Some aspects of the adaptive boundary and point control of linear distributed parameter systems

An adaptive control problem for the boundary or the point control of a linear stochastic distributed parameter system (DPS) is formulated and its solution is given. The unknown linear stochastic DPS is described by an evolution equation, in which the unknown parameters appear in the infinitesimal generator of an analytic semigroup and in the unbounded linear transformation for the boundary control. An Ito formula can be verified for smooth functions of the solution of the linear stochastic DPS boundary control considered here. The certainty equivalence adaptive control is shown to be self-tuning by noting the continuity of the solution of a stationary Riccati equation as a function of parameters in a uniform operator topology. For a quadratic cost functional of the state and the control, the certainty equivalence control is shown to be self-optimizing, i.e., the family of average costs converges to the optimal ergodic cost