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Hamilton-Jacobi-Bellman equations for the optimal control of a state equation with memory

Guillaume Carlier, Rabah Tahraoui (2010)

ESAIM: Control, Optimisation and Calculus of Variations

This article is devoted to the optimal control of state equations with memory of the form: x ˙ ( t ) = F ( x ( t ) , u ( t ) , 0 + A ( s ) x ( t - s ) d s ) , t > 0 , with initial conditions x ( 0 ) = x , x ( - s ) = z ( s ) , s > 0 . Denoting by y x , z , u the solution of the previous Cauchy problem and: v ( x , z ) : = inf u V { 0 + e - λ s L ( y x , z , u ( s ) , u ( s ) ) d s } where V is a class of admissible controls, we prove that v is the only viscosity solution of an Hamilton-Jacobi-Bellman equation of the form: λ v ( x , z ) + H ( x , z , x v ( x , z ) ) + D z v ( x , z ) , z ˙ = 0 in the sense of the theory of viscosity solutions in infinite-dimensions of Crandall and Lions.

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