EXTENDED EULER-LAGRANGE AND HAMILTONIAN CONDITIONS IN OPTIMAL - PowerPoint PPT Presentation

EXTENDED EULER-LAGRANGE AND HAMILTONIAN CONDITIONS IN OPTIMAL CONTROL OF SWEEPING PROCESSES WITH CONTROLLED MOVING SETS BORIS MORDUKHOVICH Wayne State University Talk given at the conference Optimization, State Constraints and Geometric Control Tribute to Giovanni Colombo and Franco Rampazzo Joint work with Nguyen Hoang (Univ. Concepci´ on, Chile) Padova, Italy, May 2018 Supported by NSF grants DMS-1512846 and DMS-1808978 and by Air Force grant 15RT0462

CONTROLLED SWEEPING PROCESS This talk addresses the following sweeping process � � � � x ( t ) ∈ f ˙ t, x ( t ) − N g ( x ( t )); C ( t, u ( t )) a.e. t ∈ [0 , T ] with x (0) = x 0 ∈ C (0 , u (0)), where � R n � � R m C ( t, u ) := x ∈ I � ψ ( t, x, u ) ∈ Θ ( t, u ) ∈ [0 , T ] × I , � R n → I R n → I R n × I R m → R n , g : I R n , ψ : [0 , T ] × I with f : [0 , T ] × I R s , and Θ ⊂ I R s . The feasible pairs ( u ( · ) , x ( · )) are absolutely I continuous. The normal cone is defined via the projector by R n � � � N (¯ x ; Ω) := v ∈ I � ∃ x k → ¯ x, α k ≥ 0 , w k ∈ Π( x k ; Ω) , α k ( x k − w k ) → v � if ¯ x ∈ Ω and N (¯ x ; Ω) = ∅ otherwise The major assumption is that ∇ x ψ is surjective 1

OPTIMAL CONTROL Problem ( P ) � T � � � � minimize J [ x, u ] := ϕ x ( T ) + t, x ( t ) , u ( t ) , ˙ x ( t ) , ˙ u ( t ) 0 ℓ dt over the sweeping control dynamics subject to the intrinsic pointwise state-control constraints � � t, g ( x ( t )) , u ( t ) ∈ Θ for all t ∈ [0 , T ] ψ From now on � � F = F ( t, x, u ) := f ( t, x ) − N g ( x ); C ( t, u ) 2

LOCAL MINIMIZERS DEFINITION Let the pair (¯ x ( · ) , ¯ u ( · )) be feasible to ( P ) u ( · )) be a local W 1 , 2 × W 1 , 2 -minimizer if (i) We say that (¯ x ( · ) , ¯ x ( · ) ∈ W 1 , 2 ([0 , T ]; I u ( · ) ∈ W 1 , 2 ([0 , T ]; I R n ), ¯ R m ), and ¯ u ] ≤ J [ x, u ] for all x ( · ) ∈ W 1 , 2 ([0 , T ]; I R n ) , u ( · ) ∈ W 1 , 2 ([0 , T ]; I R m ) J [¯ x, ¯ sufficiently close to (¯ x ( · ) , ¯ u ( · )) in the norm topology of the corresponding spaces (ii) Let the running cost ℓ ( · ) in do not depend on ˙ u . We u ( · )) be a local W 1 , 2 × C -minimizer if say that the pair (¯ x ( · ) , ¯ x ( · ) ∈ W 1 , 2 ([0 , T ]; I R n ), ¯ R m ), and ¯ u ( · ) ∈ C ([0 , T ]; I u ] ≤ J [ x, u ] for all x ( · ) ∈ W 1 , 2 ([0 , T ]; I R n ) , u ( · ) ∈ C ([0 , T ]; I R m ) J [¯ x, ¯ sufficiently close to (¯ x ( · ) , ¯ u ( · )) in the norm topology of the corresponding spaces 3

DISCRETE APPROXIMATIONS For local W 1 , 2 × W 1 , 2 -minimizers (¯ u ). Problem ( P 1 x, ¯ k ) x k j +1 − x k u k j +1 − u k k − 1 � � j j minimize J k [ z k ] := ϕ ( x k x k j , u k � k ) + h k ℓ j , , h k h k j =0 t k j +1 x k j +1 − x k u k j +1 − u k k − 1 2 2 � �� � � � � j j � − ˙ − ˙ � � � � + h k ¯ x ( t ) + u ( t ) ¯ dt � � � � h k h k � � � � j =0 t k j over z k := ( x k k ) ∈ ψ − 1 (Θ) and 0 , . . . , x k k , u k 0 , . . . , u k k ) s.t. ( x k k , u k x k j +1 ∈ x k j + h k F ( x k j , u k � x k 0 , u k � � � j ) , j = 0 , . . . , k − 1 , = x 0 , ¯ u (0) 0 t k j +1 x k j +1 − x k u k j +1 − u k k − 1 2 2 � dt ≤ ε �� � � � � j j � − ˙ − ˙ � � � � x ( t ) ¯ + u ( t ) ¯ � � � � 2 h k h k � � � � j =0 t k j 4

DISCRETE APPROXIMATIONS (cont.) For local W 1 , 2 × C -minimizer-minimizers (¯ u ). Problem ( P 2 x, ¯ k ) x k j +1 − x k k − 1 � � j minimize J k [ z k ] := ϕ ( x k x k j , u k � k ) + h k ℓ j , h k j =0 t k j +1 x k j +1 − x k k − 1 k 2 2 + � � � j � � � u k u ( t k � � − ˙ � � + j − ¯ j ) x ( t ) ¯ dt � � � � � h k � � j =0 j =0 t k j over z k = ( x k k ) ∈ ψ − 1 (Θ) and 0 , . . . , x k k , u k 0 , . . . , u k k ) s.t. ( x k k , u k x k j +1 ∈ x k j + h k F ( x k j , u k � x k 0 , u k � � � j ) , j = 0 , . . . , k − 1 , = x 0 , ¯ u (0) 0 t k j +1 x k j +1 − x k k − 1 k 2 2 + dt ≤ ε � � � j � � � u k u ( t k � � − ˙ � � j − ¯ j ) x ( t ) ¯ � � � � � 2 h k � � j =0 j =0 t k j 5

STRONG CONVERGENCE OF DISCRETE APPROXIMATIONS u ( · )) is a local W 1 , 2 × W 1 , 2 -minimizer x ( · ) , ¯ THEOREM (i) If (¯ for ( P ), then any sequence of piecewise linear extensions on u k ( · )) to ( P 1 x k ( · ) , ¯ [0 , T ] of the optimal solutions (¯ k ) converges to u ( · )) in the norm topology of W 1 , 2 ([0 , T ]; I R n ) × W 1 , 2 ([0 , T ]; I R m ) (¯ x ( · ) , ¯ u ( · )) is a local W 1 , 2 ×C -minimizer for ( P ), then any (ii) If (¯ x ( · ) , ¯ sequence of piecewise linear extensions on [0 , T ] of the optimal u k ( · )) to ( P 2 x k ( · ) , ¯ solutions (¯ k ) converges to (¯ x ( · ) , ¯ u ( · )) in the norm topology of W 1 , 2 ([0 , T ]; I R n ) × C ([0 , T ]; I R m ) 6

GENERALIZED DIFFERENTIATION R n → ( −∞ , ∞ ] at ¯ Subdifferential of an l.s.c. function ϕ : I x � � � ∂ϕ (¯ x ) := � ( v, − 1) ∈ N ((¯ x, ϕ (¯ x )); epi ϕ ) x ∈ dom ϕ ¯ v , � Coderivative of a set-valued mapping F � � D ∗ F (¯ � x, ¯ y )( u ) := � ( v, − u ) ∈ N ((¯ x, ¯ y ); gph F ) y ∈ F (¯ ¯ x ) v , � Generalized Hessian of ϕ at ¯ x ∂ 2 ϕ (¯ x ) := D ∗ ( ∂ϕ )(¯ x, ¯ v ) , ¯ v ∈ ∂ϕ (¯ x ) Enjoy FULL CALCULUS and PRECISELY COMPUTED in terms of the given data of ( P ) 7

FURTHER STRATEGY • For each k reduce problems ( P 1 k ) and to ( P 2 k ) a problems of mathematical programming ( MP ) with functional and increas- ingly many geometric constraints. The latter are generated by the graph of the mapping F ( z ) := f ( x ) − N ( x ; C ( u )), and so ( MP ) is intrinsically nonsmooth and nonconvex even for smooth initial data • Use variational analysis and generalized differentiation (first- and second-order) to derive necessary optimality conditions for ( MP ) and then discrete control problems ( P 1 k ) and ( P 2 k ) • Explicitly compute the coderivative of F ( z ) entirely in terms of the given data of ( P ) • By passing to the limit as k → ∞ , to derive necessary opti- mality conditions for the sweeping control problem ( P ) 8

EXTENDED EULER-LAGRANGE CONDITIONS u ( · )) is a local W 1 , 2 × W 1 , 2 -minimizer, THEOREM If (¯ x ( · ) , ¯ then there exist a multiplier λ ≥ 0, an adjoint arc p ( · ) = R n × I ( p x , p u ) ∈ W 1 , 2 ([0 , T ]; I R m ), a signed vector measure γ ∈ R n × C ∗ ([0 , T ]; I R s ), as well as pairs ( w x ( · ) , w u ( · )) ∈ L 2 ([0 , T ]; I R n × I R m ) and ( v x ( · ) , v u ( · )) ∈ L ∞ ([0 , T ]; I R m ) with I � � � � w x ( t ) , w u ( t ) , v x ( t ) , v u ( t ) u ( t ) , ˙ x ( t ) , ˙ ∈ co ∂ℓ x ( t ) , ¯ ¯ ¯ u ( t ) ¯ satisfying the collection of necessary optimality conditions • Primal-dual dynamic relationships  � �� �   ∇ 2 η ( t ) , ψ ¯ x ( t ) , ¯ u ( t ) xx � − λv x ( t ) + q x ( t ) � p ( t ) = λw ( t ) + ˙ � �� �  ∇ 2 η ( t ) , ψ x ( t ) , ¯ ¯ u ( t ) xw q u ( t ) = λv u ( t ) a.e. t ∈ [0 , T ] 9

where η ( · ) ∈ L 2 ([0 , T ]; I R s ) is uniquely defined by � ∗ η ( t ) , η ( t ) ∈ N ( ψ (¯ � ˙ x ( t ) = −∇ x ψ ¯ ¯ x ( t ) , ¯ u ( t ) x ( t ) , ¯ u ( t )); Θ) R n × I R m is of bounded variation with where q : [0 , T ] → I � ∗ dγ ( τ ) � � q ( t ) = p ( t ) − [ t,T ] ∇ ψ ¯ x ( τ ) , ¯ u ( τ ) • Measured coderivative condition : Considering the t -dependent outer limit γ ( B ) � � R s � Lim sup | B | ( t ) := y ∈ I ∃ seq. B k ⊂ [0 , 1] , t ∈ I B k , � � | B |→ 0 � | B k | → 0 , γ ( B k ) | B k | → y over Borel subsets B ⊂ [0 , 1], for a.e. t ∈ [0 , T ] we have � �� � D ∗ N Θ u ( t ))( q x ( t ) − λv x ( t )) ψ (¯ x ( t ) , ¯ u ( t )) , η ( t ) ∇ x ψ (¯ x ( t ) , ¯ γ ( B ) � Lim sup | B | ( t ) � = ∅ | B |→ 0

• Transversality condition � � � � � � � � p x ( T ) , p u ( T ) − ∈ λ ∂ϕ (¯ x ( T )) , 0 + ∇ ψ x ( T ) , ¯ ¯ u ( T ) (¯ x ( T ) , ¯ u ( T ) N Θ • Measure nonatomicity condition : Whenever t ∈ [0 , T ) with ψ (¯ x ( t ) , ¯ u ( t )) ∈ int Θ there is a neighborhood V t of t in [0 , T ] such that γ ( V ) = 0 for any Borel subset V of V t • Nontriviality condition � � p ( t ) � + � γ � � = 0 with � γ � := λ + sup sup [0 ,T ] x ( s ) dγ t ∈ [0 ,T ] � x � C ([0 ,T ] =1 • Enhanced nontriviality : If θ = 0 is the only vector satisfying � ∗ θ ∈ ∇ ψ θ ∈ D ∗ N Θ � � � � � ψ (¯ x ( T ) , ¯ u ( T )) , η ( T ) (0) , ∇ ψ x ( T ) , ¯ ¯ u ( T ) ¯ x ( T ) , ¯ u ( T ) N

then we have � � � λ + mes t ∈ [0 , T ] � q ( t ) � = 0 + � q (0) � + � q ( T ) � > 0 � u ( · )) is a local W 1 , 2 × C -minimizer, then all the (ii) If (¯ x ( · ) , ¯ above conditions hold with � w x ( t ) , w u ( t ) , v x ( t ) � � � u ( t ) , ˙ ∈ co ∂ℓ ¯ x ( t ) , ¯ ¯ x ( t )