Fixed points and exponential stability of stochastic functional partial differential equations driven by fractional Brownian motion (Q2820741)

This paper deals with the investigation of the exponential stability in mean square of a mild solution of the stochastic functional partial differential equation NEWLINE\[NEWLINE\begin{aligned} dX(t)&=[AX(t)+f(t,X_{t})]dt+g(t)dB_{Q}^{H}(t), t\geq 0, \\ X(s)&=\phi(s), -r\leq s\leq 0, r\geq 0.\end{aligned}NEWLINE\]NEWLINE Here \(B_{Q}^{H}(t)\) is a \(K\)-valued \(Q\)-cylindrical fractional Brownian motion with \(H\in (1/2,1)\) and covariance operator \(Q\); \(K,U\) are real separable Hilbert spaces; \(\phi\in C(-r,0; L^2(\Omega;U))\); \(X_{t}\in C(-r,0; L^2(\Omega;U))\), \(X_{t}(s)=X(t+s)\) for \(s\in[-r,0]\); \(A:\;\mathrm{Dom}(A)\subset U\to U\) is the infinitesimal generator of a strongly continuous semigroup \(S(\cdot)\) on \(U\); \(f:\;[0,T]\times C(-r,0;U)\to U\) is a family of nonlinear operators defined for almost every \(t\). NEWLINENEWLINENEWLINEThe authors prove the following result. Let us assume that the following conditions hold: (1) \(| S(t)|_{U}\leq Me^{-\lambda t} \forall t\geq0\), where \(M\geq1\), \(\lambda>0\); (2) There exists a constant \(C_{f}\geq0\) such that for any \(X,Y\in C(-r,T;U)\), and for all \(t\geq0\): \(\int_{0}^{t}e^{ms}| f(t,X_{s})-f(t,Y_{s})|^2_{U}ds\leq C_{f}\int_{-r}^{t}e^{ms}| X(s)-Y(s)|^2_{U}ds\, \forall \, 0\leq m\leq\lambda\), and \(\int_0^{\infty}e^{\lambda s}| f(s,0)|^2_{U}ds<\infty\); (3) For \(g:\, [0,T]\to L_{Q}^0(K,U)\), for the complete orthonormal basis \(\{e_{n}\}_{n\in N}\) in \(K\) we have \(\sum_{n=1}^{\infty}\| g Q^{1/2}e_{n}\|_{L^2([0,T];U)}<\infty\), \(\sum_{n=1}^{\infty}| g(t) Q^{1/2}e_{n}|_{U}\) is uniformly convergent for \(t\in [0,T]\), \(\int_0^{\infty}e^{\lambda s}| g(s)|^2_{L_{Q}^0(K,U)}ds<\infty\). Then the considered stochastic functional partial differential equation is exponentially stable in mean square if \(\lambda>C_{f}M^2\).