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Exponential stability of stochastic discrete-time, periodic systems in Hilbert spaces. (English) Zbl 1107.93034
Consider the linear discrete-time stochastic system \(x_k=x,x_{n+1}=A_nx_n+\xi_nB_nx_n\) with \(A_n\) and \(B_n\) bounded linear operators on a real separable Hilbert space, \(\xi_n\) independent real-valued random variables, \(\mathbb E\xi_n=0,\mathbb E|\xi_n|^2=b_n<\infty,\,n,k\in\mathbb N, n\searrow k\) an let \(X(n,k),n\searrow k\searrow 0\), be the random evolution operator associated with the system, i.e., \(X(k,k)=I,X(n,k)=(A_{n-1}+\xi_{n-1}B_{n-1}\dots A_k+\xi_kB_k),n>k\). If \(\mathbb E\| X(n,0)x\|^2=O(a^{n-k})\mathbb E\| X(k,0)x\|^2\) for some \(0<a<1,\,n\searrow k\searrow n_0\), \(n_0\) sufficiently large, and all \(x\), then the system is called exponentially stable, and if even \(\mathbb E\| X(n,0)x\|^2=O(a^{n-k})\| x\|^2\), it is called uniformly exponentially stable.
The author shows that under the assumption that \(A_n,B_n\) and \(\xi_n\) are periodic in \(n\), these two forms of stability are equivalent, but not so in general. Following two different approaches, she derives necessary and sufficient conditions for exponential stability in the periodic case.
93E03 Stochastic systems in control theory (general)
34F05 Ordinary differential equations and systems with randomness
93E15 Stochastic stability in control theory
34D20 Stability of solutions to ordinary differential equations
47B80 Random linear operators
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