J t ,t ≤ K θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i + γ ( t b,d − t a,c ) k z k ∞ ( ω ) . By definition of I T (93) and C (95), we have Z b j a j h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≥ X i ∈ C j θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i = θ ( b j − a j )3 T R h k y k ∞ (Ω) + k z k ∞ ( ω ) i > θ ( t b,d − t a,c − T R )3 T R h k y k ∞ (Ω) + k z k ∞ ( ω ) i , (107)where in the last inequality we have used (102) and (104) to get b j − a j > t b,d − t a,c − T R . By the above inequality, Lemma A.1 and (107), θ ( t b,d − t a,c − T R )6 T R h k y k ∞ (Ω) + k z k ∞ ( ω ) i + 12 Z b j a j h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ Z b j a j h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ K h J t ,t (cid:0) u T (cid:1) + k y k ∞ (Ω) i ≤ K θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i + γ ( t b,d − t a,c ) k z k ∞ ( ω ) , whence Z b j a j h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ K θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i +2 (cid:18) γ ( t b,d − t a,c ) − θ ( t b,d − t a,c − T R )6 T R (cid:19) k z k ∞ ( ω ) ≤ K θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i +2( t b,d − t a,c ) (cid:18) γ − θ T R (cid:19) k z k ∞ ( ω ) . If θ is large enough, we have γ − θ T R <
0. Hence, choosing θ large enough, weobtain the estimate Z b j a j h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ K θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i . Case 3. a j > b j = 3 T R N T .We now work in case (92) is not satisfied in [ a j , b j ], with b j = 3 T R N T . Weprovide an estimate in the final interval [ a j , T ]. As we shall see, in this case, wewill not employ the exact controllability of (2). We shall rather use the stability ofthe uncontrolled equation.Since a j >
0, we have Z a j a j − T R h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i . (108)We apply Lemma B.3 in [ a j − T R , a j ]. To this end, set c := a j − T R , d := a j and h ( t ) := k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) . By Lemma B.3, there exist t c , a j − T R < t c < a j − T R (109)such that k q T ( t c ) k ∞ (Ω) + k y T ( t c ) k ∞ (Ω) ≤ T R Z a j a j − T R h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ θT R h k y k ∞ (Ω) + k z k ∞ ( ω ) i . (110)We introduce the control u ∗ := ( u T in (0 , t c )0 in ( t c , T )Let y be the solution to (2), with initial datum y and control u and y ∗ be thesolution to (2), with initial datum y and control u ∗ . By definition of minimizer,we have J T (cid:0) u T (cid:1) ≤ J T ( u ∗ ) ≤ Z T Z ω | u ∗ | dxdt + β Z T Z ω | y ∗ − z | dxdt = 12 Z t c Z ω (cid:12)(cid:12) u T (cid:12)(cid:12) dxdt + β Z t c Z ω (cid:12)(cid:12) y T − z (cid:12)(cid:12) dxdt + β Z Tt c Z ω | y ∗ − z | dxdt, whence,12 Z Tt c Z ω (cid:12)(cid:12) u T (cid:12)(cid:12) dxdt + β Z Tt c Z ω (cid:12)(cid:12) y T − z (cid:12)(cid:12) dxdt ≤ β Z Tt c Z ω | y ∗ − z | dxdt ≤ K h k y ( t c ) k ∞ (Ω) + ( T − t c ) k z k ∞ ( ω ) i ≤ K θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i + γ ( T − t c ) k z k ∞ ( ω ) , HE TURNPIKE PROPERTY IN SEMILINEAR CONTROL 33 where we have used (110) and K θ = K θ (Ω , f, R, θ ) and γ = γ (Ω , f, R ).Now, on the one hand, by Lemma A.1 applied to the state and the adjointequation in (3), we have Z Tt c h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ K θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i + γ ( T − t c ) k z k ∞ ( ω ) . (111)On the other hand, by (109), − a j > − t c − T R and, since b j = 3 T R N T , b j ≥ T − T R . Hence, b j − a j > T − t c − T R . Then, by (93), Z Ta j h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≥ Z b j a j h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≥ X i ∈ C j θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i = θ ( b j − a j )3 T R h k y k ∞ (Ω) + k z k ∞ ( ω ) i > θ ( T − t c − T R )3 T R h k y k ∞ (Ω) + k z k ∞ ( ω ) i . By the above inequality and Lemma A.1 and (111), θ ( T − t c − T R )6 T R h k y k ∞ (Ω) + k z k ∞ ( ω ) i + 12 Z Ta j h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ Z Ta j h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ K θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i + γ ( T − t c ) k z k ∞ ( ω ) , whence Z Ta j h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ K θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i +2 (cid:18) γ ( T − t c ) − θ ( T − t c − T R )6 T R (cid:19) k z k ∞ ( ω ) ≤ K θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i +2( T − t c ) (cid:18) γ − θ T R (cid:19) k z k ∞ ( ω ) . If θ is large enough, we have γ − θ T R <
0. Hence, choosing θ large enough, weobtain the estimate Z Ta j h k q T ( t ) k ∞ (Ω) + k y T ( t ) k ∞ (Ω) i dt ≤ K θ h k y k ∞ (Ω) + k z k ∞ ( ω ) i . Step 2
Conclusion
The proof is concluded, with an application of Lemma A.2 to the state and theadjoint equation in (3). (cid:3)
Appendix C. Convergence of averages
This section is devoted to the proof of Lemma 2.1.
Proof of lemma 2.1.
Let u ∈ L ∞ (Ω) be an optimal control for (5)-(4) and let y bethe corresponding solution to (5) with control u . Following step 1 of the proof ofLemma 1.4, we obtain u ∈ C ( ω ) and k u k L ∞ ( ω ) ≤ K k z k L ∞ ( ω ) . (112) Step 1
Proof of (cid:12)(cid:12)(cid:12)(cid:12) J T ( u ) − T inf L (Ω) J s (cid:12)(cid:12)(cid:12)(cid:12) ≤ K, with K independent of T Let ˆ y be the solution to ˆ y t − ∆ˆ y + f (ˆ y ) = uχ ω in (0 , T ) × Ωˆ y = 0 on (0 , T ) × ∂ Ωˆ y (0 , x ) = y ( x ) in Ω . (113)Set η := ˆ y − y solution to η t − ∆ η + f (ˆ y ) − f ( y ) = 0 in (0 , T ) × Ω η = 0 on (0 , T ) × ∂ Ω η (0 , x ) = y ( x ) − y ( x ) in Ω . (114)By multiplying (114) by η , since f is increasing, for any t ∈ [0 , T ] we have k ˆ y ( t, · ) − y k L (Ω) ≤ exp( − λ t ) k y − y k L (Ω) , (115)where λ is the first eigenvalue of − ∆ : H (Ω) −→ H − (Ω).At this point, let us take the difference | J T ( u ) − T inf L (Ω) J s | = 12 (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)Z T Z ω h | ˆ y − z | − | y − z | i dxdt (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ Z T Z ω | ˆ y − y | dxdt + Z T Z ω | y − z | | ˆ y − y | dxdt ≤ K k y − y k (Ω) + K k y − y k L (Ω) ≤ K, (116)(117)where in appendix C we have used (115) and (112) and the constant K is indepen-dent of the time horizon T . Step 2
Conclusion
By the above reasoning, we haveinf L ((0 ,T ) × ω ) J T ≤ J T ( u )= T inf L (Ω) J s + J T ( u ) − T inf L (Ω) J s ≤ T inf L (Ω) J s + K. This finishes the proof. (cid:3)
HE TURNPIKE PROPERTY IN SEMILINEAR CONTROL 35
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