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On robust stability of singular systems with random abrupt changes. (English) Zbl 1089.34046
This paper deals with the class of \(n\)-dimensional continuous-time singular linear systems with Markovian switching described by the following dynamics \[ E\dot x(t)= (A(r(t))+ D_A(r(t)) F_A(r(t)) E_A(r(t))) x(t), \] where the Markov process \(r(\cdot)\) is a mode switching system, taking values in \(N\) modes \(\{1,2,\dots, N\}\) and the matrix \(E\) may be singular. The author defines the following stability.
1. Let \(F_A(i)\) be the zero matrix, for \(i= 1,2,\dots,N\). The solution \(x(\cdot)\) is said to be stochastically stable, if there exists a constant \(T(x_0, r_0)\) such that \[ {\mathbf E}\Biggl(\int^\infty_0\| x(t)\|^2 dt/x(0)= x_0,\;r(0)= r_0\Biggr)\leq T(x_0, r_0).\tag{\(*\)} \] 2. \(F_A\) is called admissible, if \(F_A^T(i)F_A(i)\leq\) identity matrix, \(i= \{1,2,\dots,N\}\). When the inequality \((*)\) is satisfied for all admissible \(F_A\), we say that robust stochastic stability holds.
Using Lyapunov theory and algebraic results, the author derives a sufficient condition for stochastic stability in linear matrix inequality setting. Moreover, applying this condition, he gives a sufficient condition for the robust stochastic stability.

34F05 Ordinary differential equations and systems with randomness
34D20 Stability of solutions to ordinary differential equations
60H10 Stochastic ordinary differential equations (aspects of stochastic analysis)
Full Text: DOI
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