Absolute stability approach to stochastic stability of infinite-dimensional nonlinear systems
The authors consider a system described by the semilinear stochastic evolution equation \[ dx= [Ax+ b\varphi(x)]dt+ [Cx+ D\varphi(x)] dw(t),\quad x(s)= h,\tag{1} \] where \(A\) is an operator with domain \(D(A)\subseteq H\), \(b\in {\mathcal L}(U, H)\), \(C\in {\mathcal L}(H, {\mathcal L}(W, H))\), \(D\in {\mathcal L}(U, {\mathcal L}(W, H))\) and \(\varphi: H\to U\) is a Lipschitz continuous nonlinear operator with \(\varphi(0)= 0\), \(W\), \(H\) and \(U\) are real separable Hilbert spaces and \({\mathcal L}(W, H)\) is the space of bounded linear operators \(W\to H\). To obtain sufficient conditions for absolute stability of the system (1), the authors derived a stochastic Lyapunov function and introduced the special stochastic infinite-dimensional counter-part of the Kalman-Yakubovich lemma. Examples show extensions of the Popov and circle criteria to the feedback stochastic heat and delay equations, respectively.
stochastic partial differential equations
scientific article; zbMATH DE number 819362
scientific article