Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Isentropic gas dynamic (p-system) Gas in a clinder with moving piston (in Lagrangian coord.) � ∂ t v − ∂ x u = 0 x ∈ ] 0 , h [ ∂ t u + ∂ x p ( v ) = 0 v specific volume, u speed, p pressure gas x x = 0 x = h Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Stabilization problem for gas dynamic a control acting only on speed u at x = h : u ( t , h ) = α ( t ) . a reflection condition at x = 0: u ( t , 0 ) = 0 . Pb: given v ( 0 , x ) = ¯ u ( 0 , x ) = ¯ v ( x ) , u ( x ) x ∈ ] 0 , h [ , Stabilize the system at an equilibrium ( v , u ) = ( v ∗ , 0 ) . Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Stabilization problem for gas dynamic a control acting only on speed u at x = h : u ( t , h ) = α ( t ) . a reflection condition at x = 0: u ( t , 0 ) = 0 . Pb: given v ( 0 , x ) = ¯ u ( 0 , x ) = ¯ v ( x ) , u ( x ) x ∈ ] 0 , h [ , Stabilize the system at an equilibrium ( v , u ) = ( v ∗ , 0 ) . Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Stabilization problem for gas dynamic a control acting only on speed u at x = h : u ( t , h ) = α ( t ) . a reflection condition at x = 0: u ( t , 0 ) = 0 . Pb: given v ( 0 , x ) = ¯ u ( 0 , x ) = ¯ v ( x ) , u ( x ) x ∈ ] 0 , h [ , Stabilize the system at an equilibrium ( v , u ) = ( v ∗ , 0 ) . Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Stabilization problem for gas dynamic a control acting only on speed u at x = h : u ( t , h ) = α ( t ) . a reflection condition at x = 0: u ( t , 0 ) = 0 . Pb: given v ( 0 , x ) = ¯ u ( 0 , x ) = ¯ v ( x ) , u ( x ) x ∈ ] 0 , h [ , Stabilize the system at an equilibrium ( v , u ) = ( v ∗ , 0 ) . Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Multicomponent chromatoghraphy Separate two chemical species in a fluid by selective absorption on a solid medium S 1 S 2 x x = 0 x = L INLET OUTLET � � γ c 1 ∂ x c 1 + ∂ t = 0 1 + c 1 + c 2 x ∈ ] 0 , L [ � � c 2 ∂ x c 2 + ∂ t = 0 1 + c 1 + c 2 c i concentration solute S i ( γ ∈ ] 0 , 1 ]) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Multicomponent chromatoghraphy Separate two chemical species in a fluid by selective absorption on a solid medium S 1 S 2 x x = 0 x = L INLET OUTLET � � γ c 1 ∂ x c 1 + ∂ t = 0 1 + c 1 + c 2 x ∈ ] 0 , L [ � � c 2 ∂ x c 2 + ∂ t = 0 1 + c 1 + c 2 c i concentration solute S i ( γ ∈ ] 0 , 1 ]) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Multicomponent chromatoghraphy Temple system with GNL characteristic fields Ree, Aris & Amundson (1986, 1989) control concentration solute S i entering the tube at x = 0: c i ( t , 0 ) = α i ( t ) . x ATTAINABLE SET A ( T ) L c i = c i c i = α i t 0 T Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Multicomponent chromatoghraphy Temple system with GNL characteristic fields Ree, Aris & Amundson (1986, 1989) control concentration solute S i entering the tube at x = 0: c i ( t , 0 ) = α i ( t ) . x ATTAINABLE SET A ( T ) L c i = c i c i = α i t 0 T Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Multicomponent chromatoghraphy Temple system with GNL characteristic fields Ree, Aris & Amundson (1986, 1989) control concentration solute S i entering the tube at x = 0: c i ( t , 0 ) = α i ( t ) . x ATTAINABLE SET A ( T ) L c i = c i c i = α i t 0 T Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Optimization problem for chromatography Maximize separation of solutes at time T � � x max ( c 1 ( T , ξ ) − c 2 ( T , ξ )) d ξ + x , α 0 � L � + ( c 2 ( T , ξ ) − c 1 ( T , ξ )) d ξ x � � γ c 1 ∂ x c 1 + ∂ t = 0 , 1 + c 1 + c 2 � � c 2 ∂ x c 2 + ∂ t = 0 , 1 + c 1 + c 2 x ∈ ] 0 , L [ , c i ( 0 , x ) = ¯ c i , c i ( t , 0 ) = α i ( t ) . Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Outline Introduction 1 General setting Physical Motivations Main problems Controllability & Stabilizability 2 Exact controllability Asymptotic stabilizability Optimal control problems 3 Generalized tangent vectors Linearized evolution equations Pontryagin Maximum Principle for Temple systems 4 Temple systems Evolution of first order variations Pontryagin Maximum Principle Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Two Classes of Problems 1. Controllability & Stabilizability 2. Optimal control problems (Mostly boundary controls will be considered) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Two Classes of Problems 1. Controllability & Stabilizability 2. Optimal control problems (Mostly boundary controls will be considered) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Two Classes of Problems 1. Controllability & Stabilizability 2. Optimal control problems (Mostly boundary controls will be considered) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Boundary Controllability & Stabilizability Boundary conditions (non characteristic boundary) b j ( u ( t , ψ j ( t ))) = g j ( α j ( t )) ( j = 0 , 1 ) Given: initial datum u desired terminal profile Φ (e.g. a constant state Φ( x ) ≡ u ∗ ) Do exist: boundary controls α j at x = ψ j so that solution u α ( t , x ) of corresponding IBVP satisfies: Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Boundary Controllability & Stabilizability Boundary conditions (non characteristic boundary) b j ( u ( t , ψ j ( t ))) = g j ( α j ( t )) ( j = 0 , 1 ) Given: initial datum u desired terminal profile Φ (e.g. a constant state Φ( x ) ≡ u ∗ ) Do exist: boundary controls α j at x = ψ j so that solution u α ( t , x ) of corresponding IBVP satisfies: Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Boundary Controllability & Stabilizability Boundary conditions (non characteristic boundary) b j ( u ( t , ψ j ( t ))) = g j ( α j ( t )) ( j = 0 , 1 ) Given: initial datum u desired terminal profile Φ (e.g. a constant state Φ( x ) ≡ u ∗ ) Do exist: boundary controls α j at x = ψ j so that solution u α ( t , x ) of corresponding IBVP satisfies: Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Boundary Controllability & Stabilizability Boundary conditions (non characteristic boundary) b j ( u ( t , ψ j ( t ))) = g j ( α j ( t )) ( j = 0 , 1 ) Given: initial datum u desired terminal profile Φ (e.g. a constant state Φ( x ) ≡ u ∗ ) Do exist: boundary controls α j at x = ψ j so that solution u α ( t , x ) of corresponding IBVP satisfies: Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Boundary Controllability & Stabilizability u α ( T , · ) = Φ (finite time exact controllability) t T b 0 ( u α ) = g 0 ( α 0 ) b 1 ( u α ) = g 1 ( α 1 ) u α = u x x = ψ 0 x = ψ 1 or t →∞ u α ( t , · ) = Φ ? lim (asymptotic stabilizability) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Boundary Controllability & Stabilizability u α ( T , · ) = Φ (finite time exact controllability) t T b 0 ( u α ) = g 0 ( α 0 ) b 1 ( u α ) = g 1 ( α 1 ) u α = u x x = ψ 0 x = ψ 1 or t →∞ u α ( t , · ) = Φ ? lim (asymptotic stabilizability) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Boundary Controllability & Stabilizability u α ( T , · ) = Φ (finite time exact controllability) t T b 0 ( u α ) = g 0 ( α 0 ) b 1 ( u α ) = g 1 ( α 1 ) u α = u x x = ψ 0 x = ψ 1 or t →∞ u α ( t , · ) = Φ ? lim (asymptotic stabilizability) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Optimization problem � � max J ( u , z , α ) : z ∈ Z , α ∈ A � T � + ∞ � + ∞ � � J ( u , z , α ) = L ( x , u , z ) dxdt + Φ x , u ( T , x ) dx + 0 0 0 � T � � + Ψ u ( t , 0 ) , α ( t ) dt 0 single boundary ψ 0 ≡ 0 L , Φ , Ψ smooth A ⊂ L ∞ ( 0 , T ) admissible boundary controls at x = 0 Z ⊂ L 1 loc (] 0 , + ∞ [ × R ) admissible distributed controls Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Optimization problem � � max J ( u , z , α ) : z ∈ Z , α ∈ A � T � + ∞ � + ∞ � � J ( u , z , α ) = L ( x , u , z ) dxdt + Φ x , u ( T , x ) dx + 0 0 0 � T � � + Ψ u ( t , 0 ) , α ( t ) dt 0 single boundary ψ 0 ≡ 0 L , Φ , Ψ smooth A ⊂ L ∞ ( 0 , T ) admissible boundary controls at x = 0 Z ⊂ L 1 loc (] 0 , + ∞ [ × R ) admissible distributed controls Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Optimization problem � � max J ( u , z , α ) : z ∈ Z , α ∈ A � T � + ∞ � + ∞ � � J ( u , z , α ) = L ( x , u , z ) dxdt + Φ x , u ( T , x ) dx + 0 0 0 � T � � + Ψ u ( t , 0 ) , α ( t ) dt 0 single boundary ψ 0 ≡ 0 L , Φ , Ψ smooth A ⊂ L ∞ ( 0 , T ) admissible boundary controls at x = 0 Z ⊂ L 1 loc (] 0 , + ∞ [ × R ) admissible distributed controls Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Optimization problem � � max J ( u , z , α ) : z ∈ Z , α ∈ A � T � + ∞ � + ∞ � � J ( u , z , α ) = L ( x , u , z ) dxdt + Φ x , u ( T , x ) dx + 0 0 0 � T � � + Ψ u ( t , 0 ) , α ( t ) dt 0 single boundary ψ 0 ≡ 0 L , Φ , Ψ smooth A ⊂ L ∞ ( 0 , T ) admissible boundary controls at x = 0 Z ⊂ L 1 loc (] 0 , + ∞ [ × R ) admissible distributed controls Fabio Ancona Control Problems for Hyperbolic Equations
Introduction General setting Controllability & Stabilizability Physical Motivations Optimal control problems Main problems Pontryagin Maximum Principle for Temple systems Optimization problem � � max J ( u , z , α ) : z ∈ Z , α ∈ A � T � + ∞ � + ∞ � � J ( u , z , α ) = L ( x , u , z ) dxdt + Φ x , u ( T , x ) dx + 0 0 0 � T � � + Ψ u ( t , 0 ) , α ( t ) dt 0 single boundary ψ 0 ≡ 0 L , Φ , Ψ smooth A ⊂ L ∞ ( 0 , T ) admissible boundary controls at x = 0 Z ⊂ L 1 loc (] 0 , + ∞ [ × R ) admissible distributed controls Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Exact controllability Optimal control problems Asymptotic stabilizability Pontryagin Maximum Principle for Temple systems Outline Introduction 1 General setting Physical Motivations Main problems Controllability & Stabilizability 2 Exact controllability Asymptotic stabilizability Optimal control problems 3 Generalized tangent vectors Linearized evolution equations Pontryagin Maximum Principle for Temple systems 4 Temple systems Evolution of first order variations Pontryagin Maximum Principle Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Exact controllability Optimal control problems Asymptotic stabilizability Pontryagin Maximum Principle for Temple systems Finite time exact controllability to constant states u ∗ 1. Quasilinear systems ∂ t u + A ( u ) ∂ x u = h ( u ) x ∈ ] a , b [ , with suff. small C 1 initial data u (Cirinà, 1969; T.Li, B. Rao & co, 2002-2008; M.Gugat & G. Leugering, 2003) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Exact controllability Optimal control problems Asymptotic stabilizability Pontryagin Maximum Principle for Temple systems Finite time exact controllability to constant states u ∗ 2. Nonlinear scalar convex con laws and GNL Temple systems ∂ t u + ∂ x ( f ( u )) = 0 x ∈ ] a , b [ , with initial data u ∈ L ∞ ( L 1 ) (discontinuous entropy weak solutions) (F .A., A.Marson, 1998; T. Horsin, 1998; F .A. & G.M. Coclite, 2005) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Exact controllability Optimal control problems Asymptotic stabilizability Pontryagin Maximum Principle for Temple systems Finite time exact controllability to constant states u ∗ 3. Isentropic gas dynamic (in Eulerian coord.) ∂ t ρ + ∂ x ( ρ u ) = 0 � ρ u 2 + K ρ γ � ∂ t ( ρ u ) + ∂ x = 0 with T.V. { bdr controls }≫ � u ∗ − u � ∞ (strong perturbation of the solution) (O. Glass, 2006) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Exact controllability Optimal control problems Asymptotic stabilizability Pontryagin Maximum Principle for Temple systems NO exact controllability to constant states u ∗ 4. Isentropic gas dynamic for a polytropic gas (in Eulerian coord.) ∂ t ρ + ∂ x ( ρ u ) = 0 � u 2 � K γ − 1 ρ γ − 1 ∂ t u + ∂ x 2 + = 0 ∃ initial datum so that corresponding sol. has dense set of discontinuities, whatever bdr controls are prescribed (A.Bressan & G.M.Coclite, 2002) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Exact controllability Optimal control problems Asymptotic stabilizability Pontryagin Maximum Principle for Temple systems Outline Introduction 1 General setting Physical Motivations Main problems Controllability & Stabilizability 2 Exact controllability Asymptotic stabilizability Optimal control problems 3 Generalized tangent vectors Linearized evolution equations Pontryagin Maximum Principle for Temple systems 4 Temple systems Evolution of first order variations Pontryagin Maximum Principle Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Exact controllability Optimal control problems Asymptotic stabilizability Pontryagin Maximum Principle for Temple systems 1. Stabilizability with total control on both boundaries Asymptotic stabilizability around a constant state with exponential rate (A.Bressan & G.M.Coclite, 2002) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Exact controllability Optimal control problems Asymptotic stabilizability Pontryagin Maximum Principle for Temple systems 2. Stabilizability with total control on single boundary b 0 ( u ( t , ψ 0 ( t ))) = 0 , b 1 ( u ( t , ψ 1 ( t ))) = g ( α ( t )) • Assume Dg ( α ) has full rank ⇒ full control on waves entering the domain from x = ψ 1 λ p t λ 2 λ 1 b 1 ( u ) = g ( α ) x x = ψ 0 x = ψ 1 Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Exact controllability Optimal control problems Asymptotic stabilizability Pontryagin Maximum Principle for Temple systems 2. Stabilizability with total control on single boundary b 0 ( u ( t , ψ 0 ( t ))) = 0 , b 1 ( u ( t , ψ 1 ( t ))) = g ( α ( t )) • Assume Dg ( α ) has full rank ⇒ full control on waves entering the domain from x = ψ 1 λ p t λ 2 λ 1 b 1 ( u ) = g ( α ) x x = ψ 0 x = ψ 1 Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Exact controllability Optimal control problems Asymptotic stabilizability Pontryagin Maximum Principle for Temple systems 2. Stabilizability with total control on single boundary • Assume p ≥ n − p and Db 0 ( u ) with maximum rank � � rk Db 0 · r 1 ( u ) , . . . , Db 0 · r p ( u ) = n − p t λ n use control α acting at x = ψ 1 to generate first p components of u ∗ λ p + 1 use reflections at x = ψ 0 λ p to generate remaining b 1 = g ( α ) λ 1 n − p components of u ∗ x x = ψ 0 x = ψ 1 Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Exact controllability Optimal control problems Asymptotic stabilizability Pontryagin Maximum Principle for Temple systems 2. Stabilizability with total control on single boundary • Assume p ≥ n − p and Db 0 ( u ) with maximum rank � � rk Db 0 · r 1 ( u ) , . . . , Db 0 · r p ( u ) = n − p t λ n use control α acting at x = ψ 1 to generate first p components of u ∗ λ p + 1 use reflections at x = ψ 0 λ p to generate remaining b 1 = g ( α ) λ 1 n − p components of u ∗ x x = ψ 0 x = ψ 1 Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Exact controllability Optimal control problems Asymptotic stabilizability Pontryagin Maximum Principle for Temple systems 2. Stabilizability with total control on single boundary • Assume p ≥ n − p and Db 0 ( u ) with maximum rank � � rk Db 0 · r 1 ( u ) , . . . , Db 0 · r p ( u ) = n − p t λ n use control α acting at x = ψ 1 to generate first p components of u ∗ λ p + 1 use reflections at x = ψ 0 λ p to generate remaining b 1 = g ( α ) λ 1 n − p components of u ∗ x x = ψ 0 x = ψ 1 Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Exact controllability Optimal control problems Asymptotic stabilizability Pontryagin Maximum Principle for Temple systems 2. Stabilizability with total control on single boundary • Nonlinear system ⇒ waves produced by bndr control interact with each other generating new waves (2nd generation waves) ∃ τ , bdr control α s.t. u − u ∗ | 2 T.V. u α ( τ, · ) = O ( 1 ) · | ¯ � u α ( τ, · ) − u ∗ � ∞ = O ( 1 ) · | ¯ u − u ∗ | 2 ⇓ Asymptotic stabilization to equilibrium u ∗ ( b 0 ( u ∗ ) = 0) (F .A. & A.Marson, 2007) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Exact controllability Optimal control problems Asymptotic stabilizability Pontryagin Maximum Principle for Temple systems 2. Stabilizability with total control on single boundary • Nonlinear system ⇒ waves produced by bndr control interact with each other generating new waves (2nd generation waves) ∃ τ , bdr control α s.t. u − u ∗ | 2 T.V. u α ( τ, · ) = O ( 1 ) · | ¯ � u α ( τ, · ) − u ∗ � ∞ = O ( 1 ) · | ¯ u − u ∗ | 2 ⇓ Asymptotic stabilization to equilibrium u ∗ ( b 0 ( u ∗ ) = 0) (F .A. & A.Marson, 2007) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Exact controllability Optimal control problems Asymptotic stabilizability Pontryagin Maximum Principle for Temple systems 2. Stabilizability with total control on single boundary • Nonlinear system ⇒ waves produced by bndr control interact with each other generating new waves (2nd generation waves) ∃ τ , bdr control α s.t. u − u ∗ | 2 T.V. u α ( τ, · ) = O ( 1 ) · | ¯ � u α ( τ, · ) − u ∗ � ∞ = O ( 1 ) · | ¯ u − u ∗ | 2 ⇓ Asymptotic stabilization to equilibrium u ∗ ( b 0 ( u ∗ ) = 0) (F .A. & A.Marson, 2007) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Optimization problem � T � + ∞ � + ∞ � � L ( x , u , z ) dxdt + Φ x , u ( T , x ) dx + max z ∈Z , α ∈A 0 0 0 � T � � + Ψ u ( t , 0 ) , α ( t ) dt 0 u = u z ,α ( t , x ) solution to ( ψ 0 ≡ 0): ∂ t u + ∂ x f ( u ) = h ( x , u , z ) , u ( 0 , x ) = ¯ u ( x ) , b ( u ( t , 0 )) = α ( t ) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Optimization problem � T � + ∞ � + ∞ � � L ( x , u , z ) dxdt + Φ x , u ( T , x ) dx + max z ∈Z , α ∈A 0 0 0 � T � � + Ψ u ( t , 0 ) , α ( t ) dt 0 u = u z ,α ( t , x ) solution to ( ψ 0 ≡ 0): ∂ t u + ∂ x f ( u ) = h ( x , u , z ) , u ( 0 , x ) = ¯ u ( x ) , b ( u ( t , 0 )) = α ( t ) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Goals 1. Establish existence of optimal solutions 2. Seek necessary conditions for optimality of controls � z , � α 3. Provide algorithm to construct (almost) optimal solutions Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Goals 1. Establish existence of optimal solutions 2. Seek necessary conditions for optimality of controls � z , � α 3. Provide algorithm to construct (almost) optimal solutions Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Goals 1. Establish existence of optimal solutions 2. Seek necessary conditions for optimality of controls � z , � α 3. Provide algorithm to construct (almost) optimal solutions Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Main difficulties Lack of regularity of sol’ns to cons. laws u t shock wave f ′ ( u 0 ) compression wave u 0 ( 0 , · ) u 0 ( t , · ) u 0 x → ∞ x x x 0 x 0 Non differentiability of input-to-trajectory map ( z , α ) �→ u z ,α in any natural Banach space Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Main difficulties Lack of regularity of sol’ns to cons. laws u t shock wave f ′ ( u 0 ) compression wave u 0 ( 0 , · ) u 0 ( t , · ) u 0 x → ∞ x x x 0 x 0 Non differentiability of input-to-trajectory map ( z , α ) �→ u z ,α in any natural Banach space Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Non differentiability � u 2 � u θ ( x ) . u ( 0 , x ) = ¯ ∂ t u + ∂ x = 0 , = ( 1 + θ ) x · χ [ 0 , 1 ] ( x ) 2 (1) Sol. to (1): ( 1 + θ ) x 1 + ( 1 + θ ) t · χ [ 0 , √ u θ ( t , x ) = 1 +( 1 + θ ) t ] ( x ) Notice: u θ is differentiable in L 1 at θ = 0 ¯ u θ − ¯ u 0 − θ ¯ u 0 � L 1 � ¯ lim = 0 θ θ → 0 Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Non differentiability � u 2 � u θ ( x ) . u ( 0 , x ) = ¯ ∂ t u + ∂ x = 0 , = ( 1 + θ ) x · χ [ 0 , 1 ] ( x ) 2 (1) Sol. to (1): ( 1 + θ ) x 1 + ( 1 + θ ) t · χ [ 0 , √ u θ ( t , x ) = 1 +( 1 + θ ) t ] ( x ) Notice: u θ is differentiable in L 1 at θ = 0 ¯ u θ − ¯ u 0 − θ ¯ u 0 � L 1 � ¯ lim = 0 θ θ → 0 Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Non differentiability � u 2 � u θ ( x ) . u ( 0 , x ) = ¯ ∂ t u + ∂ x = 0 , = ( 1 + θ ) x · χ [ 0 , 1 ] ( x ) 2 (1) Sol. to (1): ( 1 + θ ) x 1 + ( 1 + θ ) t · χ [ 0 , √ u θ ( t , x ) = 1 +( 1 + θ ) t ] ( x ) Notice: u θ is differentiable in L 1 at θ = 0 ¯ u θ − ¯ u 0 − θ ¯ u 0 � L 1 � ¯ lim = 0 θ θ → 0 Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Non differentiability The location of the jump in u θ ( t , · ) depends on θ u u θ ( t , · ) u 0 ( t , · ) ≈ θ · ( shift rate ) √ � x 1 + t 1 + ( 1 + θ ) t ⇒ u θ ( t , · ) is NOT diff. in L 1 at θ = 0 for t > 0 Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Non differentiability The location of the jump in u θ ( t , · ) depends on θ u u θ ( t , · ) u 0 ( t , · ) ≈ θ · ( shift rate ) √ � x 1 + t 1 + ( 1 + θ ) t ⇒ u θ ( t , · ) is NOT diff. in L 1 at θ = 0 for t > 0 Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Non differentiability u θ ( t , · ) − u 0 ( t , · ) lim θ θ → 0 yields a measure µ t with a nonzero singular part located at the √ of u 0 ( t , · ) point of jump x ( t ) = 1 + t � � � · d ( µ t ) s = ∆ u 0 ( t , x ( t )) � 1 + ( 1 + θ ) t · δ x ( t ) � � �� � d θ θ = 0 � �� � size of the jump shift rate t = 2 ( 1 + t ) · δ x ( t ) � � 1 ∆ u 0 ( t , x ( t )) = u 0 ( t , x ( t ) − ) − u 0 ( t , x ( t )+) = √ 1 + t Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Non differentiability u θ ( t , · ) − u 0 ( t , · ) lim θ θ → 0 yields a measure µ t with a nonzero singular part located at the √ of u 0 ( t , · ) point of jump x ( t ) = 1 + t � � � · d ( µ t ) s = ∆ u 0 ( t , x ( t )) � 1 + ( 1 + θ ) t · δ x ( t ) � � �� � d θ θ = 0 � �� � size of the jump shift rate t = 2 ( 1 + t ) · δ x ( t ) � � 1 ∆ u 0 ( t , x ( t )) = u 0 ( t , x ( t ) − ) − u 0 ( t , x ( t )+) = √ 1 + t Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Non differentiability u θ ( t , · ) − u 0 ( t , · ) lim θ θ → 0 yields a measure µ t with a nonzero singular part located at the √ of u 0 ( t , · ) point of jump x ( t ) = 1 + t � � � · d ( µ t ) s = ∆ u 0 ( t , x ( t )) � 1 + ( 1 + θ ) t · δ x ( t ) � � �� � d θ θ = 0 � �� � size of the jump shift rate t = 2 ( 1 + t ) · δ x ( t ) � � 1 ∆ u 0 ( t , x ( t )) = u 0 ( t , x ( t ) − ) − u 0 ( t , x ( t )+) = √ 1 + t Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Outline Introduction 1 General setting Physical Motivations Main problems Controllability & Stabilizability 2 Exact controllability Asymptotic stabilizability Optimal control problems 3 Generalized tangent vectors Linearized evolution equations Pontryagin Maximum Principle for Temple systems 4 Temple systems Evolution of first order variations Pontryagin Maximum Principle Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Generalized tangent vectors A generalized tangent vector generated by a family of � u θ � , with u θ ( t ) − u 0 ( t ) solutions ⇀ µ t , is an element θ ( v , ξ ) ∈ L 1 ( R ) × R ♯ jumps in u v (vertical displacement) takes into account of the absolutely continuous part of µ t ξ (horizontal displacement) takes into account of the singular part of µ t (no Cantor part in µ t ) (A.Bressan & A.Marson, 1995) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Generalized tangent vectors A generalized tangent vector generated by a family of � u θ � , with u θ ( t ) − u 0 ( t ) solutions ⇀ µ t , is an element θ ( v , ξ ) ∈ L 1 ( R ) × R ♯ jumps in u v (vertical displacement) takes into account of the absolutely continuous part of µ t ξ (horizontal displacement) takes into account of the singular part of µ t (no Cantor part in µ t ) (A.Bressan & A.Marson, 1995) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Generalized tangent vectors A generalized tangent vector generated by a family of � u θ � , with u θ ( t ) − u 0 ( t ) solutions ⇀ µ t , is an element θ ( v , ξ ) ∈ L 1 ( R ) × R ♯ jumps in u v (vertical displacement) takes into account of the absolutely continuous part of µ t ξ (horizontal displacement) takes into account of the singular part of µ t (no Cantor part in µ t ) (A.Bressan & A.Marson, 1995) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Generalized tangent vectors A generalized tangent vector generated by a family of � u θ � , with u θ ( t ) − u 0 ( t ) solutions ⇀ µ t , is an element θ ( v , ξ ) ∈ L 1 ( R ) × R ♯ jumps in u v (vertical displacement) takes into account of the absolutely continuous part of µ t ξ (horizontal displacement) takes into account of the singular part of µ t (no Cantor part in µ t ) (A.Bressan & A.Marson, 1995) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Vertical displacement u ≈ θ v u 0 ( t , · ) ≈ θ v u θ ( t , · ) x u θ ( t , x ) − u 0 ( t , x ) v ( t , x ) = lim θ θ → 0 Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Vertical displacement u ≈ θ v u 0 ( t , · ) ≈ θ v u θ ( t , · ) x u θ ( t , x ) − u 0 ( t , x ) v ( t , x ) = lim θ θ → 0 Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Horizontal displacement u u 0 ( t , · ) ≈ θξ β ≈ θξ α u θ ( t , · ) x x θ x 0 x 0 x θ α α β β x θ α ( t ) − x 0 α ( t ) ξ α ( t ) = lim θ θ → 0 rates of horizontal displacement of locations x θ 1 ( t ) < · · · > x θ N ( t ) of jumps in u θ ( t , · ) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Horizontal displacement u u 0 ( t , · ) ≈ θξ β ≈ θξ α u θ ( t , · ) x x θ x 0 x 0 x θ α α β β x θ α ( t ) − x 0 α ( t ) ξ α ( t ) = lim θ θ → 0 rates of horizontal displacement of locations x θ 1 ( t ) < · · · > x θ N ( t ) of jumps in u θ ( t , · ) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Admissible variations � ξ γ t ( τ 0 , η 0 ) ( τ θ , η θ ) ξ β ξ α [ x α x θ x θ x β β α x 0 � u θ ( t ) ≈ u 0 ( t ) + θ v ( t ) + ∆ u 0 ( t , x α ( t )) · χ [ x 0 ( t )+ θξ α ( t ) , x 0 ( t )] ξ α < 0 � ∆ u 0 ( t , x α ( t )) · χ [ x 0 ( t ) , x 0 ( t )+ θξ α ( t )] + ξ α > 0 Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Admissible variations � ξ γ t ( τ 0 , η 0 ) ( τ θ , η θ ) ξ β ξ α [ x α x θ x θ x β β α x 0 � u θ ( t ) ≈ u 0 ( t ) + θ v ( t ) + ∆ u 0 ( t , x α ( t )) · χ [ x 0 ( t )+ θξ α ( t ) , x 0 ( t )] ξ α < 0 � ∆ u 0 ( t , x α ( t )) · χ [ x 0 ( t ) , x 0 ( t )+ θξ α ( t )] + ξ α > 0 Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Admissible variations � ξ γ t ( τ 0 , η 0 ) ( τ θ , η θ ) ξ β ξ α [ x α x θ x θ x β β α x 0 � u θ ( t ) ≈ u 0 ( t ) + θ v ( t ) + ∆ u 0 ( t , x α ( t )) · χ [ x 0 ( t )+ θξ α ( t ) , x 0 ( t )] ξ α < 0 � ∆ u 0 ( t , x α ( t )) · χ [ x 0 ( t ) , x 0 ( t )+ θξ α ( t )] + ξ α > 0 Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Admissible variations � ξ γ t ( τ 0 , η 0 ) ( τ θ , η θ ) ξ β ξ α [ x α x θ x θ x β β α x 0 � u θ ( t ) ≈ u 0 ( t ) + θ v ( t ) + ∆ u 0 ( t , x α ( t )) · χ [ x 0 ( t )+ θξ α ( t ) , x 0 ( t )] ξ α < 0 � ∆ u 0 ( t , x α ( t )) · χ [ x 0 ( t ) , x 0 ( t )+ θξ α ( t )] + ξ α > 0 Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Outline Introduction 1 General setting Physical Motivations Main problems Controllability & Stabilizability 2 Exact controllability Asymptotic stabilizability Optimal control problems 3 Generalized tangent vectors Linearized evolution equations Pontryagin Maximum Principle for Temple systems 4 Temple systems Evolution of first order variations Pontryagin Maximum Principle Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Evolution of generalized tangent vectors If u θ (¯ t , · ) generates a generalized tangent vector discontinuities of u 0 interact at most two at the time u θ is piecewise Lipschitz with uniform in θ Lipschitz constant outside the discontinuities Then u θ ( t , · ) generates a generalized tangent vector � � for t > ¯ v ( t , · ) , ξ ( t ) t (A.Bressan & A.Marson, 1995) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Evolution of generalized tangent vectors If u θ (¯ t , · ) generates a generalized tangent vector discontinuities of u 0 interact at most two at the time u θ is piecewise Lipschitz with uniform in θ Lipschitz constant outside the discontinuities Then u θ ( t , · ) generates a generalized tangent vector � � for t > ¯ v ( t , · ) , ξ ( t ) t (A.Bressan & A.Marson, 1995) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Evolution of generalized tangent vectors If u θ (¯ t , · ) generates a generalized tangent vector discontinuities of u 0 interact at most two at the time u θ is piecewise Lipschitz with uniform in θ Lipschitz constant outside the discontinuities Then u θ ( t , · ) generates a generalized tangent vector � � for t > ¯ v ( t , · ) , ξ ( t ) t (A.Bressan & A.Marson, 1995) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Evolution of generalized tangent vectors If u θ (¯ t , · ) generates a generalized tangent vector discontinuities of u 0 interact at most two at the time u θ is piecewise Lipschitz with uniform in θ Lipschitz constant outside the discontinuities Then u θ ( t , · ) generates a generalized tangent vector � � for t > ¯ v ( t , · ) , ξ ( t ) t (A.Bressan & A.Marson, 1995) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Evolution of generalized tangent vectors If u θ (¯ t , · ) generates a generalized tangent vector discontinuities of u 0 interact at most two at the time u θ is piecewise Lipschitz with uniform in θ Lipschitz constant outside the discontinuities Then u θ ( t , · ) generates a generalized tangent vector � � for t > ¯ v ( t , · ) , ξ ( t ) t (A.Bressan & A.Marson, 1995) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Evolution of generalized tangent vectors If u θ (¯ t , · ) generates a generalized tangent vector discontinuities of u 0 interact at most two at the time u θ is piecewise Lipschitz with uniform in θ Lipschitz constant outside the discontinuities Then u θ ( t , · ) generates a generalized tangent vector � � for t > ¯ v ( t , · ) , ξ ( t ) t (A.Bressan & A.Marson, 1995) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Evolution of generalized tangent vectors Moreover v ( t , x ) is a broad solution of � � D 2 f ( u ) · v ∂ t v + Df ( u ) ∂ x v + ∂ x u = D u h ( x , u , z ) · v ξ α ( t ) satisfies an ODE along the α -th discontinuity x = x α ( t ) explicit restarting conditions at the interaction of two discontinuities (A.Bressan & A.Marson, 1995) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Evolution of generalized tangent vectors Moreover v ( t , x ) is a broad solution of � � D 2 f ( u ) · v ∂ t v + Df ( u ) ∂ x v + ∂ x u = D u h ( x , u , z ) · v ξ α ( t ) satisfies an ODE along the α -th discontinuity x = x α ( t ) explicit restarting conditions at the interaction of two discontinuities (A.Bressan & A.Marson, 1995) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Evolution of generalized tangent vectors Moreover v ( t , x ) is a broad solution of � � D 2 f ( u ) · v ∂ t v + Df ( u ) ∂ x v + ∂ x u = D u h ( x , u , z ) · v ξ α ( t ) satisfies an ODE along the α -th discontinuity x = x α ( t ) explicit restarting conditions at the interaction of two discontinuities (A.Bressan & A.Marson, 1995) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Evolution of generalized tangent vectors Moreover v ( t , x ) is a broad solution of � � D 2 f ( u ) · v ∂ t v + Df ( u ) ∂ x v + ∂ x u = D u h ( x , u , z ) · v ξ α ( t ) satisfies an ODE along the α -th discontinuity x = x α ( t ) explicit restarting conditions at the interaction of two discontinuities (A.Bressan & A.Marson, 1995) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Evolution of generalized tangent vectors Moreover v ( t , x ) is a broad solution of � � D 2 f ( u ) · v ∂ t v + Df ( u ) ∂ x v + ∂ x u = D u h ( x , u , z ) · v ξ α ( t ) satisfies an ODE along the α -th discontinuity x = x α ( t ) explicit restarting conditions at the interaction of two discontinuities (A.Bressan & A.Marson, 1995) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Necessary conditions for optimality Necessary conditions for optimality obtained by means of generalized cotangent vectors ( v ∗ , ξ ∗ ) satisfying � � v ∗ ( t , x ) · v ( t , x ) dx + ξ ∗ j ( t ) ξ j ( t ) = const j backward transported along trajectories of ∂ t u + ∂ x f ( u ) = h ( x , u , z ) (A. Bressan, A. Marson, 1995; A. Bressan, W. Shen, 2007) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Necessary conditions for optimality Necessary conditions for optimality obtained by means of generalized cotangent vectors ( v ∗ , ξ ∗ ) satisfying � � v ∗ ( t , x ) · v ( t , x ) dx + ξ ∗ j ( t ) ξ j ( t ) = const j backward transported along trajectories of ∂ t u + ∂ x f ( u ) = h ( x , u , z ) (A. Bressan, A. Marson, 1995; A. Bressan, W. Shen, 2007) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Necessary conditions for optimality Necessary conditions for optimality obtained by means of generalized cotangent vectors ( v ∗ , ξ ∗ ) satisfying � � v ∗ ( t , x ) · v ( t , x ) dx + ξ ∗ j ( t ) ξ j ( t ) = const j backward transported along trajectories of ∂ t u + ∂ x f ( u ) = h ( x , u , z ) (A. Bressan, A. Marson, 1995; A. Bressan, W. Shen, 2007) Fabio Ancona Control Problems for Hyperbolic Equations
Introduction Controllability & Stabilizability Generalized tangent vectors Optimal control problems Linearized evolution equations Pontryagin Maximum Principle for Temple systems Goal Extend variational calculus on generalized tangent and cotangent vectors to first order variations u θ that do not satisfy structural stability assumption on wave structure of reference solution u 0 uniform Lipschitzianity assumption on continuous part of reference solution u 0 Fabio Ancona Control Problems for Hyperbolic Equations
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