On some relationships between the LaSalle Invariance Principle and the Nagy—Foias canonical decomposition

Abstract

The generalization of the LaSalle Invariance Principle by Hale [l] is a powerful stability theorem for nonfinite dimensional dynamic systems. Recently Slemrod was able to apply this principle to the stabilizability problem of Hilbert space contraction semigroups [2,3]. Hilbert space contractions were extensively studied by Nagy and Foias in the last decade. Using a canonical decomposition of contraction semigroups-due to Nagy and Foias [4]-Benchimol [5] and Levan and Rigby [6] were also able to stabilize contraction semigroups on a Hilbert space. This paper will establish relationship between the above-mentioned theories-at least as far as stability and stabilizability of Hilbert space contraction semigroups are concerned. The LaSalle Invariance Principle and the Nagy-Foias canonical decomposition are recalled in Section 2. Stability and stabilizability of contraction semigroups are studied in Section 3. Here we shall combine the two theories to obtain a rather general stability result. This will then be applied to the stabilizability problem.