INVARIANT DENSITY, LYAPUNOV EXPONENT, AND ALMOST SURESTABILITY OF MARKOVIAN-REGIME-SWITCHING LINEAR SYSTEMS

doi:10.1007/s11424-011-9018-z

Journal of Systems Science and Complexity ›› 2011, Vol. 24 ›› Issue (1): 79-092.DOI: 10.1007/s11424-011-9018-z

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INVARIANT DENSITY, LYAPUNOV EXPONENT, AND ALMOST SURESTABILITY OF MARKOVIAN-REGIME-SWITCHING LINEAR SYSTEMS

Qi HE, Gang George YIN

Department of Mathematics, Wayne State University

Received:2009-01-19 Revised:1900-01-01 Online:2011-02-25 Published:2011-02-25

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Qi HE;Gang George YIN. INVARIANT DENSITY, LYAPUNOV EXPONENT, AND ALMOST SURESTABILITY OF MARKOVIAN-REGIME-SWITCHING LINEAR SYSTEMS[J]. Journal of Systems Science and Complexity, 2011, 24(1): 79-092.

This paper is concerned with stability of a class of randomly
switched systems of ordinary differential equations. The system
under consideration can be viewed as a two-component process
$(X(t),\al(t))$, where the system is linear in $X(t)$ and $\al(t)$
is a continuous-time Markov chain with a finite state space.
Conditions for almost surely exponential stability and instability
are obtained. The conditions are based on the Lyapunov exponent,
which in turn, depends on the associate invariant density.
Concentrating on the case that the continuous component is two
dimensional, using transformation techniques, differential
equations satisfied by the invariant density associated with the
Lyapunov exponent are derived. Conditions for existence and
uniqueness of solutions are derived. Then numerical solutions are
developed to solve the associated differential equations.

Key words: Invariant density, Lyapunov exponent, randomly switching ordinary differential equation.

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