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Ergodic control of semilinear stochastic equations and the Hamilton-Jacobi equation. (English) Zbl 0939.93043

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.

MSC:

93E20 Optimal stochastic control
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
93C20 Control/observation systems governed by partial differential equations
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