Second-order optimality conditions in Pontryagin form for optimal - PowerPoint PPT Presentation

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions Second-order optimality conditions in Pontryagin form for optimal control problems eric Bonnans ∗ , Xavier Dupuis ∗ , and Laurent Pfeiffer ∗ J. Fr´ ed´ ∗ INRIA Saclay and CMAP, Ecole Polytechnique (France)

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions Introduction Goal: Study of 2nd-order conditions for smooth optimal control problems of ODEs with pure and mixed constraints. Specificity: We consider strong solutions. Classical results are strengthened and expressed with Pontryagin’s multipliers. Our tools: Necessary conditions: use of relaxation Sufficient conditions: use of a decomposition principle Possible applications: shooting methods, discretization methods, characterization of local optimality...

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions 1 Generalities 2 Optimal control problems: framework 3 Weak second-order necessary optimality conditions 4 Second-order necessary conditions for Pontryagin minima 5 Second-order sufficient conditions for bounded strong minima

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions 1 Generalities 2 Optimal control problems: framework 3 Weak second-order necessary optimality conditions 4 Second-order necessary conditions for Pontryagin minima 5 Second-order sufficient conditions for bounded strong minima

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions Setting General references: [Bonnans, Shapiro ’00], [Kawazaki ’88],... Consider the abstract optimization problem x ∈ X f ( x ) subject to g ( x ) ∈ K , Min where f : X → R and g : X → Y are C 2 , X and Y are Hilbert spaces, K a convex set of K . Let ¯ x , Robinson qualification condition (RQC) holds iff ∃ ε > 0, ε B ⊂ g (¯ x ) + Dg (¯ x ) X − K , where B is the unit ball.

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions Setting For x ∈ X , λ ∈ Y ∗ , the Lagrangian is L [ λ ]( x ) = f ( x ) + � λ, g ( x ) � . The Lagrange multipliers at ¯ x are defined by Λ L = { λ ∈ Y ∗ , s.t. λ ∈ N K ( g (¯ x )) , D x L [ λ ](¯ x ) = 0 } . x ) and the quasi-radial critical cone C QR (¯ The critical cone C (¯ x ) are defined by � � C (¯ x ) = h ∈ X , s.t. Df (¯ x ) h = 0 , Dg (¯ x ) h ∈ T K (¯ x ) C QR (¯ x ) h ) = o ( θ 2 ) � � x ) = h ∈ C (¯ x ) , s.t. dist K ( g (¯ x ) + θ Dg (¯ .

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions Necessary optimality conditions Proposition Assume that ¯ x is a local optimal solution and that RQC holds, then: 1st-order necessary conditions: Λ L (¯ x ) is non-empty 2nd-order necessary conditions: for all h in cl ( C QR (¯ x )) , x ) h 2 ≥ 0 . D 2 max xx L [ λ ](¯ λ ∈ Λ L NB: we will work in the framework of the extended polyhedrecity condition, x ) = cl( C QR (¯ C (¯ x )) .

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions Proof Proof. Let h ∈ C QR (¯ x ), d ∈ X be such that x ) d + 1 x ) h 2 ∈ T K ( g (¯ 2 D 2 g (¯ Dg (¯ x )) , with a metric regularity result, we show the existence of a mapping x : [0 , 1] → X satisfying x + θ h + θ 2 d + o ( θ 2 ) . g ( x ( θ )) ∈ K and x ( θ ) = ¯ Then, x ) d + 1 � x ) h 2 � θ 2 + o ( θ 2 ) ≥ 0 . 2 D 2 f (¯ f ( x ( θ )) − f (¯ x ) = Df (¯

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions Proof It follows that the following problem  x ) d + 1 2 D 2 f (¯ x ) h 2 Min Df (¯  d ∈ X x ) h 2 ∈ T K ( g (¯ x ) d + 1 2 D 2 g (¯ s.t. Dg (¯ x )) .  has a nonnegative value. Its dual has the same value: D 2 L [ λ ](¯ x ) h 2 . Max λ ∈ Λ L The result extends to cl( C QR (¯ x )).

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions Quadratic growth The 2nd-order sufficient conditions are: for all h ∈ C (¯ x ) \ 0, x ) h 2 > 0 . λ ∈ Λ D 2 Max xx L [ λ ](¯ A quadratic form Q : X → R is said to be a Legendre form if it is weakly lower semi-continuous h n ⇀ 0 and Q ( h n ) → 0 imply that h n → 0.

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions Quadratic growth Definition The quadratic growth property holds iff there exists ε > 0 and α > 0 such that for all x ∈ X , if g ( x ) ∈ K and � x − ¯ x � ≤ ε , then x � 2 . f ( x ) − f (¯ x ) ≥ α � x − ¯ Proposition Assume that the 2nd-order sufficient condition holds and that D 2 L [ λ ](¯ x ) is a Legendre form for all λ ∈ Λ L . Then, the quadratic growth property holds.

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions Proof (by contradiction). Assume that ∃ x n → ¯ x , g ( x n ) ∈ K s.t. x � 2 ) . f ( x n ) − f (¯ x ) ≤ o ( � x n − ¯ Set h n = ( x n − ¯ x ) / � x n − ¯ x � and denote by h a weak limit point. We check that h ∈ C (¯ x ). Then, for all λ ∈ Λ L (¯ x ), f ( x n ) − f (¯ x ) ≥ f ( x n ) − f (¯ x ) + � λ, g ( x n ) − g (¯ x ) � ≥ L [ λ ]( x n ) − L [ λ ](¯ x ) x ) 2 + o ( � x n − ¯ = D 2 x � 2 ) . xx L [ λ ](¯ x )( x n − ¯ xx L [ λ ] h 2 ≤ lim inf D xx L [ λ ] h 2 Therefore, D 2 n ≤ 0 . xx L [ λ ] h 2 = 0 and h = 0. We obtain that h n → 0, Thus, sup λ ∈ Λ L D 2 in contradiction with � h n � = 1.

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions 1 Generalities 2 Optimal control problems: framework 3 Weak second-order necessary optimality conditions 4 Second-order necessary conditions for Pontryagin minima 5 Second-order sufficient conditions for bounded strong minima

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions Problem The optimal control problem is: u ∈U , y ∈Y φ ( y T ) Min subject to: y t = f ( u t , y t ) , y 0 = y 0 the dynamic: ˙ final-state constraints: Φ E ( y T ) = 0 , Φ I ( y T ) ≤ 0 pure constraints: g ( y t ) ≤ 0 for all t mixed constraints: c ( u t , y t ) ≤ 0 for a.a. t , where U = L ∞ (0 , T ; R m ) and Y = W 1 , ∞ (0 , T ; R n ). The mappings are C 2 and defined in R n , R n E , R n I , R n g , R n c . For u ∈ U , we denote by y [ u ] the solution to the state equation and set J ( u ) = φ ( y T [ u ]) .

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions Local optimal solutions A control ¯ u is said to be a weak minimum iff ∃ ε > 0 such that for all feasible u , � u − ¯ u � ∞ ≤ ε = ⇒ J ( u ) ≥ J (¯ u ) a Pontryagin minimum iff for all R > � ¯ u � ∞ , there exists ε > 0 such that for all feasible u , � u � ∞ ≤ R and � u − ¯ u � 1 ≤ ε = ⇒ J ( u ) ≥ J (¯ u ) a bounded strong minimum iff for all R > ¯ u , there exists ε > 0 such that for all feasible u , � u � ∞ ≤ R and � y [ u ] − y [¯ u ] � ∞ ≤ ε = ⇒ J ( u ) ≥ J (¯ u ) . Bounded strong min. = ⇒ Pontryagin min. = ⇒ weak min.

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions Normal cones to the constraints We set n Φ I K g = C ([0 , T ]; R n g K c = L ∞ (0 , T ; R n c K Φ = { 0 } R n Φ E × R − , − ) , − ) , and consider their normal cones at (¯ u , ¯ y ): � � Ψ = (Ψ E , Ψ I ) ∈ R n E × R n I , Ψ I Φ I (¯ N K Φ = y T ) = 0 � T � � µ ∈ M (0 , T ; R n g N K g = + ) , g ( y t ) d µ t = 0 0 � T � � ν ∈ L ∞ (0 , T ; R n c ) , N K c = c ( u t , y t ) ν t d t = 0 0 Let ¯ u be feasible, ¯ y = y [¯ u ]. The normal cone of the constraints is N = N K Φ × N K g × N K c .

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions Multipliers Assumption (Inward pointing condition) There exists v ∈ U and ε > 0 such that for a.a. t, c (¯ u t , ¯ y t ) + D u c (¯ u t , ¯ y t ) v t ≤ − ε. This assumption enables to consider multipliers (for the mixed constraint) in L ∞ (0 , T ; R n c ) instead of ( L ∞ (0 , T ; R n c )) ∗ . We define for all u ∈ R m , y ∈ R n , p ∈ R n , ν ∈ R n c the end-point Lagrangian Φ[ β, Ψ]( y T ) = βφ ( y T ) + ΨΦ( y T ), the Hamiltonian H [ p ]( u , y ) = pf ( u , y ), the augmented Hamiltonian H a [ p , ν ]( u , y ) = pf ( u , y ) + ν c ( u , y ).

Generalities Framework Weak necessary conditions Pontryagin conditions Sufficient conditions Lagrange multipliers Let λ = ( β, Ψ , µ, ν ), with β ∈ R + and (Ψ , µ, ν ) ∈ N (¯ u ), we call associated costate the solution p λ ∈ BV (0 , T ; R n ) to � D y H a [ p t , ν t ] d t + D y g ( y t ) d µ t , d p t = p T + = D y T Φ[ β, Ψ]( y T ) . We define the set Λ L of generalized Lagrange multipliers as follows: � ( β, Ψ , µ, ν ) , β ≥ 0 , (Ψ , µ, ν ) ∈ N (¯ Λ L = u ) , � D u H a [ p λ t , ν t ](¯ u t , ¯ y t ) = 0 , for a.a. t \ { (0 , 0 , 0 , 0) } .