Robust Stackelberg controllability for a heat equation Recent Advances in PDEs: Analysis, Numerics and Control

Transcription

1 Robust Stackelberg controllability for a heat equation Recent Advances in PDEs: Analysis, Numerics and Control Luz de Teresa Víctor Hernández Santamaría Supported by IN of DGAPA-UNAM

2 Robust Control

3 Robust control (Bewley, Temam, Ziane (2000)) Introduced by Temam et all in the context of fluid mechanics. 2

4 Robust control (Bewley, Temam, Ziane (2000)) Introduced by Temam et all in the context of fluid mechanics. a differential game between an engineer seeking the best control which stabilizes the flow perturbation with limited control effort and simultaneously, nature seeking maximally malevolent disturbance which destabilizes the flow perturbation with limited disturbance magnitude 2

5 Robust control (Bewley, Temam, Ziane (2000)) Introduced by Temam et all in the context of fluid mechanics. a differential game between an engineer seeking the best control which stabilizes the flow perturbation with limited control effort and simultaneously, nature seeking maximally malevolent disturbance which destabilizes the flow perturbation with limited disturbance magnitude Optimal control problem: Find a saddle point. Minimize with respect to a control, maximize with respect to the disturbance. 2

6 Saddle point

7 Robust control (Bewley, Temam, Ziane (2000)) y t y + (y )y + p = vχ O + ψ in Q, y = 0 in Q, y = 0 on Σ, y(, 0) = y 0 ( ) in Ω, (1) 4

8 Robust control (Bewley, Temam, Ziane (2000)) y t y + (y )y + p = vχ O + ψ in Q, y = 0 in Q, y = 0 on Σ, y(, 0) = y 0 ( ) in Ω, (1) Optimal control: find a saddle point for J opt (ψ, v) := µ y y d 2 dxdt y(t ) 2 L 2 (Ω) N O d (0,T ( ) + 1 ) l 2 v 2 dxdt γ 2 ψ 2 dxdt 2 Q O (0,T ) 4

9 Robust control (Bewley, Temam, Ziane (2000)) y t y + (y )y + p = vχ O + ψ in Q, y = 0 in Q, y = 0 on Σ, y(, 0) = y 0 ( ) in Ω, (1) That is: The control vχ O L 2 (Q) N, the disturbance ψ L 2 (Q) N and the solution solve the robust control problem when when J opt (ψ, v) J opt (ψ, v) J opt (ψ, v), (ψ, v) L 2 (Q) N N. 4

10 Results Bewley, Temam, Ziane (2000) Linearized Navier-Stokes and Navier Stokes. γ, l large enough. Restriction on T for 3-D case. 5

11 Results Bewley, Temam, Ziane (2000) Linearized Navier-Stokes and Navier Stokes. γ, l large enough. Restriction on T for 3-D case. Other results for fluids: Medjo (alone or with) Tebou. (2004, 2005, 2008, 2011, 2016) 5

12 Results Bewley, Temam, Ziane (2000) Linearized Navier-Stokes and Navier Stokes. γ, l large enough. Restriction on T for 3-D case. Other results for fluids: Medjo (alone or with) Tebou. (2004, 2005, 2008, 2011, 2016) Some numerical algorithms for other PDE s. 5

13 Results Bewley, Temam, Ziane (2000) Linearized Navier-Stokes and Navier Stokes. γ, l large enough. Restriction on T for 3-D case. Other results for fluids: Medjo (alone or with) Tebou. (2004, 2005, 2008, 2011, 2016) Some numerical algorithms for other PDE s. Most results in the framework of ODE s. 5

14 Hierarchic control

15 Hierarchic control Concept from game theory (Gabriel Cramer 1728, Daniel Bernouilli 1738). 7

16 Hierarchic control Concept from game theory (Gabriel Cramer 1728, Daniel Bernouilli 1738). The leader (first player) does a movement. The follower (second player) reacts trying to win or optimize the response to the leader movement. 7

17 Hierarchic control Concept from game theory (Gabriel Cramer 1728, Daniel Bernouilli 1738). The leader (first player) does a movement. The follower (second player) reacts trying to win or optimize the response to the leader movement. Historical papers due to J. von Neumann and O. Morgenstern (1943) and Nash (1950). [What is now known as Nash equilibria is due to Cournot (1838)]. 7

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20 Several concepts. 10

21 Several concepts. Two players (controls) one leader, one follower. Called Stackelberg strategy. 10

22 Several concepts. Two players (controls) one leader, one follower. Called Stackelberg strategy. Several players: Nash equilibria, Pareto strategy etc... 10

23 Several concepts. Two players (controls) one leader, one follower. Called Stackelberg strategy. Several players: Nash equilibria, Pareto strategy etc... We concentrate on Stackelberg strategy. 10

24 Hierarchic control [Lions94] We consider the heat equation y t y = hχ ω + vχ O in Q = Ω (0, T ) y = 0 on Σ y(x, 0) = y 0 (x) in Ω (Cal) h leader control v follower control Objectives: optimal control: y y d in O d (0, T ) y y d 2 + β 2 v 2, β > 0. 1 min v 2 O d (0,T ) O (0,T ) controllability find h such that y(t ) = 0. 11

25 How to solve the problem? Step 1: Fix h and obtain 1 min v 2 O d (0,T ) y y d 2 + β 2 O (0,T ) v 2. The functional is continuous, strictly convex and coercive so there is a unique minimizer characterized by v = 1 β pχ O p t p = (y y d )χ Od in Q p(x, T ) = 0 in Ω, p = 0 on Σ. (Seg) 12

26 How to solve the problem? (Cont.) Step 2: y t y = hχ ω 1 β pχ O in Q p t p = (y y d )χ Od in Q y(x, 0) = y 0 (x), p(x, T ) = 0, y = p = 0 on Σ. In [Lions94] is proved that (L-S) is aproximatelly controllable y(t ) L 2 (Ω) ε. Araruna et al. (2015) proved when ω O d and ρ 2 y d 2 < for ρ as t T, Q then (L-S) is null controllable. (L-S) 13

27 How to solve the problem? (Cont.) Prove the observability inequality ϕ(0) 2 L 2 (Ω) + ρ 2 θ 2 C Q ω (0,T ) ϕ 2 for the adjoint system ϕ t ϕ = θχ Od in Q θ t θ = 1 β ϕχ O in Q ϕ(x, T ) = ϕ T, θ(x, 0) = 0, ϕ = θ = 0 on Σ. (2) Step 3: We define h = ˆϕχ ω where ˆϕ is the solution to (2) with 1 J( ˆϕ T ) = min ϕ T L 2 2 ϕ 2 + y 0 ϕ(0) y d θ (Ω) ω (0,T ) Ω O d (0,T ) 14

28 Works related with hierarchic control Heat and wave equations: Lions, Ocean circulation models: Díaz-Lions 1997, Díaz (2002) Stokes: Guillen-González et al., Moving Domains (wave equation): IP de Jesus, 2014, Moving domains (Parabolic equations) Approximate control: Límaco, J.; Clark, H. R.; Medeiros, L. A., Linear and semilinear parabolic equations: Araruna, Fernández-Cara, Santos, 2015 Control to trajectories. Micropolar fluids: Araruna, F. D.; de Menezes, S. D. B.; Rojas-Medar, M. A approximate controllability, control in both equations. Coupled parabolic equations: Hernández-Santamaría; DeT, Pozniak (2016) 15

29 Stackelberg strategy for robust control

30 New problem: Stackelberg strategy for robust control We consider y t y = hχ ω + vχ O + ψ in Q = Ω (0, T ) y = 0 on Σ y(x, 0) = y 0 (x) in Ω h leader control v follower control r ψ perturbation Objectifs: robust control : y y d in O d (0, T ) (Cal) J r (v, ψ) = 1 2 y y d 2 L 2 (O d (0,T )) + l2 2 vχ O 2 L 2 (Q) γ2 2 ψ 2 L 2 (Q). Find ( v, ψ) such that J r ( v, ψ) = min v max ψ J r = max ψ min J r, v (saddle point) controllability: find h such that y(t ) = 0. 17

31 Stackelberg strategy for robust control Step 1: Fix h L 2 (ω (0, T )). Find the saddle point for J r (v, ψ) = 1 2 y y d 2 L 2 (O d (0,T )) + l2 2 vχ O 2 L 2 (Q) γ2 2 ψ 2 L 2 (Q). Existence, uniqueness, characterization 18

32 Stackelberg strategy for robust control Step 1: Fix h L 2 (ω (0, T )). Find the saddle point for J r (v, ψ) = 1 2 y y d 2 L 2 (O d (0,T )) + l2 2 vχ O 2 L 2 (Q) γ2 2 ψ 2 L 2 (Q). Existence, uniqueness, characterization Step 2: show the null controlability for y t y = hχ ω + v(h)χ O + ψ(h) in Q = Ω (0, T ) y = 0 on Σ y(x, 0) = y 0 (x) in Ω 18

33 Stackelberg strategy for robust control Step 1: Fix h L 2 (ω (0, T )). Find the saddle point for J r (v, ψ) = 1 2 y y d 2 L 2 (O d (0,T )) + l2 2 vχ O 2 L 2 (Q) γ2 2 ψ 2 L 2 (Q). Existence, uniqueness, characterization Step 2: show the null controlability for y t y = hχ ω + v(h)χ O + ψ(h) in Q = Ω (0, T ) y = 0 on Σ y(x, 0) = y 0 (x) in Ω Step 3: Obtain the control h. 18

34 Main result Theorem Assume that ω O d. γ, l are large enough. ρ such that O d (0,T ) ρ2 y d 2 < +, then: y 0 L 2 (Ω), null control h &! saddle point ( v, ψ). Idea of the proof: 1. γ large enough h L 2 (ω (0, T )),! saddle point v = 1 l 2 pχ O, ψ = 1 γ 2 p p t p = α(y y d )χ Od in Q p(x, T ) = 0, p = 0 on Σ. 19

35 Main result (Cont.) 2. Null controllability y t y = hχ ω 1 l pχ 2 O + 1 γ p in Q 2 p t p = (y y d )χ Od in Q y(x, 0) = y 0 (x), p(x, T ) = 0, y = p = 0 on Σ. 20

36 Main result (Cont.) 2. Null controllability y t y = hχ ω 1 l pχ 2 O + 1 γ p in Q 2 p t p = (y y d )χ Od in Q y(x, 0) = y 0 (x), p(x, T ) = 0, y = p = 0 on Σ. Observability ϕ t ϕ = θχ Od in Q θ t θ = 1 l ϕχ 2 O + 1 γ ϕ in Q 2 ϕ(x, T ) = f, θ(x, 0) = 0, ϕ = θ = 0 on Σ. We want to prove ϕ(0) 2 L 2 (Ω) + ρ 2 θ 2 C ϕ 2 Q ω (0,T ) 20

37 Carleman inequality We consider ω (ω ω d ) and η 0 = η 0 (x) [Fursikov-Imanuvilov] such that { η0 C 2 ( Ω), η > 0 in Ω, η = 0 Ω, η 0 > 0 in Ω\O, β 0 (x) = e 2C η 0 e C η 0 (x), β(x, t) = β 0(x) ξ(t) := t(t t) I (m, z) := e 2sβ (sξ) m 2 z 2 + e 2sβ (sξ) m z 2 Q Q C m ω (0,T ) s s 0 (T + T 2 ) y m R. e 2sβ (sξ) m z 2 } {{ } L ω (m,z) + Q 1 t(t t) e 2sβ (sξ) m 3 z t ± z 2. (3) 21

38 Idea of the proof Chose m = 3 and apply (3) to I (3, ϕ) + I (3, θ) C ϕ t ϕ = θχ Od, θ t θ = 1 l 2 ϕχ O + 1 γ 2 ϕ ( L ω (3, ϕ) + L ω (3, θ) + e 2sα θχ Od 2 Q ) + e 2sα 1 ϕχ l 2 O + 1 ϕ 2. γ 2 Q 22

39 Idea of the proof Chose m = 3 and apply (3) to I (3, ϕ) + I (3, θ) C For s large enough we have ϕ t ϕ = θχ Od, θ t θ = 1 l 2 ϕχ O + 1 γ 2 ϕ ( L ω (3, ϕ) + L ω (3, θ) + e 2sα θχ Od 2 Q ) + e 2sα 1 ϕχ l 2 O + 1 ϕ 2. γ 2 Q I (3, ϕ) + I (3, θ) C (L ω (3, ϕ) + L ω (3, θ)) 22

40 Idea of the proof (Cont.) ( ) I (3, ϕ) + I (3, θ) C e 2sα (sξ) 3 ( ϕ 2 + θ 2 ) ω (0,T ) 23

41 Idea of the proof (Cont.) ( ) I (3, ϕ) + I (3, θ) C e 2sα (sξ) 3 ( ϕ 2 + θ 2 ) ω (0,T ) Cutoff function: ζ C0 (Ω), ζ = 1 in ω,... e 2sα (sξ) 3 θ 2 e 2sα (sξ) 3 ζ θ 2 ω (0,T ) = O d (0,T ) Q e 2sα (sξ) 3 ζθ( ϕ t ϕ) 23

42 Idea of the proof (Cont.) ( ) I (3, ϕ) + I (3, θ) C e 2sα (sξ) 3 ( ϕ 2 + θ 2 ) ω (0,T ) I.p.p, Hölder andyoung e 2sα (sξ) 3 θ 2 ω (0,T ) εi (3, θ) + C ε ω (0,T ) e 2sα (sξ) 7 ϕ γ 2 Q e 2sα (sξ) 3 ϕ 2 23

43 Idea of the proof (Cont.) ( ) I (3, ϕ) + I (3, θ) C e 2sα (sξ) 3 ( ϕ 2 + θ 2 ) ω (0,T ) I.p.p, Hölder andyoung e 2sα (sξ) 3 θ 2 ω (0,T ) εi (3, θ) + C ε ω (0,T ) e 2sα (sξ) 7 ϕ γ 2 Q e 2sα (sξ) 3 ϕ 2 ( ) I (3, ϕ) + I (3, θ) C e 2sα (sξ) 7 ϕ 2 ω (0,T ) 23

44 Observability inequality We introduce and the weights l(t) = α(x, t) = β 0(x) l(t), ξ(t) = 1 l(t), Ω { T 2 /4 for 0 t T /2, ϕ(0) 2 + Key points: θ(x, 0) = 0 and l >> 1. t(t t) for T /2 t T, Q α (t) = max Ω ρ 2 θ 2 C α(x, t), ρ(t) := e s α. ω (0,T ) ϕ 2 24

45 Additional results Extensions Semilinear systems y t y + f (y) = hχ ω + vχ O + ψ in Q = Ω (0, T ) y = 0 on Σ, y(x, 0) = y 0 (x) in Ω! saddle point : f glob. Lipschitz, f L (R), N 6. y t y + f (y) = hχ ω 1 pχ l 2 O + 1 p in Q γ 2 p t p + f (y)p = (y y d )χ Od in Q y(x, 0) = y 0 (x), p(x, T ) = 0, y = p = 0 on Σ. Controllability to trajectories : obs. inequality for the linearized system, fixed point. 25

46 Additional results (Cont.) Extensions Case with restrictions y t y = hχ ω + vχ O + ψ in Q = Ω (0, T ) y = 0 on Σ, y(x, 0) = y 0 (x) in Ω! saddle point ( v, ψ) V ad Ψ ad (bounded,... ) : V ad = {v L 2 (O (0, T )) : v(x, t) E 1 a.e.}, Ψ ad = {v L 2 (Q) : ψ(x, t) E 2 a.e.} Null controllability : fixed point. 26

47 Additional results (Cont.) Extensions Control/boundary perturbations (V. Hernández-Santamaría work in progress) y t y = hχ ω + vχ O in Q = Ω (0, T ) y + βy = ψ over Σ, y(x, 0) = y 0 (x) in Ω n with β L (Σ).! saddle point Null controllability 27

48 Work in progress (With Cristhian Montoya) y t y + (y )y + p = hχ ω + vχ O + ψ in Q, y = 0 in Q, y = 0 on Σ, y(, 0) = y 0 ( ) in Ω, (4) With the follower functional J 1 (ψ, v; h) := µ 2 ( + 1 l 2 2 y y d 2 dxdt O d (0,T ) O (0,T ) ) v 2 dxdt γ 2 ψ 2 dxdt, Q 28

49 Thank you! Gracias! 29

50 Some words about Enrique

51 Working with Enrique 31

52 Working with Enrique 32

53 Working with Enrique 33

54 Working with Enrique 34

55 Not all is work! 35

56 Not all is work! 36

57 Not all is work! 37

58 Not all is work! 38