Some aspects of the adaptive boundary control of stochastic linear hyperbolic systems

In this paper an adaptive control problem for the boundary or point control of a stochastic linear evolution system is formulated and the solution is described. The infinitesimal generator of the evolution system generates a C₀-semigroup that can model many linear hyperbolic systems and the noise in the system is a cylindrical white noise. The solution of the algebraic Riccati equation for the ergodic control problem with a quadratic cost functional is a continuous function of parameters. A family of least squares estimates of the unknown parameters is exhibited that is strongly consistent. A certainty equivalence adaptive control is constructed that is self-optimizing, that is, the family of average costs using this control converges (almost surely) to the optimal ergodic cost