# Advances in Stabilisation of Hybrid Stochastic Differential Equations by Delay Feedback Control

## TL;DR

This paper introduces a new method for designing delay feedback control in hybrid stochastic differential equations, linking it to existing control functions without delay, and provides bounds on delay length with numerical validation.

## Contribution

It presents a novel theorem connecting delay feedback control with non-delay control functions for HSDEs, facilitating stability analysis and control design.

## Key findings

- Established a new theorem linking delay feedback control to existing control functions.
- Derived methods to determine upper bounds for delay length.
- Numerical simulations validate the theoretical results.

## Abstract

A novel approach to design the feedback control based on past states is proposed for hybrid stochastic differential equations (HSDEs). This new theorem builds up the connection between the delay feedback control and the control function without delay terms, which enables one to construct the delay feedback control using the existing results on stabilities of HSDEs. Methods to find the upper bound of the length of the time delay are also investigated. Numerical simulations are presented to demonstrate the new theorem.