Finite horizon stochastic H₂/H_∞ control with discrete and distributed delays

Abstract

Based on a Lyapunov–Krasovskii functional, a new delay-dependent $H_{\mathrm{\infty }}$ performance criterion is established for stochastic systems with discrete and distributed delays. It is shown that the $H_{\mathrm{\infty }}$-performance of the corresponding uncontrolled stochastic perturbed system is equivalent to the existence and uniqueness of solutions to the finite horizon stochastic $H_2/H_{\mathrm{\infty }}$ control problem. The resulting $H_2/H_{\mathrm{\infty }}$ controller is characterised by a kind of generalized forward–backward stochastic differential equations with delayed stochastic differential equations as forward equations and generalized anticipated backward stochastic differential equations as backward equations. A practical example is given to illustrate the proposed method.

Keywords:

• Lyapunov–Krasovskii functional
• $H_2/H_{\mathrm{\infty }}$ control
• generalized anticipated backward stochastic differential equations

Disclosure statement

No potential conflict of interest was reported by the authors.

Additional information

Funding

The authors acknowledge the financial support from the National Natural Science Foundation of China (11701214) and the Science and Technology Planning Project of University of Jinan (XKY1806).