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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.

MSC:
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)
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