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Inertia theorems for operator Lyapunov inequalities

✍ Scribed by A.J. Sasane; R.F. Curtain

Publisher
Elsevier Science
Year
2001
Tongue
English
Weight
119 KB
Volume
43
Category
Article
ISSN
0167-6911

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✦ Synopsis

We study operator Lyapunov inequalities and equations for which the inΓΏnitesimal generator is not necessarily stable, but it satisΓΏes the spectrum decomposition assumption and it has at most ΓΏnitely 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.

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A be an n X n complex matrix with inertia In(A) = (r(A), a(A), s(A)), and let H be an n x n hermitian matrix with inertia In(H) = (r(H), 6(H), 6(H)). Let K b e an n X n positive semidefinite matrix such that K = AH + HA\*. Suppose that 1 is the dimension of the controllability space of the pair (A,