## 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.

### MSC:

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

Zbl 1060.93019
Full Text:

### References:

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