Stabilization of stochastic functional differential systems by steepest descent feedback controls

Abstract

Abstract The mean‐square exponential stabilization and stochastically asymptotical stabilization for a class of stochastic functional differential systems is studied. Based on the definitions of derivatives of functionals, the generalized Itô operator for functionals and compound functions is established. Furthermore, the novel Lyapunov functionals and the induced steepest descent feedback controls are constructed to obtain such stability conditions that are weaker than the classical ones. Together with the generalized Itô operator and the steepest descent feedback controls, mean‐square exponential stabilization and stochastically asymptotical stabilization for stochastic functional differential systems are investigated, respectively. As applications, two examples are given to illustrate the derived results: one is a stochastic mass–spring–damper model and the other is a numerical example with simulation figures.

Keywords

• Mathematical analysis
• Stability in control theory
• Time‐varying control systems