On a certainty equivalence design of continuous-time stochastic systems

摘要

This paper presents a certainty equivalence design method with application for continuous-time stochastic systems with unknown parameters. The first aim is to approximate the observed process x = (x_t)_t≥_0, satisfying a stochastic differential equation with control, by choosing a control process (u_t)_t≥_0to a stable Ornstein-Uhlenbeck process (x_0 _t)_t≥_0, driven by the same Wiener process. Based on observation of x for a given ε > 0 an adaptive control process u~ε = (u~ε _t)_t≥_0 is constructed such that the corresponding observed process x(u ~ε) = (x~ε _t)_t≥_0 satisfies the relation sup_(t≥tε) E(x~ε _t - x~0)t)~2 ≤ ε. The time t_ε (ε is a threshold quantity) is found. A similar problem is solved for a stochastic delay differential equation with an unknown parameter. It is shown that the time tε has equal rates of increase ε~(-1) ln ε~(-1) in both problems. Moreover, a target inequality, which ensures the stability of the controlled processes in the sense of the L _2-norm, is established. Similar problems for more general controlled multidimensional systems can be solved using sequential estimators of unknown parameters.