On the Practical Exponential Stability in Mean Square of Stochastic Perturbed Systems Via a Lyapunov Approach

The Lyapunov method is one of the most effective methods to analyze the stability of stochastic differential equations
(SDEs). Different authors analyzed the stability of SDEs based on Lyapunov techniques when the origin can be considered as an
equilibrium point. When the origin is not necessarily an equilibrium point, it is still possible to analyze the asymptotic stability
of solutions concerning a small neighborhood of the origin. The purpose is to study the asymptotic stability of a system whose
solution behavior is a small ball of state space or close to it. Thus, all state trajectories are bounded and close to a sufficiently
small neighborhood of the origin. In this sense, the limited boundedness of solutions of random systems, or the chance of
convergence of solutions needs to be analyzed on a ball centered on the origin. This is the so called “Practical Stability”. In this
article, we mainly investigate the practical uniform exponential stability in the mean square of stochastic linear time–invariant
systems. In addition, we are developing the problem of stabilization of certain classes of perturbed stochastic systems. Our
crucial techniques include Lyapunov techniques and generalized Gronwall inequalities. Lastly, we provide a numerical example
to illustrate our theoretical findings.