Adaptive boundary and point control of linear stochastic distributed parameter systems

Abstract

An adaptive control problem for the boundary or the point control of a linear stochastic distributed parameter system is formulated and solved in this paper. The distributed parameter system is modeled by an evolution equation with an infinitesimal generator for an analytic semigroup. Since there is boundary or point control, the linear transformation for the control in the state equation is also an unbounded operator. The unknown parameters in the model appear affinely in both the infinitesimal generator of the semigroup and the linear transformation of the control. Strong consistency is verified for a family of least squares estimates of the unknown parameters. An Itô formula is established for smooth functions of the solution of this linear stochastic distributed parameter system with boundary or point control. The certainty equivalence adaptive control is shown to be self-tuning by using the continuity of th 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; that is, the family of average costs converges to the optimal ergodic cost. Some examples of stochastic parabolic problems with boundary control and a structurally damped plate with random loading and point control are described that satisfy the assumptions for the adaptive control problem solved in this paper.