Ergodic control of semilinear stochastic equations and the Hamilton-Jacobi equation
The authors consider an optimal control problem with an ergodic cost functional for a stochastic evolution equation in a Hilbert space driven by a cylindrical Brownian motion. The drift term of the controlled equation is defined as \(AX_t+ F(X_t)- u_t\), where \(A\) is the generator of a strongly continuous semigroup, \(F\) is a Lipschitz-continuous and Gateaux-differentiable dissipative mapping, \(u_t\) is a control having sufficiently small norm, and \(X_t\) is a solution of the equation. The optimal cost is characterized as a unique solution to the corresponding Hamilton-Jacobi equation in a space of polynomially increasing functions, and an optimal control is given in a feedback form. The proof is based on the analysis of an auxiliary discounted cost problem and the limit transition to the ergodic cost problem.
scientific article; zbMATH DE number 1334729