Stabilization and Robust Control Problem

Abstract

This chapter contains the essential and fundamental developments of the robust control theory of distributed parameter systems. This area concerns the investigation of the control, stability and adjoint control optimization of infinite-dimensional dissipative dynamical systems. The considered systems are derived from spatially and time-dependent partial differential equations associated with boundary-value problems. We recall that in our approach it is not assumed that the system is stabilizable or detectable. Moreover, we are interested in the robust regulation of the deviation of the systems from the desired target, by analyzing full non-linear systems, which models large perturbations to the desired target.

The objective of this chapter is to describe robust control theory in different situations. First, we use a very basic problem to explain the theory as simply as possible. Second, we present the theory for general linear evolutive problems. Then, we present an estimate parameter problem in the case of bilinear systems. Finally, we analyze full non-linear robust control problems, with particular attention to time-delay problems. In each section, we study the existence, uniqueness and optimality conditions for the optimal solution (by using adjoint problems). We develop our study for different realistic cases of observations and controls. The numerical aspect (based on the adjoint problem) is presented in the next chapter.