The problem of the asymptotic stabilizability in probability of a class of stochastic nonlinear control hybrid systems (with a linear dependence of the control) with state dependent, Markovian, and any switching rule is considered in the paper. To solve the issue, the Lyapunov technique, including a common, single, and multiple Lyapunov function, the hybrid control theory, and some results for stochastic nonhybrid systems are used. Sufficient conditions for the asymptotic stabilizability in probability for a considered class of hybrid systems are formulated. Also the stabilizing control in a feedback form is considered. Furthermore, in the case of hybrid systems with the state dependent switching rule, a method for a construction of stabilizing switching rules is proposed. Obtained results are illustrated by examples and numerical simulations.

In recent years there has been a growing interest in hybrid systems. At the same time one of the basic problems in the control theory is the problem of the stability and the stabilizability of dynamic systems. For this reason, in the paper we deal with the stabilizability problem of nonlinear stochastic hybrid systems.

It is known that, if a common Lyapunov function exists, then the hybrid system is stable for any switching [

Stability problem for stochastic hybrid systems with any switching is considered in [

In the present paper ideas of a feedback control, proposed by Florchinger for nonhybrid stochastic nonlinear control systems [

This paper is divided into three sections and is organized as follows. In the first section we recall some basic definitions related to the issue of stability of stochastic systems and some stability results of hybrid systems. In the second section we present main results regarding the asymptotic stabilizability of stochastic nonlinear hybrid systems with any, state dependent, and Markovian switching rule. Also we give two examples to illustrate obtained stabilizability criteria. In the third section we summarize the obtained results.

Throughout this paper we use the following notation. We mark

Let us consider a stochastic nonlinear hybrid system described by vector Itô differential equations as follows:

We consider three types of the switching rules

Markovian switching

In the case of Markovian switching rule we mean the switching rule with fixed and given matrix

We use the following definitions.

The null solution

stable in probability if

asymptotically stable in probability if it is stable in probability and

We say that a continuous function

For any twice differentiable function

The hybrid system

Sufficient conditions for the asymptotic stability in probability of the null solution

Let

A Lyapunov function

Using Theorem

Note that it is a known fact that if there exists a common Lyapunov function for the hybrid system (

A Lyapunov function

Note that it is a known fact that if there exists a single Lyapunov function for the hybrid system (

A family of Lyapunov functions

Since we deal only with the stability problem of the null solution in the further part of this paper we use the formulation “system is stable,” when we mean that “the null solution of system is stable.”

The aim of this paper is to establish sufficient conditions for the asymptotic stabilizability in probability for the stochastic nonlinear control hybrid system given by vector Itô differential equations as follows:

We use the following definition of the stabilizability.

The hybrid system (

Let us consider nonhybrid stochastic control nonlinear systems given by [

We introduce the following notation. The second order differential operator

Then the following corollary can be formulated from theorem given by Florchinger in [

Assume that there exists a Lyapunov function

Using Theorem of Khasminskii et al. [

Following the idea of Florchinger [

The following theorem gives sufficient conditions for the asymptotic stabilizability in probability of the stochastic hybrid system (

Assume that there exist Lyapunov functions

Using Assumption (i) we can conclude that the infinitesimal generator

Now the thesis follows from Theorem

The family of Lyapunov functions satisfying (

Let us consider a stochastic nonlinear control hybrid system with any switching rule described by the following vector Itô differential equations:

Following the idea of Florchinger [

Using this notation we formulate the next theorem that gives sufficient conditions for the asymptotic stabilizability in probability of the stochastic hybrid system (

Assume that there exists a Lyapunov function

Using Assumption (i) we can conclude that the infinitesimal generator

By means of equation (

Note that from condition (

Let us consider a particular case of hybrid system (

We look for the feedback

An examplary simulation of a trajectory of the hybrid system (

An exemplary stable path of system (

An exemplary asymptotically stable path of system (

In this section we consider a stochastic nonlinear control hybrid system with state dependent switching rule described by vector Itô differential equations as follows:

Following the methodology introduced in [

A set of regions

The interior of

Let us denote the domain of the active-region set of system (

Following the idea of Florchinger [

The differential operator

The following theorem gives sufficient conditions for the asymptotic stabilizability of the stochastic hybrid system (

Assume that there exists a Lyapunov function

Using Assumption (i) similarly as in the proof of Theorem

We note that in the assumptions of Theorems

Note that in general, the presence of input constraints inherently limits the set of initial conditions from which stability can be achieved (the so-called null controllable region) [

Let us consider a particular case of the hybrid system (

We look for the stabilizing switching rule

We obtain

From (

We note that

An examplary simulation of a trajectory of the hybrid system (

Regions

An exemplary stable path of system (

An exemplary asymptotically stable path of system (

In this paper stochastic nonlinear control hybrid systems, consisting of unstable and stable subsystems described by Itô stochastic differential equations, have been analyzed in terms of the stabilizability. The asymptotic stabilizability in probability problem, for considered class of hybrid systems with any, state dependent, and Markovian switching rules, has been discussed. It has been assumed that the trivial solution of unforced hybrid system is stable in probability while some subsystems of unforced hybrid systems still can be unstable. By applying the obtained control the trivial solution of hybrid system becomes asymptotically stable in probability.

To find sufficient stabilizability conditions and to obtain the control law in a feedback form, the Lyapunov function technique (including a common, single, and a multiple Lyapunov function), the hybrid control theory, and some results of Florchinger [

The author declares that there is no conflict of interests regarding the publication of this paper.

The author gratefully acknowledges the research support from Cardinal Stefan Wyszyński University in Warsaw.