Adaptive control of some stochastic semilinear equations

In this paper an adaptive control problem for some stochastic semilinear systems is formulated and the solution is described. The linear part of the semilinear equation is a negative, self-adjoint generator of an analytic semigroup. The semilinear equation has a unique invariant measure that is shown to be continuous with respect to parameters. The optimal control for the stochastic control of the known semilinear equation with a cylindrical noise is a continuous function of parameters. A family of least squares estimates is strongly consistent for a class of adaptive controls. A certainty equivalence adaptive control law is self-optimizing, that is, the family of average costs using this adaptive control converges (almost surely) to the optimal ergodic cost