#### Abstract

The problem of th mean exponential stability and stabilizability of a class of stochastic nonlinear and bilinear hybrid systems with unstable and stable subsystems is considered. Sufficient conditions for the th mean exponential stability and stabilizability under a feedback control and stabilizing switching rules are derived. A method for the construction of stabilizing switching rules based on the Lyapunov technique and the knowledge of regions of decreasing the Lyapunov functions for subsystems is given. Two cases, including single Lyapunov function and a a single Lyapunov-like function, are discussed. Obtained results are illustrated by examples.

#### 1. Introduction

The problem of stability and stabilization of dynamic systems is one of the basic problems in the control theory. It is well known that there are classes of control systems which cannot be stabilized by a single feedback control [1]. In this case and in the case of the hybrid control systems, switched controls can assure the stability.

Liberzon and Morse in [2] mention that one of the basic problems for dynamic systems is the construction of stabilizing switching laws [1, 3–6]. It is known that if a common Lyapunov function exists, then the hybrid system is stable for any switching. In the absence of the common Lyapunov function, stability properties of the hybrid system in general depend on the switching signal, and in this case the hybrid system is not stable for any switching rules but only for the so-called stabilizing switching rules [1]. In this case, more generally a single Lyapunov and a single Lyapunov-like functions have been introduced [1, 3, 7]. Some results for the linear stochastic hybrid systems are given in [8] and for the nonlinear deterministic hybrid systems are given in [9]. Recent results for the deterministic hybrid systems are collected and summarized in [10].

In the present paper, ideas of a feedback control, proposed by Florchinger for nonhybrid stochastic nonlinear control systems [11, 12], are used and combined with the concept of stabilizing switching rules for hybrid systems to derive the results for the th mean exponential stabilizability of stochastic nonlinear and bilinear hybrid systems consisting of unstable and stable structures. The authors propose also a design method for stabilizing switching rules, which is based on the knowledge of regions of decreasing the Lyapunov functions for subsystems. Similar methods were used for deterministic hybrid systems, for example, in [3, 6–8]. In this paper, we extend them to stochastic hybrid case.

#### 2. Mathematical Preliminaries

Throughout this paper, we use the following notation. Let be the Euclidean norm. By we denote the eigenvalue of the matrix , and and denote the smallest and the biggest eigenvalues of the matrix , respectively. We denote by the transposition of matrix . We mark , . Let be a complete probability space with a filtration satisfying usual conditions. Let , , be the -dimensional standard Wiener process defined on the probability space . Let be the set of states, and let be the stochastic switching rule. We denote switching times as and assume that there is a finite number of switches on every finite time interval. We assume that processes and are both adapted. We say that a proper twice differentiable function is a Lyapunov function if and , for any .

Let us consider the stochastic hybrid system described by the vector Itô differential equation where is the state vector and is an initial condition, . Functions and are locally Lipschitz such that , , . The local Lipschitz conditions together with these enforced on the switching rule ensure that there exists a unique solution to the hybrid system (1).

For any twice differentiable with respect to and once differentiable with respect to function (i.e., ), the th process has a generator (the Itô operator for the th subsystem of the system (1)) given in every structure by

We use the following definitions.

*Definition 1. *The null solution of the stochastic differential equation (1) is said to be th mean exponentially stable, , if there exists a pair of positive scalars , such that
where is called the decay rate.

*Definition 2. *The hybrid system (1) is said to be stabilizable if there exist a switching signal and the associated linear feedback control law such that the hybrid system (1) is th mean exponentially stable for some .

*Definition 3. *A Lyapunov function satisfying
is called a common Lyapunov function for the hybrid system (1).

Note that it is a known fact that if there exists a common Lyapunov function for the hybrid system (1), then the null solution of (1) is asymptotically stable for any switching.

*Definition 4. *A Lyapunov function satisfying
for some switching rule is called a single Lyapunov function for the hybrid system (1).

Lemma 5 ([13] (Itô formula)). *If , then for any switching times ,
**
if the integrations involved exist and are finite.*

Following the methodology introduced in [6], for deterministic hybrid systems, we assume that the hybrid state space is partitioned into regions , and . We consider a special class of switching rule given by Note that a switching rule given by (7) is a stochastic switching rule because of its dependence on the stochastic process . Our aim is to find a special partition defined by (7) such that every switching rule is a stabilizing switching rule for a considered class of stochastic hybrid systems.

#### 3. Stability of Nonlinear Stochastic Hybrid Systems

First, we study the problem of the stability of the nonlinear stochastic hybrid system (1). Two cases, single Lyapunov and single Lyapunov-like functions, are considered.

##### 3.1. Single Lyapunov Functions

We formulate a theorem which establishes sufficient conditions for the th mean exponential stability of the nonlinear hybrid system (1).

Theorem 6. *If the following conditions hold:*(1)*there exist a Lyapunov function and positive constants such that
*(2)*there exists a Lebesgue-measurable function such that
**Then the null solution of the stochastic hybrid system (1) is th mean exponentially stable under the stabilizing switching rule . *

*Proof. *From assumptions for the Lyapunov function , we obtain
From (11) and the Itô formula 6, we obtain
and by Gronwall’s inequality
Now from (10) and (13), we obtain the following inequality:
Hence, the thesis follows.

Notice that function is a single Lyapunov function for the hybrid system (1).

##### 3.2. Single Lyapunov-Like Functions

In the case when condition (2) of Theorem 6 is not satisfied, then one can look for a single Lyapunov-like function. We assume in this case that the hybrid state space is partitioned into regions and , which can be separated into two disjoint subregions: a stable subregion and an unstable region , (the upper scripts “s” and “us” denote stable and unstable regions, resp.) that is,

Let us denote by the sum of time intervals of the residence of the system (1) in the regions , , and by the sum of time intervals of the residence of the system (1) in the regions , .

In this case we cannot construct a single Lyapunov function, but we can look for a single Lyapunov-like function defined as follows.

*Definition 7. *A Lyapunov function is called a single Lyapunov-like function if there exist positive constants and such that
for some switching rule .

Using this definition, the following theorem can be formulated.

Theorem 8. *Let one assume that the following conditions hold: *(1)* there exist a Lyapunov function and positive constants such that
*(2)* there exist Lebesgue-measurable functions and such that partition (15) satisfies the conditions
*(3)* there exists a Lebesguemeasurable function such that , where
**Then the null solution of (1) is th mean exponentially stable under the stabilizing switching rule .*

*Proof. *From assumptions for , it follows that
Let us consider the switching strategy described by (19). Then we obtain
Further proof is similar to the proof of Theorem 6. Hence, the thesis follows.

Notice that function is a single Lyapunov-like function for the hybrid system (1).

*Example 9. *Let us consider a special case of the system (1) with the two subsystems () given as follows:
where

Let us consider the Lyapunov function . Since , , then
Note that and functions and are constant and are given as follows , .

Condition
is satisfied for . Exemplary simulations are shown in Figures 1, 2, and 3.

**(a)**

**(b)**

From Theorem 8, it follows that is the single Lyapunov-like function for the system (22), and it is exponentially stable in mean square with a decay rate for the switching strategy given by

#### 4. Stabilizability of Stochastic Hybrid Systems

In this section, we discuss the stabilizability problem of the stochastic nonlinear and bilinear hybrid systems. We formulate sufficient conditions for the th mean exponential stabilizability, and we find a control of feedback form for the considered class of systems.

##### 4.1. Stabilizability of Nonlinear Stochastic Hybrid Systems

Let us consider the stochastic control hybrid system described by the vector Itô differential equations where is the state vector, is a measurable —a real-valued control vector law, is an initial condition, and . Functions and are the locally Lipschitz, , , , , .

The local Lipschitz condition together with these enforced on the switching rule ensures that there is a unique solution to the hybrid system (27).

The aim of this part of the paper is to establish sufficient conditions under which one can design a state feedback control law so that the null solution of the stochastic hybrid control system (27) is th mean exponentially stable. We extend the results of Florchinger for the stochastic nonhybrid systems [11, 12] to the hybrid systems. Some results for asymptotic stability and stabilizability for the hybrid system (27) with Markovian or any switchings under a feedback control have been proposed in [14].

We introduce the following notation of operators and , , , for :

Then, the following stabilization result for the control hybrid system (27) holds.

Theorem 10. *Suppose that the following conditions hold:*(1)* there exist a Lyapunov function and positive constants such that
*(2)* there exists a Lebesgue-measurable function such that partition determined by (7) satisfies conditions**Then the control law given as follows:
**
together with the stabilizing switching rule renders the null solution of the stochastic hybrid system (27) the th mean exponentially stable. *

* Proof. *Applying the infinitesimal operator defined by (2) to the hybrid system (27), we find that
Now the thesis follows from Theorem 6.

We can formulate a more general theorem in a case when a Lyapunov-like function exists as follows.

Theorem 11. *Suppose that the following conditions hold:*(1)* there exist a Lyapunov function and positive constants such that
*(2)* there exist Lebesgue-measurable functions and such that partition (15) satisfies conditions
*(3)* there exists a Lebesgue-measurable function such that , where
**Then the control law given as follows
**
together with the stabilizing switching rule renders the null solution of the stochastic hybrid system (27) the th mean exponentially stable. *

*Proof. *The thesis follows from Theorem 8.

##### 4.2. Stabilizability of the Bilinear Hybrid Systems

Let us consider a special class of the system (27) given by a bilinear stochastic hybrid system as follows: where is the state vector, , is the control vector, , is an initial condition, and , , , , are for every constant matrices of dimension .

For this particular case, we can combine the above results with the theorem given by Mao [15] for the stochastic linear systems and formulate the theorems which can be obtained directly from Theorems 10 and 11. Sufficient conditions for the th mean exponential stabilizability for the linear hybrid systems are formulated in [16].

Operators (28) reduce to the following ones: where and denote gradient and Hessian of the function , respectively.

Theorem 12. *Suppose that there exist symmetric positive definite matrix , constant , and positive constants , such that the following conditions are satisfied:
**
Then control of a form
**
together with the stabilizing switching rule makes the hybrid system (37) the th mean exponentially stable for *(a)* if , *(b)* if . *

*Proof. *The thesis of the theorem follows from Theorem 10. Let us choose a Lyapunov function of a form
Notice that satisfies assumption of Theorem 10 for and . Then using assumptions (39), we obtain
where
From Theorem 10, it follows that the control chosen as follows:
together with the stabilizing switching rule makes the system (37) the th mean exponentially stable for (a) if , (b) if . Hence, the thesis follows.

Notice that function is a single Lyapunov function for the hybrid system (37). See [16] for the details of the proof.

*Remark 13. *In a particular case of , we obtain the following criterion.

*Criterion 14. *Suppose that there exists a symmetric positive definite matrix such that
Then the control of a form
together with the stabilizing switching rule exponentially in mean-square stabilizes the bilinear hybrid system (37).

We formulate now a more general theorem which formulates sufficient conditions of th mean exponential stabilizability for the stochastic bilinear hybrid system (37) in a case when a single Lyapunov-like function exists. Exemplary simulations are shown in Figures 4 and 5.

**(a) Trajectory of the hybrid system**

**(b) Trajectory of the Lyapunov function**

Theorem 15. *Suppose that the following conditions are satisfied: *(1)* there exist symmetric positive definite matrix , constants , and positive constants such that
*(2)* there exists constant such that , where
**Then the control of a form
**
together with the stabilizing switching rule makes the hybrid system (37) the th mean exponentially stable for *(a)* if , *(b)* if .*

*Proof. *The thesis follows from Theorem 11. Let us choose a Lyapunov function as follows:
Function satisfies assumption of Theorem 11 for and .

Furthermore,
Since condition (51) is satisfied, assumption of Theorem 11 also holds. Assumption of Theorem 11 follows directly. Now using Theorem 11, we obtain that control can be chosen as follows:
Hence, the thesis follows.

Notice that function is a single Lyapunov-like function for the hybrid system (37). See [16] for the details of the proof.

*Example 16. *Let us consider a special case of the hybrid system (37) given as follows:
where
We study the problem of the th mean exponential stabilizability for the hybrid system (53) for . We look for a control vector of a form
Let us choose the Lyapunov function of a form . Then regions , , are given as follows:
Condition (39) of Theorem 12 is satisfied for . Function is a single Lyapunov function for the system (53). From Theorem 12, it follows that the control
together with stabilizing switching rule given by
exponentially th mean, stabilizes the system (53) for .

#### 5. Conclusions

In this paper, nonlinear and the bilinear hybrid systems parametrically excited by a white noise, consisted of unstable and stable subsystems the described by the Itô stochastic differential equations, have been analyzed. To find sufficient conditions for the th mean exponential stability and stabilizability, the Lyapunov function techniques and the hybrid control theory have been used. We have found the control of a feedback form, which is a generalization of a feedback control proposed by Florchinger for the nonhybrid systems [11, 12], and the stabilizing switching rule, which is constructed on the basis of the knowledge of the regions of decreasing of the Lyapunov functions for subsystems.

Results for the asymptotic stabilizability under a feedback control of the stochastic nonlinear and bilinear hybrid systems with Markovian or any switching rule have been discussed in [14] and for the th mean exponential stability and stabilizability of the stochastic linear hybrid system in [4, 16]. The obtained results have been illustrated by an example.

The proposed criteria of the th mean exponentially stability and stabilizability can be generalized to the hybrid systems parametrically excited by Gaussian colored and non-Gaussian noises.

The presented results cannot be compared because they relate to different systems, class. For some class of systems stability conditions obtained using single Lyapunov function approach are the same as those obtained using single Lyapunov-like function approach, while for a wide class of systems, for which Lyapunov-like functions can be used, generally we cannot use single Lyapunov functions. The relationship between the obtained results can be summarized in Table 1.

#### Acknowledgment

The authors gratefully acknowledge research support from Cardinal Stefan Wyszyński University in Warsaw.