Infinite horizon stochastic optimal control for Volterra equations with completely monotone kernels

摘要

The aim of the paper is to study an optimal control problem on infinitehorizon for an infinite dimensional integro-differential equation withcompletely monotone kernelskernels, where we assume that the noise enters thesystem when we introduce a control. We start by reformulating the stateequation into a semilinear evolution equation which can be treated by semigroupmethods. The application to optimal control provide other interesting resultand require a precise descriprion of the properties of the generated semigroup.The main tools consist in studying the differentiability of theforward-backward system with infinite horizon corresponding with thereformulated problem and the proof of existence and uniqueness of of mildsolutions to the corresponding HJB equation.