Introduction Another approach Necessary conditions Approximation of optimal control problems for conservation laws via vanishing viscosity and relaxation Andrea Marson University of Padova Work in progress with F . Ancona (Padova) and M. Herty (Aachen) International Workshop on Hyperbolic and Kinetic Problems: Theory and Applications Andrea Marson Approximation of optimal control problems for conservation laws
Introduction Another approach Necessary conditions Outline Introduction 1 Optimal control problem for conservation laws What is known Another approach 2 Γ convergence Approximation via vanishing viscosity Approximation via relaxation Necessary conditions 3 Linearized and adjoint equations An alternative attempt Andrea Marson Approximation of optimal control problems for conservation laws
Introduction Optimal control problem for conservation laws Another approach What is known Necessary conditions Outline Introduction 1 Optimal control problem for conservation laws What is known Another approach 2 Γ convergence Approximation via vanishing viscosity Approximation via relaxation Necessary conditions 3 Linearized and adjoint equations An alternative attempt Andrea Marson Approximation of optimal control problems for conservation laws
Introduction Optimal control problem for conservation laws Another approach What is known Necessary conditions A story started 27 years ago... Andrea Marson Approximation of optimal control problems for conservation laws
Introduction Optimal control problem for conservation laws Another approach What is known Necessary conditions A story started 27 years ago... � ∂ t u + ∂ x f ( u ) = 0 x ∈ R , t ∈ [ 0 , T ] (1) u ( 0 , x ) = u 0 ( x ) Andrea Marson Approximation of optimal control problems for conservation laws
Introduction Optimal control problem for conservation laws Another approach What is known Necessary conditions A story started 27 years ago... � ∂ t u + ∂ x f ( u ) = 0 x ∈ R , t ∈ [ 0 , T ] (1) u ( 0 , x ) = u 0 ( x ) u = u ( t , x ) ∈ R n Andrea Marson Approximation of optimal control problems for conservation laws
Introduction Optimal control problem for conservation laws Another approach What is known Necessary conditions A story started 27 years ago... � ∂ t u + ∂ x f ( u ) = 0 x ∈ R , t ∈ [ 0 , T ] (1) u ( 0 , x ) = u 0 ( x ) u = u ( t , x ) ∈ R n f : R n → R n , smooth flux Andrea Marson Approximation of optimal control problems for conservation laws
Introduction Optimal control problem for conservation laws Another approach What is known Necessary conditions A story started 27 years ago... � ∂ t u + ∂ x f ( u ) = 0 x ∈ R , t ∈ [ 0 , T ] (1) u ( 0 , x ) = u 0 ( x ) u = u ( t , x ) ∈ R n f : R n → R n , smooth flux u 0 control variable � � u 0 ∈ L 1 ( R ) : TV u 0 ≤ δ u 0 ∈ U ad = , set of admissible controls Andrea Marson Approximation of optimal control problems for conservation laws
Introduction Optimal control problem for conservation laws Another approach What is known Necessary conditions A story started 27 years ago... � ∂ t u + ∂ x f ( u ) = 0 x ∈ R , t ∈ [ 0 , T ] (1) u ( 0 , x ) = u 0 ( x ) u = u ( t , x ) ∈ R n f : R n → R n , smooth flux u 0 control variable � � u 0 ∈ L 1 ( R ) : TV u 0 ≤ δ u 0 ∈ U ad = , set of admissible controls We would like to solve � � � min J ( x , u ( T , x )) dx : u = u ( t , x ) fulfills (1) , u 0 ∈ U ad R where J : R × R n → R is a smooth lagrangian Andrea Marson Approximation of optimal control problems for conservation laws
Introduction Optimal control problem for conservation laws Another approach What is known Necessary conditions Other optimal control problems Boundary control � ∂ t u + ∂ x f ( u ) = 0 x ≥ 0 , t ∈ [ 0 , T ] (2) u ( 0 , x ) = u 0 ( x ) , “ u ( t , 0 ) = u b ( t )” u b ∈ BV ( 0 , T ) control variable Andrea Marson Approximation of optimal control problems for conservation laws
Introduction Optimal control problem for conservation laws Another approach What is known Necessary conditions Other optimal control problems Boundary control � ∂ t u + ∂ x f ( u ) = 0 x ≥ 0 , t ∈ [ 0 , T ] (2) u ( 0 , x ) = u 0 ( x ) , “ u ( t , 0 ) = u b ( t )” u b ∈ BV ( 0 , T ) control variable Distributed control � � � ∂ t u + ∂ x f ( u ) = h t , x , u , z ( t , x ) x ∈ R , t ∈ [ 0 , T ] (3) u ( 0 , x ) = u 0 ( x ) z ∈ L ∞ ([ 0 , T ] × R ) control variable Andrea Marson Approximation of optimal control problems for conservation laws
Introduction Optimal control problem for conservation laws Another approach What is known Necessary conditions Other optimal control problems Boundary control � ∂ t u + ∂ x f ( u ) = 0 x ≥ 0 , t ∈ [ 0 , T ] (2) u ( 0 , x ) = u 0 ( x ) , “ u ( t , 0 ) = u b ( t )” u b ∈ BV ( 0 , T ) control variable Distributed control � � � ∂ t u + ∂ x f ( u ) = h t , x , u , z ( t , x ) x ∈ R , t ∈ [ 0 , T ] (3) u ( 0 , x ) = u 0 ( x ) z ∈ L ∞ ([ 0 , T ] × R ) control variable � � � min J ( x , u ( T , x )) dx : u = u ( t , x ) fulfills (2) or (3) R Andrea Marson Approximation of optimal control problems for conservation laws
Introduction Optimal control problem for conservation laws Another approach What is known Necessary conditions Other optimal control problems Boundary control � ∂ t u + ∂ x f ( u ) = 0 x ≥ 0 , t ∈ [ 0 , T ] (2) u ( 0 , x ) = u 0 ( x ) , “ u ( t , 0 ) = u b ( t )” u b ∈ BV ( 0 , T ) control variable Distributed control � � � ∂ t u + ∂ x f ( u ) = h t , x , u , z ( t , x ) x ∈ R , t ∈ [ 0 , T ] (3) u ( 0 , x ) = u 0 ( x ) z ∈ L ∞ ([ 0 , T ] × R ) control variable � � � min J ( x , u ( T , x )) dx : u = u ( t , x ) fulfills (2) or (3) R We can also consider a mixed of (2) and (3) Andrea Marson Approximation of optimal control problems for conservation laws
Introduction Optimal control problem for conservation laws Another approach What is known Necessary conditions Problems Existence of a solution: via direct method of calculus of variation, provided the compactness of the set of admissible controls Andrea Marson Approximation of optimal control problems for conservation laws
Introduction Optimal control problem for conservation laws Another approach What is known Necessary conditions Problems Existence of a solution: via direct method of calculus of variation, provided the compactness of the set of admissible controls Compute the optimal solution: difficult task Andrea Marson Approximation of optimal control problems for conservation laws
Introduction Optimal control problem for conservation laws Another approach What is known Necessary conditions Problems Existence of a solution: via direct method of calculus of variation, provided the compactness of the set of admissible controls Compute the optimal solution: difficult task Need of necessary conditions for optimality Andrea Marson Approximation of optimal control problems for conservation laws
{ f. nothyo multi ) Fiat .)+o(w ) .)+wrH mltitutt •n , , w pm , oo| . ) optimal trajectory ulti no t Introduction Optimal control problem for conservation laws Another approach What is known Necessary conditions An hitchikers’ guide to necessary conditions Assume to have an optimal solution u = u ( t , x ) � � � � J ( x , u ( T , x )) dx = min J ( x , u ( T , x )) dx : u fulfills ∂ t u + ∂ x f ( u ) = 0 , R R Andrea Marson Approximation of optimal control problems for conservation laws
{ f. nothyo multi ) Fiat .)+o(w ) .)+wrH mltitutt •n , , w pm , oo| . ) optimal trajectory ulti no t Introduction Optimal control problem for conservation laws Another approach What is known Necessary conditions An hitchikers’ guide to necessary conditions Assume to have an optimal solution u = u ( t , x ) � � � � J ( x , u ( T , x )) dx = min J ( x , u ( T , x )) dx : u fulfills ∂ t u + ∂ x f ( u ) = 0 , R R Perturbe the optimal initial datum, u ( 0 , · ) = u 0 + hv 0 Andrea Marson Approximation of optimal control problems for conservation laws
Introduction Optimal control problem for conservation laws Another approach What is known Necessary conditions An hitchikers’ guide to necessary conditions Assume to have an optimal solution u = u ( t , x ) � � � � J ( x , u ( T , x )) dx = min J ( x , u ( T , x )) dx : u fulfills ∂ t u + ∂ x f ( u ) = 0 , R R Perturbe the optimal initial datum, u ( 0 , · ) = u 0 + hv 0 Transport the tangent vector v = v ( t , x ) along the optimal solution by means of linearized equations { f. nothyo multi ) Fiat .)+o(w ) mltitutt .)+wrH •n , , w pm , oo| . ) optimal trajectory ulti no t Andrea Marson Approximation of optimal control problems for conservation laws
Introduction Optimal control problem for conservation laws Another approach What is known Necessary conditions An hitchikers’ guide to necessary conditions At the final time T � ∇ J ( x , u ( T , x )) · v ( T , x ) dx ≥ 0 R Andrea Marson Approximation of optimal control problems for conservation laws
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