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Averaging of the Hamilton-Jacobi equation in infinite dimensions and an application. (English) Zbl 0945.49022

Extending to Hilbert spaces previous results on finite-dimensional spaces of F. Chaplais [SIAM J. Control Optimization 25, 767-780 (1987; Zbl 0616.49019)] and E. N. Barron [SIAM J. Control Optimization 31, No. 6, 1630-1652 (1993; Zbl 0791.49033)], the authors prove in their main result (Theorem 2.2) that the unique viscosity solution, \(V_\varepsilon (.,.)\), of the problem: \[ V_t(t,x)+H(t,t/\varepsilon,x,V_x(t,x))=0, \;(t,x)\in [0,T)\times X, \;V(T,x)=g(x) \] is converging uniformly on bounded subsets of \([0,T]\times X\) as \(\varepsilon \to 0_+\) to the unique viscosity solution, \(V(.,.)\), of the problem: \[ V_t(t,x)+\overline H(t,x,V_x(t,x))=0, \;(t,x)\in [0,T)\times X, \;V(T,x)=g(x) \] where the “averaged” Hamiltonian, \(\overline H\), is given by: \[ \overline H(t,x,p):=\int_0^1H(t,s,x,p)ds, \;(t,x,p)\in [0,T]\times X\times X^* \] if \(H(t,.,x,p)\) are \(1-periodical\) and satisfy usual hypotheses in the theory of viscosity solutions.
The main result is then applied to the value functions of optimal control problems that consist in minimizing cost-functionals of the form: \[ J_{t,x}^\varepsilon (u(.)):=\int_t^Tf^0(r,r/\varepsilon,x(r),u(r))dr + g(x(T)) \] subject to: \[ x'(r)=A.x(r)+f(r,r/\varepsilon,x(r),u(r)), \;r\in [0,T], \;x(t)=x\in X \] over the class of measurable controls, \(u(.):[0,T] \to U\), taking values in a given metric space.

MSC:

49L25 Viscosity solutions to Hamilton-Jacobi equations in optimal control and differential games
49L20 Dynamic programming in optimal control and differential games
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35B37 PDE in connection with control problems (MSC2000)
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