ADAPTIVE BOUNDARY AND POINT CONTROL OF LINEAR STOCHASTIC DISTRIBUTED-PARAMETER SYSTEMS

An adaptive control problem for the boundary or the point control of a linear stochastic distributed parameter system is formulated and solv ed in this paper. The distributed parameter system is modeled by an ev olution equation with an infinitesimal generator for an analytic semig roup. Since there is boundary or point control, the linear transformat ion for the control in the state equation is also an unbounded operato r. The unknown parameters in the model appear affinely in both the inf initesimal generator of the semigroup and the linear transformation of the control. Strong consistency is verified for a family of least squ ares estimates of the unknown parameters. An Ito formula is establishe d for smooth functions of the solution of this linear stochastic distr ibuted parameter system with boundary or point control. The certainty equivalence adaptive control is shown to be self-tuning by using the c ontinuity of the solution of a stationary Riccati equation as a functi on of parameters in a uniform operator topology. For a quadratic cost functional of the state and the control, the certainty equivalence con trol is shown to be self-optimizing, that is, the family of average co sts converges to the optimal ergodic cost. Some examples of stochastic parabolic problems with boundary control and a structurally damped pl ate with random loading and point control are described that satisfy t he assumptions for the adaptive control problem solved in this paper.