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Optimal control problems for stochastic reaction-diffusion systems with non-Lipschitz coefficients. (English) Zbl 0987.60073
The authors consider stochastic reaction-diffusion systems with distributed parameter controls in bounded domains of \(R^d\) with \(d\leq 3\). The system contains uniformly elliptic second order differential operator with regular real coefficients and reaction term \(f\) having polynomial growth together with its derivatives. The term \(f\) fulfills also appropriate dissipativity conditions. The control problem is studied using the dynamic programming approach. The cost functional is constructed in such a way that its value function satisfies Hamilton-Jacobi-Bellman (HJB) equation containing Hamiltonian \(K\) that is not assumed to be Lipschitz continuous. It permits to cover the important case of quadratic Hamiltonians. The HBJ equation is first resolved by a fixed point argument in a small time interval and is then extended to arbitrary time intervals by suitable a priori estimates. The main ingredient in the proof is the smoothing effect of the transition semigroup associated with the uncontrolled system. Using Galerkin argument, the authors get good estimates for the solution of HJB equation.

MSC:
60H15 Stochastic partial differential equations (aspects of stochastic analysis)
93C20 Control/observation systems governed by partial differential equations
93E20 Optimal stochastic control
60J35 Transition functions, generators and resolvents
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