Stabilisation of hybrid stochastic differential equations by delay feedback control

Abstract

This paper is concerned with the exponential mean-square stabilisation of hybrid stochastic differential equations (also known as stochastic differential equations with Markovian switching) by delay feedback controls. Although the stabilisation by nondelay feedback controls for such equations has been discussed by several authors, there is so far little on the stabilisation by delay feedback controls and our aim here is mainly to close the gap. To make our theory more understandable as well as to avoid complicated notations, we will restrict our underlying hybrid stochastic differential equations to a relatively simple form. However our theory can certainly be developed to cope with much more general equations without any difficulty.

Stabilization of stochastic differential equations with Markovian switching by feedback control based on discrete-time state observation with a time delay

“…The method of linear matrix inequalities (LMIs) [24,28] is one of the commonly used technique in the study of stability of linear systems. Although there are other methods (see the survey paper [29]), we shall adopt the LMIs in this paper.…”

“…There are many papers devoted to this field with emphasis on the asymptotic analysis of stability, and we just mention some of the works [14,15,8,13,19,17,18,20,21,22,23,24] and the references therein. One classical problem of this field is that for some given unstable SDE system d(x(t)) = f (x(t), r(t), t)dt + g(x(t), r(t), t)dw(t),…”

Stabilisation of hybrid stochastic differential equations by feedback control based on discrete‐time observations of state and mode

“…To use our new theory, we need to know how to design the control function u : R n × S × R + → R n such that the hybrid SDE (10) is mean-square exponentially stable, namely Assumption 2 is satisfied. There is an intensive literature in the area of stability of hybrid SDEs, for example, [5,11,13,19,20]. For the convenience of the reader, we state a lemma which follows easily from [14, Thereom 4.5 on page 166].…”

“…We here use the structure control of the form F (r(δ t ))G(r(δ t ))x(δ t ) instead of the general form u(x(δ t ), r(δ t ), t) is because that such type of controls is one of the most popular feedback controls for linear systems (see e.g. [1,4,5,11,15]). We will write F (i) = F i and G(i) = G i .…”

“…Instead of using a classical continuoustime feedback control u(x(t), r(t), t) which requires continuous observations of the state x(t) for all time t ≥ 0 (see e.g. [5,11,13,19,20]), Mao proposed in [10] to design a feedback control u(x(δ(t, t 0 , τ )), r(t), t) based on the discrete-time observations of the state in order to make the controlled system…”