𝑝-Stability and 𝑝-Stabilizability of Stochastic Nonlinear and Bilinear Hybrid Systems under Stabilizing Switching Rules

Abstract

The problem of 𝑝th 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 𝑝th 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 𝑝th 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 𝜆(𝐀) we denote the eigenvalue of the matrix 𝐀, and 𝜆_min(𝐀) and 𝜆_max(𝐀) denote the smallest and the biggest eigenvalues of the matrix 𝐀, respectively. We denote by 𝐀^𝑇 the transposition of matrix 𝐀. We mark 𝕋=[𝑡₀,∞), 𝑡₀≥0. Let Ξ=(Ω,ℱ,{ℱ_𝑡}_𝑡≥0,ℙ) be a complete probability space with a filtration {ℱ_𝑡}_𝑡≥0 satisfying usual conditions. Let 𝐰(𝐭)=[𝑤₁(𝑡),…,𝑤_𝑀(𝑡)]^𝑇, 𝑡≥0, be the 𝑚-dimensional standard Wiener process defined on the probability space Ξ. Let 𝓢={1,…,𝑁} be the set of states, and let 𝜎(⋅)∶[𝑡₀,∞)→𝓢 be the stochastic switching rule. We denote switching times as 𝜏₁,𝜏₂,… and assume that there is a finite number of switches on every finite time interval. We assume that processes 𝑤_𝑘 and 𝜎 are both {ℱ_𝑡}_𝑡≥0 adapted. We say that a proper twice differentiable function 𝑉∶ℛ^𝑛→ℛ₊=[0,∞) is a Lyapunov function if 𝑉(𝟎)=0 and 𝑉(𝐱)>0, for any 𝐱≠0.

Let us consider the stochastic hybrid system described by the vector Itô differential equation 𝑑𝐱(𝑡)=𝐟(𝐱(𝑡),𝑡,𝜎)𝑑𝑡+^𝑀_𝑘=1𝐠_𝑘(𝐱(𝑡),𝑡,𝜎)𝑑𝑤_𝑘(𝑡),𝜎𝑡₀,𝐱𝑡₀=𝜎₀,𝐱₀, (1) where 𝐱∈ℛ^𝑛 is the state vector and (𝜎₀,𝐱₀) is an initial condition, 𝑡∈𝕋. Functions 𝐟∶ℛ^𝑛×𝕋×𝓢→ℛ^𝑛 and 𝐠_𝑘∶ℛ^𝑛×𝕋×𝓢→ℛ^𝑛 are locally Lipschitz such that ∀𝑙∈𝓢, 𝑡∈𝕋𝐟(𝟎,𝑡,𝑙)=𝐠_𝑘(𝟎,𝑡,𝑙)=𝟎, 𝑘=1,…,𝑀. The local Lipschitz conditions together with these enforced on the switching rule 𝜎 ensure that there exists a unique solution to the hybrid system (1).

For any twice differentiable with respect to 𝐱∈ℛ^𝑛 and once differentiable with respect to 𝑡∈𝕋 function 𝜙(𝐱,𝑡,𝑙) (i.e., 𝜙(𝐱,𝑡,𝑙)∈𝐶^2,1(ℛ^𝑛×𝕋×𝓢;[0,∞))), the 𝑙th process has a generatorℒ_𝑙⁽¹⁾ (the Itô operator for the 𝑙th subsystem of the system (1)) given in every structure by ℒ_𝑙⁽¹⁾𝜙()=()𝐱,𝑡,𝑙𝜕𝜙𝐱,𝑡,𝑙+𝜕𝑡^𝑛_𝜇=1𝑓^𝜇()()𝐱,𝑡,𝑙𝜕𝜙𝐱,𝑡,𝑙𝜕𝑥_𝜇+12^𝑛𝑀_{𝑟,𝑠=1𝑘=1}𝑔^𝑟_𝑘(𝐱,𝑡,𝑙)𝑔^𝑠_𝑘()𝜕𝐱,𝑡,𝑙²𝜙(𝐱,𝑡,𝑙)𝜕𝑥_𝑟𝜕𝑥_𝑠,𝑙∈𝓢. (2)

We use the following definitions.

Definition 1. The null solution 𝐱≡𝟎 of the stochastic differential equation (1) is said to be 𝑝th mean exponentially stable, 𝑝>0, if there exists a pair of positive scalars 𝜅, 𝑐 such that ∀𝐱₀,𝑡₀∈ℛ^𝑛×[),𝔼𝐱0,∞𝑡,𝐱₀,𝑡₀^𝑝≤𝑐𝔼𝐱₀^𝑝exp−𝜅𝑡−𝑡₀,𝑡≥𝑡₀, (3) where 𝜅 is called the decay rate.

Definition 2. The hybrid system (1) is said to be stabilizable if there exist a switching signal 𝜎∶𝕋→𝓢 and the associated linear feedback control law 𝐮∶ℛ^𝑛×𝕋×𝓢→ℛ^𝑚 such that the hybrid system (1) is 𝑝th mean exponentially stable for some 𝑝>0.

Definition 3. A Lyapunov function 𝑉(𝐱)∈𝐶²(ℛ^𝑛;ℛ₊) satisfying ∀𝑙∈𝓢,∀𝐱≠0ℒ_𝑙⁽¹⁾𝑉(𝐱)<0 (4) 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 𝑉(𝐱)∈𝐶²(ℛ^𝑛;ℛ₊) satisfying ∀𝐱≠0ℒ_𝜎(𝑡)⁽¹⁾𝑉(𝐱)<0 (5) for some switching rule 𝜎 is called a single Lyapunov function for the hybrid system (1).

Lemma 5 ([13] (Itô formula)). If 𝜙(𝐱,𝑡,𝑙)∈𝐶^2,1(ℛ^𝑛×𝕋×𝓢;[0,∞)), then for any switching times 0≤𝜏₁≤𝜏₂<∞, ∀𝑙∈𝓢𝔼𝜙𝐱𝜏₂,𝜏₂,𝑙=𝔼𝜙𝐱𝜏₁,𝜏₁,𝑙+𝔼^𝜏2_𝜏1ℒ_𝑙𝜙(𝐱(𝑠),𝑠,𝑙)𝑑𝑠 (6) 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 Ω_𝑙,𝑙∈𝓢, and ⋃_𝑙∈𝓢Ω_𝑙=ℛ^𝑛. We consider a special class of switching rule 𝒮𝒲 given by 𝒮𝒲=𝜎∶(𝜎(𝑡)=𝑙)⇒𝐱(𝑡)∈Ω_𝑙,𝑙∈𝓢. (7) Note that a switching rule 𝜎∈𝒮𝒲 given by (7) is a stochastic switching rule because of its dependence on the stochastic process 𝐱(𝑡). Our aim is to find a special partition Ω_𝑙 defined by (7) such that every switching rule 𝜎∈𝒮𝒲 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 𝑝th mean exponential stability of the nonlinear hybrid system (1).

Theorem 6. If the following conditions hold:(1)there exist a Lyapunov function 𝑉∶ℛ^𝑛→ℛ₊ and positive constants 𝐶_min,𝐶_max such that 𝐶_min|𝐱|^𝑝≤𝑉(𝐱)≤𝐶_max|𝐱|^𝑝,𝑝>0, (8)(2)there exists a Lebesgue-measurable function 𝜅∶𝕋→(0,∞) such that _𝑙∈𝓢Ω_𝑙=ℛ^𝑛Ω,𝑤ℎ𝑒𝑟𝑒_𝑙=𝐱∈ℛ^𝑛∶∀𝑡∈𝕋ℒ_𝑙⁽¹⁾𝑉(𝐱)≤−𝜅(𝑡)𝑉(𝐱)𝑓𝑜𝑟𝑙∈𝓢. (9)Then the null solution 𝐱≡𝟎 of the stochastic hybrid system (1) is 𝑝th mean exponentially stable under the stabilizing switching rule 𝜎_st∈𝒮𝒲.

Proof. From assumptions for the Lyapunov function 𝑉∶ℛ^𝑛→ℛ₊, we obtain 𝐶_min|𝐱|^𝑝≤𝑉(𝐱)≤𝐶_max|𝐱|^𝑝,𝑝>0,𝔼ℒ_𝜎⁽¹⁾_st𝑉(𝐱)≤−𝜅(𝑡)𝔼[𝑉(𝐱)]. (10)(11) From (11) and the Itô formula 6, we obtain 𝔼[𝑉(𝐱)]=𝔼𝑉𝐱₀+𝔼^𝑡_𝑡0ℒ_𝜎⁽¹⁾_st𝑉(𝐱(𝑠𝑉𝐱))𝑑𝑠≤𝔼₀−^𝑡_𝑡0𝜅(𝑡)𝔼[𝑉(𝐱(𝑠))]𝑑𝑠 (12) and by Gronwall’s inequality 𝐸[𝑉(𝐱)]≤𝐸𝑉𝐱₀exp−^𝑡_𝑡0𝜅(𝑡)𝑑𝑠. (13) Now from (10) and (13), we obtain the following inequality: 𝐸|𝐱(𝑡)|^𝑝≤𝐶_max𝐶_min𝐸𝐱₀^𝑝exp−^𝑡_𝑡0𝜅(𝑡)𝑑𝑠,𝑝>0. (14) Hence, the thesis follows.

Notice that function 𝑉 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 Ω_𝑙 and 𝑙∈𝓢, which can be separated into two disjoint subregions: a stable subregion Ω^s_𝑙 and an unstable region Ω^us_𝑙, 𝑙∈𝓢 (the upper scripts “s” and “us” denote stable and unstable regions, resp.) that is, Ω_𝑙=Ω^s_𝑙∪Ω^us_𝑙,Ω^s_𝑙∩Ω^us_𝑙=∅. (15)

Let us denote by 𝕋_s the sum of time intervals of the residence of the system (1) in the regions Ω^s_𝑙, 𝑙∈𝓢, and by 𝕋_us the sum of time intervals of the residence of the system (1) in the regions Ω^us_𝑙,  𝑙∈𝓢.

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 𝑉(𝐱)∈𝐶²(ℛ^𝑛;ℛ₊) is called a single Lyapunov-like function if there exist positive constants 𝜅_s and 𝜅_us such that ℒ_𝜎(𝑡)⁽¹⁾𝑉(𝐱)≤−𝜅_s𝑉(𝐱)for𝐱∈Ω^s_𝜎(𝑡),𝜅_us𝑉(𝐱)for𝐱∈Ω^us_𝜎(𝑡), (16) for some switching rule 𝜎.

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 𝑉(𝐱)∈𝐶²(ℛ^𝑛;ℛ₊) and positive constants 𝐶_min,𝐶_max such that 𝐶_min|𝐱|^𝑝≤𝑉(𝐱)≤𝐶_max|𝐱|^𝑝,𝑝>0, (17)(2) there exist Lebesgue-measurable functions 𝜅_s∶𝕋→(0,∞) and 𝜅_us∶𝕋→(0,∞) such that partition (15) satisfies the conditions _𝑙∈𝓢Ω_𝑙=ℛ^𝑛Ω,𝑤ℎ𝑒𝑟𝑒_s𝑙=𝐱∈ℛ^𝑛∶∀𝑡∈𝕋ℒ_𝑙⁽¹⁾𝑉(𝐱)≤−𝜅_s(𝑡)𝑉(𝐱Ω)𝑓𝑜𝑟𝑙∈𝓢,_us𝑙=𝐱∈ℛ^𝑛∶∀𝑡∈𝕋ℒ_𝑙⁽¹⁾𝑉(𝐱)≤𝜅_us(𝑡)𝑉(𝐱)𝑓𝑜𝑟𝑙∈𝓢, (18)(3) there exists a Lebesguemeasurable function 𝜅∶𝕋→(0,∞) such that 𝒯≠∅, where 𝒯=𝜎∈𝒮𝒲∶_𝑇u𝑠𝜅_us(𝑠)+𝜅(𝑠)𝑑𝑠≤_𝑇s𝜅_s(𝑠)−𝜅(𝑠)𝑑𝑠. (19)Then the null solution 𝐱≡𝟎 of (1) is 𝑝th mean exponentially stable under the stabilizing switching rule 𝜎_st∈𝒯.

Proof. From assumptions for 𝜎_st∈𝒮𝒲, it follows that 𝔼[𝑉(𝐱)]≤𝔼𝑉𝐱₀exp_𝑇us𝜅_us(𝑠)𝑑𝑠−_𝑇s𝜅_s(𝑠)𝑑𝑠. (20) Let us consider the switching strategy 𝜎_st∈𝒯 described by (19). Then we obtain 𝔼[𝑉(𝐱)]≤𝔼𝑉𝐱₀exp−^𝑡_𝑡0𝜅(𝑠)𝑑𝑠. (21) Further proof is similar to the proof of Theorem 6. Hence, the thesis follows.

Notice that function 𝑉 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 (𝓢={1,2}) given as follows: (𝑡)=[𝐀(𝜎)𝐱(𝑡)((𝜎)(𝑡),𝜎𝑡𝑑𝐱+𝐟𝐱,𝜎)]𝑑𝑡+𝐆𝑑𝑤₀,𝐱𝑡₀=𝜎₀,𝐱₀∈𝓢×ℛ^𝑛, (22) where 𝐀(1)=(2)=,𝐟()=0𝑓2.182−4−2.82,𝐀−1.9054−21.095𝐱,1₂𝑥₂()=0,1𝑓,𝐟𝐱,2₂𝑥₂,𝑓,2₂𝑥₂,1=0.8𝑥₂−0.2for𝑥₂𝑥<−1,₂for𝑥₂∈[−1,1],0.8𝑥₂+0.2for𝑥₂𝑓>1,₂𝑥₂,2=0.9𝑥₂−0.1for𝑥₂𝑥<−1,₂for𝑥₂∈[−1,1],0.9𝑥₂+0.1for𝑥₂𝐆(1)=(2)=.>1,0.2000.2,𝐆0.1000.1 (23)
Let us consider the Lyapunov function 𝑉(𝐱)=𝐱^𝑇𝐱. Since 0≤𝑓₂(𝑥₂,𝑙)𝑥₂≤𝑥²₂, 𝑙=1,2, then ℒ₁⁽²²⁾𝑉(𝐱)=2.2𝑥²₁−2𝑥₁𝑥₂−2.8𝑥²₂+𝑓(𝐱,1)^𝑇𝐱≤−0.8𝑥²₁+𝑥²₂forΩ^s₁=𝐱∈ℛ²∶𝑥₂+3𝑥₁𝑥₁−𝑥₂,ℒ≤0₂₍₂₂₎𝑉(𝐱)=−1.9𝑥²₁+2𝑥₁𝑥₂+1.1𝑥²₂+𝑓(𝐱,2)^𝑇𝐱≤−0.8𝑥²₁+𝑥²₂forΩ^s₂=𝐱∈ℛ²∶−1.1𝑥²₁+2𝑥₁𝑥₂+2.9𝑥²₂,ℒ≤0₂₍₂₂₎𝑉(𝐱)=−1.9𝑥²₁+2𝑥₁𝑥₂+1.1𝑥²₂+𝑓(𝐱,2)^𝑇𝐱≤1.1𝑥²₁+𝑥²₂forΩ^us₂=𝐱∈ℛ²∶𝑥₂+3𝑥₁𝑥₂−𝑥₁≤0⧵Ω^s₁∪Ω^s₂. (24) Note that ⋃_𝑙∈𝓢Ω_𝑙=ℛ² and functions 𝜅_s and 𝜅_us are constant and are given as follows 𝜅_s=0.8, 𝜅_us=1.1.
Condition 𝕋_us𝕋_s≤𝜅_s−𝜅𝜅_us+𝜅≈0.61 (25) is satisfied for 𝜅=0.08. Exemplary simulations are shown in Figures 1, 2, and 3.

(a) Unstable path of the system (22) for 𝑙=1

(b) Unstable path of the system (22) for 𝑙=2

Figure 1 

Trajectories of the subsystems of (22).

(a) Stable path of the hybrid system (22) with stabilizing switching 𝜎st given by  (26)

(b) Lyapunov function 𝑉=𝐱𝑇𝐱 for the hybrid system  (22)

Figure 2 

Trajectories of the hybrid system (22).

(a)

(b)

Figure 3 

Stable path of the hybrid system (22) with stabilizing switching 𝜎st given by (26).

From Theorem 8, it follows that 𝑉(𝐱)=𝐱^𝑇𝐱 is the single Lyapunov-like function for the system (22), and it is exponentially stable in mean square with a decay rate 𝜅=0.08 for the switching strategy 𝜎_st∈𝒯 given by 𝜎_st()=1𝐱,𝑡if𝐱(𝑡)∈Ω₁,2if𝐱(𝑡)∈Ω₂. (26)

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 𝑝th 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 𝑑𝐱(𝑡)=𝐟₀(𝐱(𝑡),𝑡,𝜎)𝑑𝑡+^𝑝_𝑖=1𝑢_𝑖(𝐱,𝑡,𝜎)𝐟_𝑖(𝐱(𝑡))+,𝑡,𝜎_𝑀𝑘=1𝐠_𝑘(𝐱(𝑡),𝑡,𝜎)𝑑𝑤_𝑘(𝑡),𝜎𝑡₀,𝐱𝑡₀=𝜎₀,𝐱₀, (27) where 𝐱∈ℛ^𝑛 is the state vector, 𝐮=[𝑢₁,…,𝑢_𝑚]^𝑇 is a measurable ℛ^𝑚—a real-valued control vector law, (𝜎₀,𝐱₀)∈𝓢×ℛ^𝑛 is an initial condition, and 𝑡∈𝕋. Functions 𝐟_𝑖∶ℛ^𝑛×𝕋×𝓢→ℛ^𝑛 and 𝐠_𝑘∶ℛ^𝑛×𝕋×𝓢→ℛ^𝑛 are the locally Lipschitz, 𝐟_𝑖(𝟎,𝑡,𝑙)=𝐠_𝑘(𝟎,𝑡,𝑙)=𝟎,  ∀𝑙∈𝓢,  𝑡∈𝕋,  𝑖=0,…,𝑚,  𝑘=1,…,𝑀.

The local Lipschitz condition together with these enforced on the switching rule 𝜎 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 𝐱≡𝟎 of the stochastic hybrid control system (27) is 𝑝th 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 𝐿_𝑙 and 𝐿^𝑖_𝑙,  𝑙∈𝓢,  𝑖=1,…,𝑚, for 𝜙(𝐱,𝑡,𝑙)∈𝐶^2,1(ℛ^𝑛×𝕋×𝓢;ℛ₊): 𝐿_𝑙𝜙()=()𝐱,𝑡,𝑙𝜕𝜙𝐱,𝑡,𝑙+𝜕𝑡^𝑛_𝜇=1𝑓^𝜇₀()()𝐱,𝑡,𝑙𝜕𝜙𝐱,𝑡,𝑙𝜕𝑥_𝜇+12^𝑛_𝑟,s^𝑀_=1𝑘=1𝑔^𝑟_𝑘(𝐱,𝑡,𝑙)𝑔^s_𝑘()𝜕𝐱,𝑡,𝑙²𝜙(𝐱,𝑡,𝑙)𝜕𝑥_𝑟𝜕𝑥_s,𝐿^𝑖_𝑙𝜙(𝐱,𝑡,𝑙)=^𝑛_𝜇=1𝑓^𝜇_𝑖()()𝐱,𝑡,𝑙𝜕𝜙𝐱,𝑡,𝑙𝜕𝑥_𝜇,𝑖=1,…,𝑚,𝑙∈𝓢. (28)

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 𝑉(𝐱)∈𝐶²(ℛ^𝑛;ℛ₊) and positive constants 𝐶_min,𝐶_max such that 𝐶_min|𝐱|^𝑝≤𝑉(𝐱)≤𝐶_max|𝐱|^𝑝,𝑝>0, (29)(2) there exists a Lebesgue-measurable function 𝜅∶𝕋→(0,∞) such that partition determined by (7) satisfies conditions_𝑙∈𝓢Ω_𝑙=ℛ^𝑛Ω,𝑤ℎ𝑒𝑟𝑒_𝑙=𝐱∈ℛ^𝑛∶∀𝑡∈𝕋𝐿_𝑙𝑉(𝐱)≤−𝜅(𝑡)𝑉(𝐱)𝑓𝑜𝑟𝑙∈𝓢. (30)Then the control law 𝐮∶ℛ^𝑛×𝕋×𝓢→ℛ^𝑚 given as follows: 𝑢_𝑖(𝐱,𝑡,𝑙)=−𝐿^𝑖_𝑙𝑉(𝐱),𝑖=1,2,…,𝑚,𝑙∈𝓢 (31) together with the stabilizing switching rule 𝜎_st∈𝒮𝒲 renders the null solution of the stochastic hybrid system (27) the 𝑝th mean exponentially stable.

Proof. Applying the infinitesimal operator ℒ defined by (2) to the hybrid system (27), we find that ∀𝑙∈𝓢ℒ_𝑙⁽²⁷⁾𝑉(𝐱)=𝐿_𝑙𝑉(𝐱)−^𝑚_𝑖=1𝐿^𝑖_𝑙𝑉(𝐱)²≤−𝜅(𝑡)𝑉(𝐱),for𝐱∈Ω_𝑙. (32) 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 𝑉(𝐱)∈𝐶²(ℛ^𝑛;ℛ₊) and positive constants 𝐶_min,𝐶_max such that 𝐶_min|𝐱|^𝑝≤𝑉(𝐱)≤𝐶_max|𝐱|^𝑝,𝑝>0, (33)(2) there exist Lebesgue-measurable functions 𝜅_s∶𝕋→(0,∞) and 𝜅_us∶𝕋→(0,∞) such that partition (15) satisfies conditions _𝑙∈𝓢Ω_𝑙=ℛ^𝑛Ω,𝑤ℎ𝑒𝑟𝑒_s𝑙=𝐱∈ℛ^𝑛∶∀𝑡∈𝕋𝐿_𝑙𝑉(𝐱)≤−𝜅_s(𝑡)𝑉(𝐱Ω)𝑓𝑜𝑟𝑙∈𝓢,_us𝑙=𝐱∈ℛ^𝑛∶∀𝑡∈𝕋𝐿_𝑙𝑉(𝐱)≤𝜅_us(𝑡)𝑉(𝐱)𝑓𝑜𝑟𝑙∈𝓢, (34)(3) there exists a Lebesgue-measurable function 𝜅∶𝕋→ℛ₊ such that 𝒯≠∅, where 𝒯=𝜎∈𝒮𝒲∶_𝑇us𝜅_us(𝑠)+𝜅(𝑠)𝑑𝑠≤_𝑇s𝜅_s(𝑠)−𝜅(𝑠)𝑑𝑠. (35)Then the control law 𝐮∶ℛ^𝑛×𝕋×𝓢→ℛ^𝑚 given as follows 𝑢_𝑖(𝐱,𝑡,𝑙)=−𝐿^𝑖_𝑙𝑉(𝐱),𝑖=1,2,…,𝑚 (36) together with the stabilizing switching rule 𝜎_st∈𝒯 renders the null solution of the stochastic hybrid system (27) the 𝑝th 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: 𝑑𝐱(𝑡)=𝐀(𝜎(𝑡))𝐱(𝑡)+^𝑀_𝑖=1𝑢_𝑖(𝐱(𝑡),𝜎(𝑡))𝐂_𝑖(𝜎(𝑡𝐱(𝑡+)))𝑑𝑡_𝑀𝑘=1𝐁_𝑘(𝜎(𝑡))𝐱(𝑡)𝑑𝑤_𝑘(𝑡),𝜎𝑡₀,𝐱𝑡₀=𝜎₀,𝐱₀, (37) where 𝐱∈ℛ^𝑛 is the state vector, 𝐮=[𝑢₁,…,𝑢_𝑚]^𝑇∈ℛ^𝑚,𝑢_𝑖∶ℛ^𝑛×𝓢→ℛ, is the control vector, 𝑡∈𝕋, (𝜎₀,𝐱₀)∈𝓢×ℛ^𝑛 is an initial condition, and 𝐀(𝑙), 𝐂_𝑖(𝑙), 𝐁_𝑘(𝑙),  𝑖=1,…,𝑚,  𝑘=1,…,𝑀 are for every 𝑙∈𝓢 constant matrices of dimension 𝑛×𝑛.

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 𝑝th mean exponential stabilizability for the linear hybrid systems are formulated in [16].

Operators (28) reduce to the following ones: 𝐿_𝑙𝜙(𝐱,𝑙)=𝐱^𝑇𝐀^𝑇(𝑙)()+1∇𝜙𝐱,𝑙2tr𝐁_𝑘(𝑙)𝐱𝐱^𝑇𝐁_𝑘(𝑙)^𝑇∇²𝜙(),𝐿𝐱,𝑙𝑙∈𝓢,_𝑖𝑙(𝐱,𝑙)=𝐱^𝑇𝐂^𝑇_𝑖(𝑙)∇𝜙(𝐱,𝑙),𝑖=1,…,𝑚,𝑙∈𝓢, (38) where ∇𝜙(⋅,𝑞) and ∇²𝜙(⋅,𝑞) denote gradient and Hessian of the function 𝜙(⋅,𝑞), respectively.

Theorem 12. Suppose that there exist symmetric positive definite matrix 𝐇, constant 𝛼₁, and positive constants 𝛼₂, 𝛼₃ such that the following conditions are satisfied: _𝑙∈𝓢Ω_𝑙=ℛ^𝑛Ω,𝑤ℎ𝑒𝑟𝑒_𝑙=𝐱∈ℛ^𝑛∶𝐱^𝑇𝐀^𝑇(𝑙)1𝐇𝐱+2^𝑀_𝑘=1𝐱^𝑇𝐁^𝑇_𝑘(𝑙)𝐇𝐁_𝑘(𝑙)𝐱≤𝛼₁𝐱^𝑇𝐇𝐱∧𝛼₂𝐱^𝑇≤𝐱𝐇𝐱_𝑇𝐇^𝑀_𝑘=1𝐁_𝑘(𝑙)𝐱≤𝛼₃𝐱^𝑇𝐇𝐱𝑙∈𝓢. (39) Then control 𝐮∶ℛ^𝑛×𝓢→ℛ^𝑚 of a form 𝑢_𝑖(𝐱,𝑙)=−𝑝𝐱^𝑇𝐇𝐱^(𝑝/2)−1𝐱^𝑇𝐂^𝑇_𝑖(𝑙)𝐇𝐱,𝑖=1,2,…,𝑚,𝑙∈𝓢 (40) together with the stabilizing switching rule 𝜎_𝑠𝑡∈𝒮𝒲 makes the hybrid system (37) the 𝑝th mean exponentially stable for (a)2≤𝑝<2+2|𝛼₁|/(𝛼₃)² if 𝛼₁<0, (b)0<𝑝<2−2𝛼₁/(𝛼₂)² if 0≤𝛼₁<𝛼²₂.

Proof. The thesis of the theorem follows from Theorem 10. Let us choose a Lyapunov function of a form 𝑉(𝐱)=𝐱^𝑇𝐇𝐱^(𝑝/2),𝑝>0. (41) Notice that 𝑉 satisfies assumption (1) of Theorem 10 for 𝐶_min=𝜆^(𝑝/2)_min(𝐇) and 𝐶_max=𝜆^(𝑝/2)_max(𝐇). Then using assumptions (39), we obtain ∀𝑙∈𝓢𝐿_𝑙𝑉(𝐱)≤−𝜅𝑉(𝐱)𝜅>0for𝐱∈Ω_𝑙, (42) where 𝑝𝛼𝜅=¹−𝑝2−1𝛼²₃𝑉(𝐱),for2𝛼2≤𝑝<2+¹𝛼₃²,𝛼₁𝑝<0,−𝛼₁−𝑝2−1𝛼²₂𝑉(𝐱),for0<𝑝<2−2𝛼₁𝛼₂²,0≤𝛼₁<𝛼₂². (43) From Theorem 10, it follows that the control chosen as follows: 𝑢_𝑖(𝐱,𝑙)=−𝐿^𝑖_𝑙𝑉(𝐱)=−𝑝𝐱^𝑇𝐇𝐱^(𝑝/2)−1𝐱^𝑇𝐂^𝑇_𝑖(𝑙)𝐇𝐱,𝑖=1,2,…,𝑚,𝑙∈𝓢 (44) together with the stabilizing switching rule 𝜎_st∈𝒮𝒲 makes the system (37) the 𝑝th mean exponentially stable for (a)2≤𝑝<2+2|𝛼₁|/(𝛼₃)² if 𝛼₁<0, (b)0<𝑝<2−2𝛼₁/(𝛼₂)² if 0≤𝛼₁<𝛼²₂. Hence, the thesis follows.

Notice that function 𝑉(𝐱)=(𝐱^𝑇𝐇𝐱)^(𝑝/2) 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 𝑝=2, we obtain the following criterion.

Criterion 14. Suppose that there exists a symmetric positive definite matrix 𝐇 such that _𝑙∈𝓢Ω_𝑙=ℛ^𝑛,whereΩ_𝑙=𝐱∈ℛ^𝑛∶𝐱^𝑇𝐀^𝑇(𝑙)1𝐇𝐱+2^𝑀_𝑘=1𝐱^𝑇𝐁^𝑇_𝑘(𝑙)𝐇𝐁_𝑘(𝑙)𝐱<0,𝑙∈𝓢. (45) Then the control 𝐮∶ℛ^𝑛×𝓢→ℛ^𝑚 of a form 𝑢_𝑖(𝐱,𝑙)=−2𝐱^𝑇𝐂^𝑇_𝑖(𝑙)𝐇𝐱,𝑖=1,2,…,𝑚,𝑙∈𝓢 (46) together with the stabilizing switching rule 𝜎_st∈𝒮𝒲 exponentially in mean-square stabilizes the bilinear hybrid system (37).

We formulate now a more general theorem which formulates sufficient conditions of 𝑝th 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

Figure 4 

Simulation of the hybrid system (53) with stabilizing switching 𝜎=𝜎st for 𝑝=1.

Figure 5 

Simulation of the hybrid system (53) with stabilizing switching 𝜎=𝜎st for 𝑝=1.

Theorem 15. Suppose that the following conditions are satisfied: (1) there exist symmetric positive definite matrix 𝐇, constants 𝛼^s₁,𝛼^us₁, and positive constants 𝛼₂,𝛼₃ such that _𝑙∈𝓢Ω_𝑙=ℛ^𝑛Ω,𝑤ℎ𝑒𝑟𝑒_s𝑙=𝐱∈ℛ^𝑛∶𝐱^𝑇(𝑙)1𝐇𝐀𝐱+2^𝑀_𝑘=1𝐱^𝑇𝐁^𝑇_𝑘(𝑙)𝐇𝐁_𝑘(𝑙)𝐱≤𝛼^s₁𝐱^𝑇𝐇𝐱,𝛼^s₁<𝛼₂²∧𝛼₂𝐱^𝑇≤𝐱𝐇𝐱_𝑇𝐇^𝑀_𝑘=1𝐁_𝑘(𝑙)𝐱≤𝛼₃𝐱^𝑇Ω𝐇𝐱𝑙∈𝓢,_us𝑙=𝐱∈ℛ^𝑛∶𝛼^s₁𝐱^𝑇𝐇𝐱<𝐱^𝑇(𝑙)𝐱+1𝐇𝐀2^𝑀_𝑘=1𝐱^𝑇𝐁^𝑇_𝑘(𝑙)𝐇𝐁_𝑘(𝑙)𝐱≤𝛼^us₁𝐱^𝑇𝐇𝐱,∧𝛼₂𝐱^𝑇≤𝐱𝐇𝐱_𝑇𝐇^𝑀_𝑘=1𝐁_𝑘(𝑙)𝐱≤𝛼₃𝐱^𝑇𝐇𝐱𝑙∈𝓢, (47)(2) there exists constant 𝜅>0 such that 𝒯≠∅, where 𝕋𝒯=𝜎∈𝒮𝒲∶^us𝕋_s≤𝜅_s−𝜅𝜅_us,𝜅+𝜅_s=𝑝𝛼^s₁−𝑝2−1𝛼²₃dla𝛼2≤𝑝<2+2^s1𝛼₃²,𝛼₁𝑝<0,−𝛼^s₁−𝑝2−1𝛼²₂dla𝛼0<𝑝<2−2^s1𝛼₂²,0≤𝛼^s₁<𝛼₂²,𝜅_us=𝑝𝛼^us₁+𝑝2−1𝛼²₃dla𝑝≥2,𝛼^us₁𝑝𝛼≥0,_us1+𝑝2−1𝛼²₂dla0<𝑝<2,𝛼^us₁≥𝛼₂². (48)Then the control 𝐮∶ℛ^𝑛×𝓢→ℛ^𝑚 of a form 𝑢_𝑖(𝐱,𝑙)=−𝑝𝐱^𝑇𝐇𝐱^(𝑝/2)−1𝐱^𝑇𝐶^𝑇_𝑖(𝑙)𝐇𝐱,𝑖=1,2,…,𝑚,𝑙∈𝓢 (49) together with the stabilizing switching rule 𝜎_st∈𝒯 makes the hybrid system (37) the 𝑝th mean exponentially stable for (a)2≤𝑝<2+2|𝛼^s₁|/(𝛼₃)² if 𝛼^s₁<0, (b)0<𝑝<2−2𝛼^s₁/(𝛼₂)² if 0≤𝛼^s₁<𝛼²₂.

Proof. The thesis follows from Theorem 11. Let us choose a Lyapunov function as follows: 𝑉(𝐱)=𝐱^𝑇𝐇𝐱^(𝑝/2),𝑝>0. (50) Function 𝑉 satisfies assumption (1) of Theorem 11 for 𝐶_min=𝜆^(𝑝/2)_min(𝐇) and 𝐶_max=𝜆^(𝑝/2)_max(𝐇).
Furthermore, 𝐿_𝑙𝑉(𝐱)≤−𝜅_s𝑉(𝐱),for𝐱∈Ω^s_𝑙,𝜅_us𝑉(𝐱),for𝐱∈Ω^us_𝑙. (51) Since condition (51) is satisfied, assumption (2) of Theorem 11 also holds. Assumption (3) of Theorem 11 follows directly. Now using Theorem 11, we obtain that control 𝐮∶ℛ^𝑛×𝓢→ℛ^𝑚 can be chosen as follows: 𝑢_𝑖(𝐱,𝑙)=−𝐿^𝑖_𝑙𝑉(𝐱)=−𝑝𝐱^𝑇𝐇𝐱^(𝑝/2)−1𝐱^𝑇𝐂^𝑇_𝑖(𝑙)𝐇𝐱,𝑖=1,2,…,𝑚,𝑙∈𝓢. (52) Hence, the thesis follows.

Notice that function 𝑉(𝐱)=(𝐱^𝑇𝐇𝐱)^(𝑝/2) 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: 𝑑𝐱(𝑡)=𝐀(𝜎(𝑡))𝐱(𝑡)+𝑢₁(𝜎(𝑡),𝐱(𝑡))𝐂₁(𝜎(𝑡))𝐱(𝑡)+𝑢₂(𝜎(𝑡),𝐱(𝑡))𝐂₂(𝜎(𝑡𝐱(𝑡(𝜎(𝑡𝐱(𝑡)(𝑡),(𝜎(0)(0=𝜎)))𝑑𝑡+𝐁))𝑑𝑤,𝐱))₀,𝐱₀, (53) where 𝐀(1)=−0.024−2−0.02,𝐂₁(1)=,𝐂02−10₂(1)=(1)=,𝐀(2)=10−20,𝐁0.2000.2−0.0052−4−0.005,𝐂₁(2)=,𝐂−1001₂(2)=(2)=.0001,𝐁0.1000.1 (54) We study the problem of the 𝑝th mean exponential stabilizability for the hybrid system (53) for 𝑝∈(0,2). We look for a control vector of a form 𝐮∶𝓢×ℛ²→ℛ²,𝐮=𝑢₁,𝑢₂^𝑇. (55) Let us choose the Lyapunov function of a form 𝑉(𝐱)=𝐱^𝑇𝐱. Then regions Ω_𝑙,  𝑙∈𝓢, are given as follows: Ω₁=𝐱∈ℛ²∶𝐱^𝑇𝐀(1)+12𝐁^𝑇(1)𝐁(1=)𝐱≤0𝐱∈ℛ²∶𝑥₁𝑥₂,Ω≤0₂=𝐱∈ℛ²∶𝐱^𝑇𝐀(2)+12𝐁^𝑇(2)𝐁(2=)𝐱≤0𝐱∈ℛ²∶𝑥₁𝑥₂Ω≥0₁∪Ω₂=ℛ². (56) Condition (39) of Theorem 12 is satisfied for 𝛼₁=0. Function 𝑉(𝐱)=𝐱^𝑇𝐱 is a single Lyapunov function for the system (53). From Theorem 12, it follows that the control 𝐮(1,𝐱)=−𝑝𝐱^𝑇𝐇𝐱^(𝑝/2)−1𝑥₁𝑥₂,−𝑝𝐱^𝑇𝐇𝐱^(𝑝/2)−1𝑥²₁−2𝑥₁𝑥₂𝐮(2,𝐱)=−𝑝𝐱^𝑇𝐇𝐱^(𝑝/2)−1𝑥²₂−𝑥²₁,−𝑝𝐱^𝑇𝐇𝐱^(𝑝/2)−1𝑥²₂ (57) together with stabilizing switching rule 𝜎_st given by 𝜎_st(𝑡)=1gdy𝐱(𝑡)∈Ω₁,2gdy𝐱(𝑡)∈Ω₂, (58) exponentially 𝑝th mean, stabilizes the system (53) for 𝑝∈(0,2).

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 𝑝th 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 𝑝th 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 𝑝th 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.

Table 1 

The relationship between the theorems proposed in the paper.

Acknowledgment

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