The Linear Quadratic Optimal Control Problem Over an Infinite Time Horizon for a Class of Distributed Parameter Systems

Abstract

This paper surveys currently available theory for linear infinite dimensional systems the evolution of which can be described by semigroups of operators of class C0. For systems of linear ordinary differential equations the infinite-time quadratic cost problem is well-studied (cf.R.BROCKETT [1], R.E.KALMAN [1], [2], J .C.WILLEMS [1], W.M.WONHAM [1]). This problem has also been studied for certain classes of infinite-dimensional systems (cf. J.L.LIONS [1], LUKES-RUSSELL [1], R.DATKO [3],[5],[7]).This paper makes use of the work of R.DATKO [2] on Stability Theory in Hilbert spaces and J.ZABCZYK [1] on the concept of detectability in Hilbert spaces. In doing this, we insist on an approach which clarifies the system-theoretic relationship between controllability, stabilizability, stability and existence of a solution of an associated equation of Riccati type.This theory covers certain classes of distributed controls; it also covers hereditary systems which can be looked at as distributed parameter systems with boundary control. At this time, it does not seem possible to systematically deal with boundary control problems. However some results are now available (cf. H.O.FATTORINI [1],[2],[3],[4], H.L.KOH [1], J.L.LIONS [1])