Lecture 1: Some illustrative optimal control problems Enrique FERN ´ ANDEZ-CARA Dpto. E.D.A.N. - Univ. of Sevilla Four interesting optimal control problems Results and open questions They lead to new theoretical results . . . They are connected to applications . . . E. Fern´ andez-Cara Optimal control problems
Optimal control problems for PDEs Structure The state equation: � A ( y ) = B ( v ) ( S ) + . . . The cost: ( v , y ) �→ J ( v , y ) The constraints: v ∈ V ad , y ∈ Y ad The optimal control problem: Minimize J ( v , y ) Subject to v ∈ V ad , y ∈ Y ad , ( v , y ) satisfies ( S ) Main questions: ∃ /uniqueness/multiplicity, characterization, computation, . . . E. Fern´ andez-Cara Optimal control problems
Outline Optimal control of a capacitor 1 The problem The main results and their proofs Control on the coefficients, homogenization, optimal materials 2 The original problem The relaxed problem Optimal design for Navier-Stokes flow 3 The problem An optimality result Optimal control oriented to therapies for tumor growth models 4 The problem The results E. Fern´ andez-Cara Optimal control problems
Optimal control of a capacitor The problem Ω ⊂ R N bounded, regular, connected, open; Γ = ∂ Ω . ω ⊂⊂ Ω non-empty, open; 1 ω : the characteristic function The state system: � − ∆ y = 1 ω u in Ω , (1) y = 0 on Γ , y = y ( x ) : electric potential; the density of charge is 1 ω u ; E = −∇ y is the associated electric field Question: How to choose u to have y as good as possible? For instance, for given a , b > 0, y d ∈ L 2 (Ω) and U ad ⊂ L 2 ( ω ) : Minimize J ( u ) = a | y − y d | 2 dx + b � � | u | 2 dx (2) 2 2 Ω ω Subject to u ∈ U ad , (1) where a , b > 0. E. Fern´ andez-Cara Optimal control problems
Optimal control of a capacitor The main results and their proofs Theorem 1: existence, uniqueness Assume: U ad ⊂ L 2 ( ω ) is non-empty, closed, convex. Then: ∃ ! optimal ˆ u Theorem 2: characterization (optimality) Same hypotheses. Then: ∃ ˆ y , ˆ p with � − ∆ˆ y = ˆ u 1 ω in Ω (3) ˆ y = 0 on Γ � − ∆ˆ p = ˆ y − y d in Ω (4) ˆ p = 0 on Γ � ( a ˆ p + b ˆ u )( u − ˆ u ) dx ≥ 0 ∀ u ∈ U ad (5) ω E. Fern´ andez-Cara Optimal control problems
Optimal control of a capacitor The main results and their proofs P ROOF OF T HEOREM 1: Recall: J ( u ) = a � | y − y d | 2 dx + b � | u | 2 dx ∀ u ∈ U ad 2 2 Ω ω u �→ J ( u ) is strictly convex, coercive and continuous (hence weakly lsc) in L 2 ( ω ) U ad is closed and convex Hence . . . ✷ QUESTIONS: a = 0? b = 0? Interpretations? P ROOF OF T HEOREM 2: Try to write that � J ′ (ˆ u ) , u − ˆ u � ≥ 0 u ∈ U ad , with ˆ u ∈ U ad � If ˆ p solves ( 4 ) , then � J ′ (ˆ u ) , u − ˆ ( a ˆ p + b ˆ u ) ( u − ˆ u � = u ) dx Consequently, . . . ✷ ω Remark In this case, the reciprocal holds: if ( 3 ) − ( 5 ) holds, then ˆ u is the optimal control E. Fern´ andez-Cara Optimal control problems
Optimal control of a capacitor The main results and their proofs Remark � From the previous argument: � J ′ ( u ) , v � = ( ap + bu ) v dx , ω with − ∆ p = y − y d in Ω , p = 0 on Γ (the adjoint state) Useful! QUESTIONS: First suggested iterative method: − ∆ y n = u n − 1 1 ω in Ω , y n = 0 on Γ − ∆ p n = y n − y d in Ω , p n = 0 on Γ ω ( ap n + bu n )( u − u n ) dx ≥ 0 ∀ u ∈ U ad , u n ∈ U ad � Convergence? Other iterates based on gradient computation? Many possible generalizations . . . QUESTIONS: Similar optimal control problems for other PDEs? y t − ∆ y = u 1 ω in Ω × ( 0 , T ) + . . . or y tt − ∆ y = u 1 ω , iy t + ∆ y = u 1 ω . . . similar nonlinear PDEs, etc. Existence/uniqueness? Characterization? Convergent algorithms? E. Fern´ andez-Cara Optimal control problems
Control on the coefficients and homogenization The original problem Assume: in Ω we find two different dielectric materials, with permeability coefficients α and β (0 < α < β ). How can we determine an optimal distribution? The electrostatic potential y = y ( x ) for a partition { G 1 , G 2 } of Ω : −∇ · ( a ( x ) ∇ y ) = f ( x ) in Ω , y = 0 on Γ where a ( x ) = α in G 1 , a ( x ) = β in G 2 ( f is given; a is the control and y is the state) Question: How to choose a to have y as good as possible? For instance, for given y d ∈ L 2 (Ω) : � | y − y d | 2 dx Minimize j ( a ) = 1 2 (6) Ω Subject to a ∈ A ad = { a ∈ L ∞ (Ω) : a ( x ) ∈ { α, β } a.e. } Even beter: � | y − y d | 2 dx Minimize j ( a ) = 1 2 Ω (7) � Subject to a ∈ A ad , a dx ≤ I Ω E. Fern´ andez-Cara Optimal control problems
Control on the coefficients and homogenization The original problem We assume N = 2 (for simplicity) IN GENERAL, �∃ SOLUTION: Minimizing { a n } , a n → a 0 weakly- ∗ , y n → y weakly, but . . . (typical for control on the coefficients) Notation: A ( α, β ) is the family of 2 × 2 matrices A such that A ( x ) ξ · ξ ≥ α | ξ | 2 , ( A ( x )) − 1 ξ · ξ ≥ 1 β | ξ | 2 ∀ ξ ∈ R 2 , x a.e. in Ω If A n , A 0 ∈ A ( α, β ) , A n H -converges to A 0 if ∀O ⊂ Ω , ∀ g the corresponding solutions satisfy y n → y 0 weakly in H 1 0 and A n ∇ y n → A 0 ∇ y 0 weakly in L 2 [Murat and Tartar, 1978 . . . ] Theorem 3: compactness The family A ( α, β ) is compact for the H -convergence The key point: we can have A n = a n I ∀ n and non-diagonal A 0 Explicit examples; thus, no solution for (6) E. Fern´ andez-Cara Optimal control problems
Control on the coefficients and homogenization The relaxed problem What can be done? Relaxation: (Q) is the relaxed problem of (P) if (a) ∃ solutions to (Q) (b) Solutions to (Q) ≡ weak limits of minimizing sequences of (P) Notation: ˜ A ad is the family of all symmetric A ∈ A ( α, β ) with αβ α ≤ λ 1 ( x ) ≤ λ 2 ( x ) ≤ β, α + β − λ 2 ( x ) ≤ λ 1 ( x ) a.e. in Ω A new problem: � | Y − y d | 2 dx Minimize j ( A ) := 1 2 (8) Ω Subject to A ∈ ˜ A ad , −∇ · ( A ( x ) ∇ Y ) = f ( x ) in Ω , . . . Theorem 4: relaxation A ∈ ˜ A ad ⇔ A is the H -limit of some a n I , with a n ∈ A ad Hence, the relaxed problem of (6) is (8) Physical interpretation: a composite anisotropic material E. Fern´ andez-Cara Optimal control problems
Control on the coefficients and homogenization The relaxed problem QUESTIONS: Optimality systems for (6) and (8)? Convergent iterates? Numerics? QUESTIONS: The H -closure of A ad for N -dimensional problems ( N ≥ 3)? Similar results for parabolic and hyperbolic PDEs? Nonlinearities? In view of the difficulty: periodic structures Many results under these conditions for many related problems E. Fern´ andez-Cara Optimal control problems
Optimal design for Navier-Stokes flow The problem Assume: Ω is filled with a Navier-Stokes fluid We try to find the optimal shape of a body travelling in Ω : � | Dy | 2 dx Minimize T ( B , y ) := 2 ν (9) Ω \ B Subject to B ∈ B ad , ( y , π ) solves NS in Ω \ B − ν ∆ y + ( y · ∇ ) y + ∇ π = 0 , ∇ · y = 0 in Ω \ B y = y ∞ on Γ (10) y = 0 on ∂ B B ad is the family of admissible bodies For instance: B ∈ B ad ⇔ B = O for some connected open O with D 0 ⊂ O ⊂ D 1 , ∂ O ∈ W 1 , ∞ We are minimizing the drag, subject to B ∈ B ad , since � T ( B , y ) = − C 0 y ∞ · ( σ ( y , π ) · n ) d Γ Γ E. Fern´ andez-Cara Optimal control problems
Optimal design for Navier-Stokes flow The problem In general: NO WAY TO PROVE ∃ , unless ARTIFICIAL CONDITIONS ARE IMPOSED TO B ad (typical for optimal design) Explanation: a minimizing sequence { B n , y n } . Then: � y n � H 1 is uniformly bounded, whence y n → y weakly in H 1 B n → B 0 in the Haussdorf distance sense But: there is no reason to have y = y 0 ! This would be the case if all B ∈ B ad are uniformly W 1 , ∞ . But . . . QUESTIONS: Minimal uniform regularity hypotheses for existence? A “natural” condition on B ad ensuring that y = y 0 ? E. Fern´ andez-Cara Optimal control problems
Optimal design for Navier-Stokes flow An optimality result Assume ∃ . We look for a “body variations” formula: D (ˆ B + u ) = D (ˆ B ) + D ′ (ˆ B + u = { x = ( I + u )( ξ ) : ξ ∈ ˆ ˆ B ; u ) + o ( u ) , B } (differentiating u �→ D (ˆ B + u ) ; ˆ B is a reference body shape) Theorem 5: optimality B , Γ ∈ W 2 , ∞ and u ∈ W 2 , ∞ . Then: Assume: ∂ ˆ � ∂ w � ∂ n − ∂ y � · ∂ y D ′ (ˆ B ; u ) = ∂ n ( u · n ) d σ, ∂ n ∂ ˆ B where ( w , q ) is the associated adjoint state: � − ν ∆ w i + � j ∂ i y j w j − � j y j ∂ j w i + ∂ i q = − 2 ν ∆ y i ∇ · w = 0 , etc. Again very useful! QUESTIONS: A sequence { B n } “converging” to a solution? Second-order derivatives and applications? E. Fern´ andez-Cara Optimal control problems
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