Sensitivity analysis for relaxed optimal control problems with - PowerPoint PPT Presentation

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control Sensitivity analysis for relaxed optimal control problems with final-state constraints eric Bonnans ∗ , Laurent Pfeiffer ∗ , and Oana Serea † J. Fr´ ed´ ∗ INRIA Saclay and CMAP, Ecole Polytechnique (France), † University of Perpignan (France) May 2012

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control Introduction We consider... a family of optimal control problems (OCP), parametrized by a nonnegative variable θ close to 0, its value function V ( θ ), a strong solution ( u , y ) for the value θ = 0. Our goal: computing a second-order expansion of V ( θ ) near 0.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control Introduction The main difficulty: large perturbations in uniform norm for the control variable may occur. Our tools: the abstract methodology of sensitivity analysis, relaxation with Young measures.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control 1 Relaxation of optimal control problems Strong solutions Relaxation 2 Methodology for sensitivity analysis Upper estimate Lower estimate 3 Application to optimal control Multipliers Linearizations of the problem Decomposition principle

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control 1 Relaxation of optimal control problems Strong solutions Relaxation 2 Methodology for sensitivity analysis Upper estimate Lower estimate 3 Application to optimal control Multipliers Linearizations of the problem Decomposition principle

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control Strong solutions For a control u in L ∞ ([0 , T ] , R m ) and θ ≥ 0, consider the trajectory y [ u , θ ] solution of the following differential system: � y t = ˙ f ( u t , y t , θ ) , for a. a. t in [0 , T ] , y 0 . y 0 = Set K = { 0 n E } × R n I + . Our family of OCP’s is u ∈ L ∞ φ ( y T [ u , θ ]) , Min s.t. Φ( y T [ u ] , θ ) ∈ K . Reference problem: θ = 0.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control Strong solutions For a control u in L ∞ ([0 , T ] , R m ) and θ ≥ 0, consider the trajectory y [ u , θ ] solution of the following differential system: � y t = ˙ f ( u t , y t , θ ) , for a. a. t in [0 , T ] , y 0 . y 0 = Set K = { 0 n E } × R n I + . Our family of OCP’s is u ∈ L ∞ φ ( y T [ u , θ ]) , Min s.t. Φ( y T [ u ] , θ ) ∈ K . Reference problem: θ = 0.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control Strong solutions A control u is a R − strong solution for the reference problem if there exists η > 0 such that u is solution to the localized problem:  || u || ∞ ≤ R φ ( y T [ u , 0]) , Min   s.t. Φ( y T [ u ] , 0) ∈ K , || y [ u , 0] − y || ∞ ≤ η. Now, we fix u , R , and η .

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control Relaxation Set: B R , the ball of R m of radius R P R , the set of probabilities on B R , M R , the set of Young measures on [0 , T ] × B R , defined by L ∞ ([0 , T ] , P R ). For µ ∈ M R , denote by y [ µ, θ ] the solution to � � y t = ˙ B R f ( u , y t , θ ) d µ t ( u ) , y 0 . y 0 =

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control Relaxation Set: B R , the ball of R m of radius R P R , the set of probabilities on B R , M R , the set of Young measures on [0 , T ] × B R , defined by L ∞ ([0 , T ] , P R ). For µ ∈ M R , denote by y [ µ, θ ] the solution to � � y t = ˙ B R f ( u , y t , θ ) d µ t ( u ) , y 0 . y 0 =

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control Relaxation Example: for µ t = 1 t + 1 2 δ u 1 2 δ u 2 t , the dynamic is: y t [ µ, θ ] = 1 t , y [ µ, θ ] , θ ) + 1 2 f ( u 1 2 f ( u 2 ˙ t , y [ µ, θ ] , θ ) . and can be approximated by this control: � � (�) with probability 1/2 � � (�) with probability 1/2 0 T

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control Relaxation Example: for µ t = 1 t + 1 2 δ u 1 2 δ u 2 t , the dynamic is: y t [ µ, θ ] = 1 t , y [ µ, θ ] , θ ) + 1 2 f ( u 1 2 f ( u 2 ˙ t , y [ µ, θ ] , θ ) . and can be approximated by this control: � � (�) with probability 1/2 v (�) � � (�) with probability 1/2 0 T

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control Relaxation We relax the OCP’s as follows  V ( θ ) = Min µ ∈M R φ ( y T [ µ, θ ]) ,  s.t. Φ( y T [ µ ] , θ ) ∈ K , || y [ µ, θ ] − y || ∞ ≤ η.  Theorem Under some qualification assumptions, the relaxed and the classical OCP’s have the same value. Therefore, µ t = δ u t is a relaxed solution for θ = 0.

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control 1 Relaxation of optimal control problems Strong solutions Relaxation 2 Methodology for sensitivity analysis Upper estimate Lower estimate 3 Application to optimal control Multipliers Linearizations of the problem Decomposition principle

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control Upper estimate Consider the family of optimization problems: V ( θ ) = Min x ∈ H f ( x , θ ) s.t. g ( x , θ ) ∈ K , where K stands for inequalities and equalities. The Lagrangian is L ( x , λ, θ ) = f ( x , θ ) + � λ, g ( x , θ ) � . Consider x an optimal solution for θ = 0, Λ the set of Lagrange multipliers associated with x .

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control Upper estimate Consider the family of optimization problems: V ( θ ) = Min x ∈ H f ( x , θ ) s.t. g ( x , θ ) ∈ K , where K stands for inequalities and equalities. The Lagrangian is L ( x , λ, θ ) = f ( x , θ ) + � λ, g ( x , θ ) � . Consider x an optimal solution for θ = 0, Λ the set of Lagrange multipliers associated with x .

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control Upper estimate Let d and h be in H , set y θ = x + d θ + h θ 2 , then, f ( y θ , θ ) = f ( x , 0) + Df ( x , 0)( d , 1) θ � D x f ( x , 0) h + 1 2 D 2 f ( x , 0)( d , 1) 2 � θ 2 + o ( θ 2 ) , + g ( y θ , θ ) = g ( x , 0) + Dg ( x , 0)( d , 1) θ � D x g ( x , 0) h + 1 2 D 2 g ( x , 0)( d , 1) 2 � θ 2 + o ( θ 2 ) . +

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control Upper estimate By a regularity metric theorem, if Dg ( x , 0)( d , 1) ∈ T K ( g ( x , 0)) , and D x g ( x , 0) h + 1 � � 2 D 2 g ( x , 0)( d , 1) ∈ T 2 g ( x , 0) , Dg ( x , 0)( d , 1) , K then, there exists x θ = x + d θ + h θ 2 + o ( θ 2 ) ˜ such that g (˜ x θ , θ ) = 0 .

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control Upper estimate By a regularity metric theorem, if Dg ( x , 0)( d , 1) ∈ T K ( g ( x , 0)) , and D x g ( x , 0) h + 1 � � 2 D 2 g ( x , 0)( d , 1) ∈ T 2 g ( x , 0) , Dg ( x , 0)( d , 1) , K then, there exists x θ = x + d θ + h θ 2 + o ( θ 2 ) ˜ such that g (˜ x θ , θ ) = 0 .

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control Upper estimate This justifies the two linearized problems: at the first order, s.t. Dg ( x , 0)( d , 1) ∈ T K ( ... ) . Min d ∈ H Df ( x , 0)( d , 1) ( LP ) Its dual is: Max λ ∈ Λ D θ L ( x , λ, 0) . ( LD )

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control Upper estimate at the second order, for d in S ( LP ),  D x f ( x , 0) h + 1 2 D 2 f ( x , 0)( d , 1) 2 Min  h ∈ H 2 D 2 g ( x , 0)( d , 1) 2 ∈ T 2 D x g ( x , 0) h + 1 s.t. K ( ... ) .  ( QP ( d )) Its dual is: λ ∈ S ( LD ) D 2 L ( x , λ, 0)( d , 1) 2 . Max ( QD ( d )) Finally, � � θ 2 + o ( θ 2 ) . V ( θ ) ≤ V (0) + Val( LP ) θ + Min Val( QP ( d )) d ∈ S ( LP )

Relaxation of optimal control problems Methodology for sensitivity analysis Application to optimal control Upper estimate at the second order, for d in S ( LP ),  D x f ( x , 0) h + 1 2 D 2 f ( x , 0)( d , 1) 2 Min  h ∈ H 2 D 2 g ( x , 0)( d , 1) 2 ∈ T 2 D x g ( x , 0) h + 1 s.t. K ( ... ) .  ( QP ( d )) Its dual is: λ ∈ S ( LD ) D 2 L ( x , λ, 0)( d , 1) 2 . Max ( QD ( d )) Finally, � � θ 2 + o ( θ 2 ) . V ( θ ) ≤ V (0) + Val( LP ) θ + Min Val( QP ( d )) d ∈ S ( LP )