No-gap Second-order Optimality Conditions for State Constrained Optimal Control Problems J. Fr´ ed´ eric Bonnans Audrey Hermant INRIA Rocquencourt, France Workshop on Advances in Continuous Optimization Reykjavik, Iceland, 30 June and 1 July 2006 J. Fr´ ed´ eric Bonnans, Audrey Hermant No-gap Second-order Optimality Conditions for State Constrained
Outline 1 Presentation of the problem, motivations 2 Definitions and assumptions 3 Main result 4 Application to the shooting algorithm J. Fr´ ed´ eric Bonnans, Audrey Hermant No-gap Second-order Optimality Conditions for State Constrained
The Optimal Control Problem � T ( P ) min ℓ ( u ( t ) , y ( t )) d t + φ ( y ( T )) subject to: ( u , y ) ∈U×Y 0 y ( t ) = f ( u ( t ) , y ( t )) a.e. on [0 , T ] ; y (0) = y 0 ˙ g ( y ( t )) ≤ 0 on [0 , T ] . Control and state spaces: U := L ∞ (0 , T ; R ), Y := W 1 , ∞ (0 , T ; R n ). Assumptions: (A0) The mappings ℓ : R × R n → R , φ : R n → R , f : R × R n → R n and g : R n → R are C ∞ ; f is Lipschitz continuous. (A1) The initial condition satisfies g ( y 0 ) < 0. J. Fr´ ed´ eric Bonnans, Audrey Hermant No-gap Second-order Optimality Conditions for State Constrained
Why study Second-Order Optimality Conditions ? Second-order Sufficient Conditions: analysis of convergence of numerical algorithms, stability and sensitivity analysis. Strong second-order sufficient conditions known, in e.g. [Malanowski-Maurer et al. 1997,1998,2001,2004] ... To weaken the sufficient condition, find a Second-order Sufficient Condition as close as possible to the Second-order Necessary Condition (no gap). No-gap Second-order conditions known for mixed control-state constraints [Milyutin-Osmolovskii 1998], [Zeidan 1994] ... J. Fr´ ed´ eric Bonnans, Audrey Hermant No-gap Second-order Optimality Conditions for State Constrained
Abstract formulation of Optimal Control Problem State mapping U → Y , u �→ y u , where y u is the solution of: y u ( t ) = f ( u ( t ) , y u ( t )) ˙ for a.a. t ∈ [0 , T ]; y u (0) = y 0 Cost and constraint mappings J : U → R , G : U → C [0 , T ]: � T J ( u ) = ℓ ( u ( t ) , y u ( t )) d t + φ ( y u ( T )) ; G ( u ) = g ( y u ) . 0 Abstract formulation of ( P ) is: min u ∈U J ( u ); G ( u ) ∈ K , where K is the cone of nonpositive continuous functions C − [0 , T ]. J. Fr´ ed´ eric Bonnans, Audrey Hermant No-gap Second-order Optimality Conditions for State Constrained
Definitions (1/3) Structure of a trajectory State constraint: g ( y u ( t )) ≤ 0 , ∀ t ∈ [0 , T ] . Contact set: I ( g ( y u )) := { t ∈ [0 , T ] ; g ( y u ( t )) = 0 } . boundary arc [ τ en , τ ex ] isolated contact point { τ to } → entry and exit points → touch points Junction points: T := ∂ I ( g ( y u )) . J. Fr´ ed´ eric Bonnans, Audrey Hermant No-gap Second-order Optimality Conditions for State Constrained
Definitions (2/3) Order of the state constraint Order of the state constraint q : smallest number of time-derivation of the function t �→ g ( y u ( t )) , so that an explicit dependence in the control variable u appears. g ( j ) ( u , y ) := g ( j − 1) ( u , y ) ∈ R × R n ( y ) f ( u , y ) 1 ≤ j ≤ q , y g ( j ) g ( q ) ≡ 0 , 0 ≤ j ≤ q − 1 and �≡ 0 . u u Example of a state constraint of order q : y ( q ) ( t ) = u ( t ) ; y ( t ) ≤ 0 . J. Fr´ ed´ eric Bonnans, Audrey Hermant No-gap Second-order Optimality Conditions for State Constrained
Definitions (3/3) ( P ) min J ( u ) ; G ( u ) ∈ K Lagrangian L : U × M [0 , T ] → R L ( u , η ) := J ( u ) + � η, G ( u ) � � T � T = ℓ ( u ( t ) , y u ( t )) d t + φ ( y u ( T )) + g ( y u ( t )) d η ( t ) 0 0 Hamiltonian H : R × R n × R n ∗ → R , H ( u , y , p ) := ℓ ( u , y ) + pf ( u , y ) . Costate p u ,η : the solution in BV ([0 , T ]; R n ∗ ) of: − d p u ,η = H y ( u , y u , p u ,η ) d t + g y ( y u ) d η ; p u ,η ( T ) = φ y ( y u ( T )) . J. Fr´ ed´ eric Bonnans, Audrey Hermant No-gap Second-order Optimality Conditions for State Constrained
Assumptions (1/2) (A2) Strong convexity of the Hamiltonian w.r.t. the control variable: ∃ α > 0 such that α ≤ H uu ( w , y u ( t ) , p u ,η ( t − )) for all w ∈ R and t ∈ [0 , T ] . (A3) Constraint Regularity: ∃ γ, ε > 0 such that γ ≤ | g ( q ) u ( u ( t ) , y u ( t )) | for a.a. t , dist { t ; I ( g ( y u )) } ≤ ε. (A4) Finite set of junctions points T , and g ( y u ( T )) < 0. J. Fr´ ed´ eric Bonnans, Audrey Hermant No-gap Second-order Optimality Conditions for State Constrained
Junctions conditions results u ∈ U is a stationary point of ( P ), if there exists a Lagrange multiplier η ∈ M + [0 , T ] such that � D u L ( u , η ) = H u ( u ( · ) , y u ( · ) , p u ,η ( · )) = 0 , a.e. on [0 , T ] η ∈ N K ( G ( u )) . J. Fr´ ed´ eric Bonnans, Audrey Hermant No-gap Second-order Optimality Conditions for State Constrained
Junctions conditions results u ∈ U is a stationary point of ( P ), if there exists a Lagrange multiplier η ∈ M + [0 , T ] such that � D u L ( u , η ) = H u ( u ( · ) , y u ( · ) , p u ,η ( · )) = 0 , a.e. on [0 , T ] η ∈ N K ( G ( u )) . Proposition (Jacobson, Lele and Speyer, 1971) Let ( u , η ) ∈ U × M + [0 , T ] a stationary point and its (unique) Lagrange multiplier, satisfying (A2)-(A4). Then: u and η are C ∞ on [0 , T ] \ T ⇒ d η = η 0 d t + � τ ∈T ν τ δ τ u , . . . , u ( q − 2) are continuous at junctions times; If q is odd, u ( q − 1) and η are continuous at entry/exit times; If q = 1 , η is continous at touch points. J. Fr´ ed´ eric Bonnans, Audrey Hermant No-gap Second-order Optimality Conditions for State Constrained
Junctions conditions results Proposition (Jacobson, Lele and Speyer, 1971) Let ( u , η ) ∈ U × M + [0 , T ] a stationary point and its (unique) Lagrange multiplier, satisfying (A2)-(A4). Then: u and η are C ∞ on [0 , T ] \ T ⇒ d η = η 0 d t + � τ ∈T ν τ δ τ u , . . . , u ( q − 2) are continuous at junctions times; If q is odd, u ( q − 1) and η are continuous at entry/exit times; If q = 1 , η is continous at touch points. Consequence: the time-derivatives of t �→ g ( y u ( t )) are continuous at entry/exit points until order ˆ q , with ˆ q := 2 q − 2 if q is even, and ˆ q = 2 q − 1 if q is odd. A touch point τ is said to be essential, if τ ∈ supp ( η ) (equivalently, if ν τ � = 0 or if η is discontinuous at τ ). J. Fr´ ed´ eric Bonnans, Audrey Hermant No-gap Second-order Optimality Conditions for State Constrained
Assumptions (2/2) (A5)(i) Non-Tangentiality condition at entry/exit points: q +1 d ˆ q +1 d ˆ q +1 ( − 1) ˆ q +1 g ( y u ( t )) | t = τ − en < 0 ; q +1 g ( y u ( t )) | t = τ + ex < 0 d t ˆ d t ˆ (A5)(ii) Reducibility Condition at essential touch points ( q ≥ 2): d 2 to = g (2) ( u ( τ ess to ) , y u ( τ ess d t 2 g ( y u ( t )) | t = τ ess to )) < 0 (A6) Strict Complementarity on boundary arcs: int I ( g ( y u )) = ∪ [ τ en , τ ex ] ⊂ supp ( η ) J. Fr´ ed´ eric Bonnans, Audrey Hermant No-gap Second-order Optimality Conditions for State Constrained
Main Result For u ∈ U and v ∈ L 2 (0 , T ), the linearized state z u , v is the solution in H 1 (0 , T ; R n ) of z u , v = f u ( u , y u ) v + f y ( u , y u ) z u , v on [0 , T ] ; z u , v (0) = 0 . ˙ Note that ( DG ( u ) v )( t ) = g y ( y u ( t )) z u , v ( t ). For u a stationary solution with multiplier η , the critical cone is C 2 ( u ) := { v ∈ L 2 ; DG ( u ) v ∈ T K ( G ( u )) ; DJ ( u ) v ≤ 0 } = { v ∈ L 2 ; DG ( u ) v ∈ T K ( G ( u )) ; supp ( η ) ⊂ I 2 u , v } with the second-order contact set: I 2 u , v := { t ∈ I ( g ( y u )) ; g y ( y u ( t )) z u , v ( t ) = 0 } . J. Fr´ ed´ eric Bonnans, Audrey Hermant No-gap Second-order Optimality Conditions for State Constrained
Main Result Theorem (No-gap Second-order Necessary Condition) Let u ∈ U a local optimal solution of ( P ) , with (unique) Lagrange multiplier η , satisfying (A1)-(A6). Denote by T ess the set of to essential touch points of the trajectory ( u , y u ) and ν τ = [ η ( τ )] . Then, for all v ∈ C 2 ( u ) : ( g (1) y ( y u ( τ )) z u , v ( τ )) 2 D 2 � uu L ( u , η )( v , v ) − ≥ 0 . ν τ g (2) ( u ( τ ) , y u ( τ )) τ ∈T ess to J. Fr´ ed´ eric Bonnans, Audrey Hermant No-gap Second-order Optimality Conditions for State Constrained
Main Result Theorem (No-gap Second-order Necessary Condition) Let u ∈ U a local optimal solution of ( P ) , with (unique) Lagrange multiplier η , satisfying (A1)-(A6). Denote by T ess the set of to essential touch points of the trajectory ( u , y u ) and ν τ = [ η ( τ )] . Then, for all v ∈ C 2 ( u ) : ( g (1) y ( y u ( τ )) z u , v ( τ )) 2 D 2 � uu L ( u , η )( v , v ) − ≥ 0 . ν τ g (2) ( u ( τ ) , y u ( τ )) τ ∈T ess to Additional term (in blue), called the curvature term [Kawasaki, 1988]. Only essential touch points have a contribution to the curvature term (the contribution of boundary arcs is null). J. Fr´ ed´ eric Bonnans, Audrey Hermant No-gap Second-order Optimality Conditions for State Constrained
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