Adaptive control or semilinear stochastic systems

An adaptive, ergodic cost stochastic control problem for a partially known, semilinear, stochastic system in an infinite dimensional space is formulat ed and solved. The solutions of the Hamilton Jacobi Bellman equations for t he discounted cost and the ergodic cost stochastic control problems require some special interpretations because they do not typically exist in the us ual sense. The solutions of the parameter dependent ergodic Hamilton Jacobi Bellman equations are obtained from some corresponding discounted cost con trol problems as the discount rate tends to zero. The solutions of the ergo dic Hamilton Jacobi Bellman equations are shown to depend continuously on t he parameter. A certainty equivalence adaptive control is given that is bas ed on the optimal controls from the solutions of the ergodic Hamilton Jacob i Bellman equations and a strongly consistent family of estimates of the un known parameter. This adaptive control is shown to achieve the optimal ergo dic cost for the known system.