Diffusion approximation and optimal stochastic control
By the same goal with the previous paper of these authors [SIAM J. Control Optimization 34, No. 1, 161-178 (1996; Zbl 0867.93085)] but considering the case of a stochastic control model admitted a diffusion approximation, they show in the present paper that an optimal Lipschitz feedback control of the limit model NEWLINE\[NEWLINEdX_t = [A_0 (t,X_t)+ a_1 (t,X_t)u_t] dt + B(t, X_t) dW_tNEWLINE\]NEWLINE with the cost \( J(u)= E \{ \text{int}_0^T [p(t,X_t) +q(t,u_t)] dt +r(X_T) \}\) remains asymptotically optimal feedback for a family of controlled stochastic processes (prelimit models), indexed by a small parameter and associated with the same cost, whose dynamics are perturbed by additive stationary elements (wideband noises and martingales), a fast-fluctuating external process that contaminates some of the coefficients, as well as noises inducing singularly perturbed processes.NEWLINENEWLINENEWLINEHere, \( A_0\) and \(B^2\) depend on the invariant measure of the external process and the asymptotic intensity of the quadratic variation of the martingale; asymptotical optimality means \( \lim_{ \varepsilon \to 0} J^{\varepsilon}(u) = \lim_{ \varepsilon \to 0} \{ J_0^{\varepsilon} =\inf_{u^{\varepsilon}} J^{\varepsilon} (u^{\varepsilon}) \}\). Moreover, the statement holds for the case of \( \delta\)-optimality in the sense that \( J(u^{\delta}) \leq \{J_0 = \inf_u J(u) \} + {\delta}\) implies \(\limsup_{\varepsilon \to 0} ( J^{\varepsilon} (u^{\delta}) - J_0^{\varepsilon})\leq \delta \). The statement also remains true in the case of Markov feedback controls. All mentioned results and the estimation \(\liminf_{\varepsilon \to 0} J_0^{\varepsilon} \geq J_0 \) are obtained by using the technique of weak convergence of stochastic processes. Properties involving the prelimit model are described in Section 3 (relative compactness) and Section 5 (Lipschitz feedback control). Section 4 is devoted to problems of the limit model: feedback controls, existence of an optimal feedback control, existence of a \(\delta\)-optimal Lipschitz and Markov feedback control. The main results of the paper are presented in Section 1, the assumptions in Section 2, the proofs in Section 6, and all the technical details in Section 7.
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