Stochastic uniform observability of linear differential equations with multiplicative noise. (English) Zbl 1147.60043

Let \(W=(W^1,\dots,W^m)\) be an \(m\)-dimensional Brownian motion. Given two real Hilbert spaces \(H\) and \(V\) and a \(\tau\)-periodic, closed unbounded linear operator \(A_t, t\in \mathbb{R}_+\), with constant domain in \(H\), the author considers the stochastic differential equation (SDE) \[ dY^{s,x}_t=(A_t+B_t)Y^{s,x}_tdt+\sum_{i=1}^mG_t^iY^{s,x}_tdW_t^i,\quad t\geq s,\;Y^{s,x}_s=x\in H. \] Assuming that \(B,G^1,\dots,G^m\) are strongly continuous \(L(H)\)-valued mappings, and being given another, \(L(H,V)\)-valued strongly continuous mapping \(C\), the author studies an equivalent deterministic characterization for the stochastic uniform observability of the system \(\{A,B,G^1,\dots,G^m;C\}\), i.e., a criterion for the existence of reals \(\sigma>0,\gamma>0\) such that \(E[\int_t^{t+\sigma}| C_rY_r^{t,x}|^2\,dr]\geq \gamma| x|^2\), for all \(t\in\mathbb R_+,x\in H,\) by using Lyapunov functions.
For the proof of this criterion the author establishes a series of results concerning a class of perturbed evolution operators and gives a new representation of the covariance operators associated with the mild solution of the investigated SDEs. The author’s results extend the work by V. Dragan and T. Morozan on stochastic observability [IMA J. Math. Control Inf. 21, No. 3, 323–344 (2004; Zbl 1060.93019)] to the infinite-dimensional case.


60H15 Stochastic partial differential equations (aspects of stochastic analysis)
93B07 Observability
35B40 Asymptotic behavior of solutions to PDEs


Zbl 1060.93019
Full Text: DOI


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