Pointwise second-order necessary optimality conditions and sensitivity relations Nonlinear Partial Differential Equations and Applications Daniel Hoehener MIT joint work with H´ el` ene Frankowska Paris, June 20, 2016
O PTIMAL C ONTROL P ROBLEM Problem : ϕ ( x ( 1 )) Minimize ˙ x ( t ) = f ( t , x ( t ) , u ( t )) ∀ t u ( t ) ∈ U ( t ) ∀ t x ( 0 ) ∈ K 0 .
O PTIMAL C ONTROL P ROBLEM Problem : Assumptions : ◮ f satisfies standard ϕ ( x ( 1 )) Minimize assumptions and is twice ˙ x ( t ) = f ( t , x ( t ) , u ( t )) ∀ t differentiable in x with u ( t ) ∈ U ( t ) ∀ t Lipschitz derivative ◮ ϕ is differentiable x ( 0 ) ∈ K 0 . ◮ K 0 and U ( t ) are closed and nonempty
O PTIMAL C ONTROL P ROBLEM Problem : Assumptions : ◮ f satisfies standard ϕ ( x ( 1 )) Minimize assumptions and is twice ˙ x ( t ) = f ( t , x ( t ) , u ( t )) ∀ t differentiable in x with u ( t ) ∈ U ( t ) ∀ t Lipschitz derivative ◮ ϕ is differentiable x ( 0 ) ∈ K 0 . ◮ K 0 and U ( t ) are closed and nonempty Value function V ( t , x 0 ) = inf { ϕ ( x ( 1 )) | x solution of problem starting at ( t , x 0 ) } .
I NTRODUCTION Maximum Principle Let (¯ x , ¯ u ) be an optimal solution. The solution ¯ p of � − ˙ u ( t )) T ¯ ¯ p ( t ) = f x ( t , ¯ x ( t ) , ¯ p ( t ) − ¯ p ( 1 ) = ∇ ϕ (¯ x ( 1 )) , satisfies for all t ∀ k 0 ∈ T ♭ i) � ¯ p ( 0 ) , k 0 � ≤ 0 K 0 (¯ x ( 0 )) ii) � ¯ p ( t ) , f ( t , ¯ x ( t ) , ¯ u ( t )) � = max u ∈ U ( t ) � ¯ p ( t ) , f ( t , ¯ x ( t ) , u ) � .
I NTRODUCTION Maximum Principle Let (¯ x , ¯ u ) be an optimal solution. The solution ¯ p of � − ˙ u ( t )) T ¯ ¯ p ( t ) = f x ( t , ¯ x ( t ) , ¯ p ( t ) − ¯ p ( 1 ) = ∇ ϕ (¯ x ( 1 )) , satisfies for all t ∀ k 0 ∈ T ♭ i) � ¯ p ( 0 ) , k 0 � ≤ 0 K 0 (¯ x ( 0 )) ii) � ¯ p ( t ) , f ( t , ¯ x ( t ) , ¯ u ( t )) � = max u ∈ U ( t ) � ¯ p ( t ) , f ( t , ¯ x ( t ) , u ) � . ◮ Pointwise necessary condition
I NTRODUCTION Maximum Principle Let (¯ x , ¯ u ) be an optimal solution. The solution ¯ p of � − ˙ ¯ p ( t ) = H x [ t ]( t ) − ¯ p ( 1 ) = ∇ ϕ (¯ x ( 1 )) , satisfies ∀ k 0 ∈ T ♭ 1. � ¯ p ( 0 ) , k 0 � ≤ 0 K 0 (¯ x ( 0 )) 2. H [ t ] = max u ∈ U ( t ) H ( t , ¯ x ( t ) , ¯ p ( t ) , u ) . ◮ Pointwise necessary condition ◮ H ( t , x , p , u ) = � p , f ( t , x , u ) � ◮ [ t ] = ( t , ¯ x ( t ) , ¯ p ( t ) , ¯ u ( t ))
I NTRODUCTION Maximum Principle Let (¯ x , ¯ u ) be an optimal solution. The solution ¯ p of � − ˙ ¯ p ( t ) = H x [ t ]( t ) − ¯ p ( 1 ) = ∇ ϕ (¯ x ( 1 )) , satisfies p ( 0 ) ∈ N ♭ 1. ¯ K 0 (¯ x ( 0 )) 2. H [ t ] = max u ∈ U ( t ) H ( t , ¯ x ( t ) , ¯ p ( t ) , u ) . ◮ Pointwise necessary condition ◮ H ( t , x , p , u ) = � p , f ( t , x , u ) � ◮ N ♭ K ( x ) = � � q , k � ≤ 0 � � ∀ k ∈ T ♭ � K ( x ) q
I NTRODUCTION Maximum Principle Sensitivity Relations Let (¯ x , ¯ Let ¯ u ) be an optimal solution. x be an optimal solution. The The solution ¯ p of solution ¯ p of � � − ˙ − ˙ ¯ ¯ p ( t ) = H x [ t ]( t ) p ( t ) = H x [ t ]( t ) − ¯ p ( 1 ) = ∇ ϕ (¯ − ¯ p ( 1 ) = ∇ ϕ (¯ x ( 1 )) , x ( 1 )) , satisfies satisfies p ( 0 ) ∈ N ♭ p ( t ) ∈ ∂ + − ¯ x V ( t , ¯ 1. ¯ K 0 (¯ x ( 0 )) x ( t )) ∀ t . 2. H [ t ] = max u ∈ U ( t ) H ( t , ¯ x ( t ) , ¯ p ( t ) , u ) . ◮ ∂ + ◮ Pointwise necessary condition x is the superdifferential in x . ◮ H ( t , x , p , u ) = � p , f ( t , x , u ) � ◮ N ♭ K ( x ) = � � q , k � ≤ 0 � � ∀ k ∈ T ♭ � K ( x ) q
I NTRODUCTION II 2nd-order pointwise conditions ◮ Goh conditions ◮ U is time independent ◮ Requires structural assumptions on U ◮ Jacobson conditions ◮ continuous optimal control ◮ optimal control in interior of U ◮ U is time independent
I NTRODUCTION II 2nd-order pointwise conditions 2nd-order sensitivity relations ◮ Goh conditions ◮ Investigated using matrix ◮ U is time independent Riccati equations ◮ Requires structural ◮ Relationship with regularity assumptions on U of the value function ◮ Jacobson conditions ◮ continuous optimal control ◮ optimal control in interior of U ◮ U is time independent
I NTRODUCTION II 2nd-order pointwise conditions 2nd-order sensitivity relations ◮ Goh conditions ◮ Investigated using matrix ◮ U not time dependent Riccati equations ◮ Requires structural ◮ Relationship with regularity assumptions on U of the value function ◮ Jacobson conditions ◮ Link to higher order ◮ continuous optimal control necessary conditions? ◮ optimal control in interior of U ◮ U not time dependent
P RELIMINARIES Tangent cone Normal cone � � � K − x N ♭ ∀ u ∈ T ♭ T ♭ K ( x ) = q � � q , u � ≤ 0 K ( x ) � K ( x ) = Liminf h h → 0 + 2nd-order normal 2nd-order tangent � � K − hu − x � � q , v � + 1 N ♭ ( 2 ) T ♭ ( 2 ) � ( x , q ) = Q ∈ S 2 � Qu , u � ≤ 0 ( x , u ) = Liminf � K K h 2 h → 0 + � ∀ u ∈ T ♭ K ( x ) ∩ { q } ⊥ , v ∈ T ♭ ( 2 ) ( x , u ) K
P RELIMINARIES Tangent cone Normal cone � � � K − x N ♭ ∀ u ∈ T ♭ T ♭ K ( x ) = q � � q , u � ≤ 0 K ( x ) � K ( x ) = Liminf h h → 0 + 2nd-order normal 2nd-order tangent � � K − hu − x � � q , v � + 1 N ♭ ( 2 ) T ♭ ( 2 ) � ( x , q ) = Q ∈ S 2 � Qu , u � ≤ 0 ( x , u ) = Liminf � K K h 2 h → 0 + � ∀ u ∈ T ♭ K ( x ) ∩ { q } ⊥ , v ∈ T ♭ ( 2 ) ( x , u ) K Superjet � � � � f ( y ) − f ( x ) ≤ � q , y − x � + 1 J 2 , + f ( x ) = � 2 � Q ( y − x ) , y − x � + o ( | y − x | 2 ) , ( q , Q ) ∀ y � Subjet � � � J 2 , − f ( x ) = � ( q , Q ) ≥ � �
S ECOND - ORDER MAXIMUM PRINCIPLE Theorem. Let (¯ x , ¯ u , ¯ p ) be an optimal solution and adjoint state. Then for x ( 1 )) , Ψ) ∈ J 2 , + ϕ (¯ every Ψ ∈ S such that ( ∇ ϕ (¯ x ( 1 )) , the solution W of � ˙ W ( t ) = −H px [ t ] W ( t ) − W ( t ) H xp [ t ] − H xx [ t ] , W ( 1 ) = − Ψ , satisfies i) W ( 0 ) ∈ N ♭ ( 2 ) K 0 (¯ x ( 0 ); ¯ p ( 0 )) M T ¯ � � ii) max p ( t ) + W ( t ) v , v = 0 , a.e. in [ 0 , 1 ] . ( v , M ) ∈ ¯ F ( t ) Notation ¯ � u ∈ ¯ � � ( f ( t , ¯ x ( t ) , u ) , f x ( t , ¯ � F ( t ) = co x ( t ) , u )) − ( f [ t ] , f x [ t ])) U ( t )) � � � ¯ � z ∈ arg max u ∈ U ( t ) H ( t , ¯ x ( t ) , ¯ U ( t ) = z ∈ U ( t ) p ( t ) , u ) �
S ECOND - ORDER SENSITIVITY RELATIONS Theorem (Backward propagation). Let ¯ x be optimal with ¯ x ( t 0 ) = x 0 and ¯ p , W and Ψ be as in the maximum principles. Then p ( t ) , − W ( t )) ∈ J 2 , + ( − ¯ V ( t , ¯ x ( t )) , ∀ t ∈ [ t 0 , 1 ] . x
S ECOND - ORDER SENSITIVITY RELATIONS Theorem (Backward propagation). Let ¯ x be optimal with ¯ x ( t 0 ) = x 0 and ¯ p , W and Ψ be as in the maximum principles. Then p ( t ) , − W ( t )) ∈ J 2 , + ( − ¯ V ( t , ¯ x ( t )) , ∀ t ∈ [ t 0 , 1 ] . x Theorem (Forward propagation). Let ¯ x and ¯ p be as above. If for some W 0 ∈ S we have p ( t 0 ) , − W 0 ) ∈ J 2 , − ( − ¯ V ( t 0 , x 0 ) . Then for x � ˙ W ( t ) + H px [ t ] W ( t ) + W ( t ) H xp [ t ] + H xx [ t ] = 0 , W ( t 0 ) = W 0 , the following sensitivity relation holds true: p ( t ) , − W ( t )) ∈ J 2 , − ( − ¯ V ( t , ¯ x ( t )) , ∀ t ∈ [ t 0 , 1 ] . x
J ACOBSON TYPE NECESSARY CONDITIONS I Relaxed Assumption : ◮ f ( t , · , · ) is twice differentiable with Lipschitz derivatives ◮ ϕ is twice differentiable
J ACOBSON TYPE NECESSARY CONDITIONS I Relaxed Assumption : ◮ f ( t , · , · ) is twice differentiable with Lipschitz derivatives ◮ ϕ is twice differentiable Objectives : � � � 1. Replace z ∈ U ( t ) � z ∈ arg max u ∈ U ( t ) H ( t , ¯ x ( t ) , ¯ p ( t ) , u ) � with more “explicit” set (tangents) 2. Pass to the limit in the expression � � M T ¯ max p ( t ) + W ( t ) v , v = 0 , a.e. in [ 0 , 1 ] ( v , M ) ∈ ¯ F ( t )
J ACOBSON TYPE NECESSARY CONDITIONS II Theorem. Let ¯ x , ¯ u , ¯ p and W be as in the second-order maximum principle with Ψ := ϕ ′′ (¯ x ( 1 )) . For a.e. t ∈ [ 0 , 1 ] and for every u ∈ T ♭ U ( t ) (¯ u ( t )) such that either (i) H u [ t ] � = 0, H u [ t ] u = 0 and H u [ t ] v + 1 2 u T H uu [ t ] u = 0 for some v ∈ T ♭ ( 2 ) U ( t ) (¯ u ( t ) , u ) , or (ii) H u [ t ] = 0 and u T H uu [ t ] u = 0, we have � f u [ t ] T ( H ux [ t ] + W ( t ) f u [ t ]) u , u � ≤ 0 .
I NEQUALITY CONSTRAINTS Assumption : ◮ U ( t ) = � s � � c j ( t , u ) ≤ 0 � � u , j = 1 ◮ c j twice differentiable in u ◮ {∇ u c j ( t , ¯ u ( t )) } s j = 1 are linearly independent
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