Exponential Stability of Infinite Dimensional Linear Stochastic Systems with Time Delay

In this paper, exponential stability of infinite dimensional stochastic systems with delay is considered. Firstly, we construct the suitable Lyapunov functions. And then sufficient conditions for exponential stability of infinite dimensional linear stochastic systems with delay are derived by using Ito formula and Poincare inequality in partial differential equations. The conditions are formulated as linear operator inequality, where the decision variables are operators. Finally, being applied to a heat equations and to a wave equations, these conditions are reduced to standard Linear Matrix Inequalities.