Approximation in optimal control of diffusion processes (Q2722268)

The control problem in which the state evolves according to the \(d\)-dimensional stochastic differential equation NEWLINE\[NEWLINEx_{t}=x+\int_{0}^{t}b(s,x_{s},u_{s})ds+\int_{0}^{t}\sigma(s,x_{s }) dB_{s}NEWLINE\]NEWLINE is considered on some filtered probability space, where \(b, \sigma\) are deterministic functions, \((B_{t}; t\geq 0)\) is a \(d\)-dimensional Brownian motion, \(x\) is the initial state and \(u_{t}\) stands for the control variable. The expected cost on the time interval \([0,1]\) has the form NEWLINE\[NEWLINEJ(u)=E\int_{0}^{t}h(t,x_{t},u_{t}) dt+g(x_{1}).NEWLINE\]NEWLINE The aim of control theory is to optimize this expected cost over the class \(U\) of admissible controls which are progressively measurable processes with values in a compact Polish space \(A.\) Without the Fillipov-type convexity condition an optimal control does not necessarily exist in \(U.\) The idea is then to introduce a new class of admissible controls in which the controller chooses at time \(t\) a probability measure \(\mu_{t}(da)\) on the control set \(A\) rather than an element \(u_{t}\in A.\) These controls are called relaxed controls. The goal of this paper is to show that each relaxed diffusion is a strong limit of a sequence of diffusions associated with ordinary controls, under any condition on the coefficients ensuring pathwise uniqueness of the controlled equations. As a sequence, it is proved that the value functions of both relaxed and original problems are equal. The relaxed control problem which is a generalization of the original problem is studied, where admissible controls are measure-valued processes. The only assumption imposed is the uniform ellipticity of the diffusion matrix which leads to the existence of a weak solution for the controlled equation associated with constant controls. The proof is inspired from the method used by the authors and \textit{Y. Ouknine} [in: Séminaire de probabilités XXXII. Lect. Notes Math. 1686, 166-187 (1998; Zbl 0910.60049)] where various strong stability results for stochastic differential equations with non-Lipschitz coefficients have been derived.