Some results on the adaptive control of stochastic semilinear systems

Abstract

An adaptive, ergodic cost stochastic control problem for a partially known, semilinear, stochastic system in an infinite dimensional space is formulated and solved. Since the results for discounted cost and ergodic cost stochastic control problems for known semilinear stochastic systems are only relatively recent, it seems that this is the first work on adaptive control for semilinear systems. The solutions of the Hamilton-Jacobi-Bellman equations for these stochastic control problems require some special interpretations because they do not typically exist in the usual sense. The solutions of the parameter dependent ergodic Hamilton-Jacobi-Bellman equations are obtained from some corresponding discounted cost control problems as the discount rate tends to zero. The solutions of the ergodic Hamilton-Jacobi-Bellman equations are shown to depend continuously on the parameter. A certainty equivalence adaptive control is given that is based on the optimal controls from the solutions of the ergodic Hamilton-Jacobi-Bellman equations and a strongly consistent family of estimates of the unknown parameter. This adaptive control is shown to achieve the optimal ergodic cost for the known system.