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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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