Elsevier

Systems & Control Letters

Volume 43, Issue 2, 15 June 2001, Pages 127-132
Systems & Control Letters

Inertia theorems for operator Lyapunov inequalities

https://doi.org/10.1016/S0167-6911(01)00083-4Get rights and content

Abstract

We study operator Lyapunov inequalities and equations for which the infinitesimal generator is not necessarily stable, but it satisfies the spectrum decomposition assumption and it has at most finitely many unstable eigenvalues. Moreover, the input or output operators are not necessarily bounded, but are admissible. We prove an inertia result: under mild conditions, we show that the number of unstable eigenvalues of the generator is less than or equal to the number of negative eigenvalues of the self-adjoint solution of the operator Lyapunov inequality.

Section snippets

Introduction and the main result

The inertia of a square matrix A∈Cn×n is the triple (ν(A),ζ(A),π(A)), whereν(A)=numberofeigenvaluesofAinC,ζ(A)=numberofeigenvaluesofAontheimaginaryaxis,π(A)=numberofeigenvaluesofAinC+,where C={z∈C|Re(z)<0} and C+={z∈C|Re(z)>0}. Inertia theorems for matrices concern relations between the inertia of Hermitian solutions Q of the Lyapunov equationAQ+QA=−CCand the matrix A. The fundamental result was by Ostrowski and Schneider [12], and later contributions can be found in [15], [2]. We shall

Preliminaries on indefinite inner products and proof of the main result

The proof of our main theorem relies on the fact that any self-adjoint solution of the operator Lyapunov inequality gives rise to a natural indefinite inner product space. So, we will first state a few preliminaries and results about indefinite inner product spaces which will be used in the proof. For more details, see [1].

Let V be a vector space over C. An indefinite inner product [·,·] on V is a map [·,·]:V×VC satisfying:

  • 1.

    [αx1+βx2,y]=α[x1,y]+β[x2,y], ∀x1,x2,y∈V and ∀α,β∈C.

  • 2.

    [x,y]=[y,x] for all

Corollaries

In this section, we give a few corollaries of our main theorem applied to Lyapunov equations with possibly unbounded observation operators.

Throughout this section, we assume that X is a Hilbert space and A:D(A)→X is the infinitesimal generator of a C0-semigroup {T(t)}t⩾0 on X.

Definition 3.1

Let us denoteσ+(A)σ(A)∩C+,σ(A)σ(A)∩C.A satisfies the spectrum decomposition assumption if σ+(A) is a bounded set which is separated from σ(A) in such a way that a rectifiable, simple closed curve, Γ, can be drawn so

Acknowledgements

The authors thank the referee for his careful reading of the manuscript and, in particular, for pointing out a gap in the original manuscript.

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