ABSTRACT
Introduction We consider the problem of calculating feedback controls for systems modeled by partial differential or delay differential equations. In these systems the state x(t) lies in an infinite-dimensional space. A classical controller design objective is to find a control u(t) so that the objective function ∫ ∞
0 〈Cx(t), Cx(t)〉+ u∗(t)Ru(t)dt (1)
is minimized where R is a positive definite matrix and the observation C ∈ L(X,Rp). The theoretical solution to this problem for many infinite-dimensional systems parallels the theory for finite-dimensional
systems [10, 17, 18, e.g.]. In practice, the control is calculated through approximation. This leads to solving an algebraic Riccati equation
A∗P + PA− PBR−1B∗P = −C∗C (2)
for a feedback operator
K = −R−1B′P. (3)
The matrices A, B, C arise in a finite-dimensional approximation of the infinite-dimensional system. Let n indicate the order of the approximation, m the number of control inputs, and p the number of observations. Thus, A is n × n, B is n ×m, and C is p × n. There have been many papers written describing conditions under which approximations lead to approximating controls that converge to the control for the original infinite-dimensional system [3, 11, 14, 17, 18, e.g.]. In this paper we will assume that an approximation has been chosen so that a solution to the Riccati equation (2) exists for sufficiently large n and also that the approximating feedback operators converge.
