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

Authors: Ewelina Seroka; Lesław Socha
Journal: Journal of Applied Mathematics
 (2013)

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## Abstract

The problem ofpth 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.


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## Body

## 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 λmin(A) and λmax(A) denote the smallest and the biggest eigenvalues of the matrix A, respectively. We denote by AT the transposition of matrix A. We mark 𝕋=[t0,∞), t0≥0. Let Ξ=(Ω,ℱ,{ℱt}t≥0,ℙ) be a complete probability space with a filtration {ℱt}t≥0 satisfying usual conditions. Let w(t)=[w1(t),…,wM(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 wk and σ are both {ℱt}t≥0 adapted. We say that a proper twice differentiable function V:ℛn→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(1)dx(t)=f(x(t),t,σ)dt+∑k=1M‍gk(x(t),t,σ)dwk(t),(σ(t0),x(t0))=(σ0,x0),
where x∈ℛn is the state vector and (σ0,x0) is an initial condition, t∈𝕋. Functions f:ℛn×𝕋×S→ℛn and gk:ℛn×𝕋×S→ℛn are locally Lipschitz such that ∀l∈S, t∈𝕋f(0,t,l)=gk(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 tox∈ℛn and once differentiable with respect to t∈𝕋 function ϕ(x,t,l) (i.e., ϕ(x,t,l)∈C2,1(ℛn×𝕋×S;[0,∞))), the lth process has a generatorℒl(1) (the Itô operator for the lth subsystem of the system (1)) given in every structure by
(2)ℒl(1)ϕ(x,t,l)=∂ϕ(x,t,l)∂t+∑μ=1n‍fμ(x,t,l)∂ϕ(x,t,l)∂xμ+12∑r,s=1n‍∑k=1M‍gkr(x,t,l)gks(x,t,l)∂2ϕ(x,t,l)∂xr∂xs,l∈S.We use the following definitions.Definition 1.
The null solutionx≡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
(3)∀(x0,t0)∈ℛn×[0,∞),𝔼[|x(t,x0,t0)|p]≤c𝔼[|x0|p]exp{-κ(t-t0)},t≥t0,
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:ℛn×𝕋×S→ℛm such that the hybrid system (1) is pth mean exponentially stable for some p>0.Definition 3.
A Lyapunov functionV(x)∈C2(ℛn;ℛ+) satisfying
(4)∀l∈S,∀x≠0ℒl(1)V(x)<0
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 functionV(x)∈C2(ℛn;ℛ+) satisfying
(5)∀x≠0ℒσ(t)(1)V(x)<0
for some switching rule σ is called a single Lyapunov function for the hybrid system (1).Lemma 5 ([13] (Itô formula)).
Ifϕ(x,t,l)∈C2,1(ℛn×𝕋×S;[0,∞)), then for any switching times 0≤τ1≤τ2<∞,
(6)∀l∈S𝔼[ϕ(x(τ2),τ2,l)]=𝔼[ϕ(x(τ1),τ1,l)]+𝔼[∫τ1τ2‍ℒlϕ(x(s),s,l)ds]
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 Ωl,l∈S, and ⋃l∈S‍Ωl=ℛn. We consider a special class of switching rule 𝒮𝒲 given by
(7)𝒮𝒲={σ:(σ(t)=l)⇒(x(t)∈Ωl),l∈S}.
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 Ωl 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 thepth mean exponential stability of the nonlinear hybrid system (1).Theorem 6.
If the following conditions hold:(1)
there exist a Lyapunov functionV:ℛn→ℛ+ and positive constants Cmin,Cmax such that
(8)Cmin|x|p≤V(x)≤Cmax|x|p,p>0,(2)
there exists a Lebesgue-measurable functionκ:𝕋→(0,∞) such that
(9)⋃l∈S‍Ωl=ℛn,whereΩl={x∈ℛn:∀t∈𝕋ℒl(1)V(x)≤-κ(t)V(x)}forl∈S.Then the null solution x≡0 of the stochastic hybrid system (1) is pth mean exponentially stable under the stabilizing switching rule σst∈𝒮𝒲.Proof.
From assumptions for the Lyapunov functionV:ℛn→ℛ+, we obtain
(10)Cmin|x|p≤V(x)≤Cmax|x|p,p>0,(11)𝔼[ℒσst(1)V(x)]≤-κ(t)𝔼[V(x)].
From (11) and the Itô formula 6, we obtain
(12)𝔼[V(x)]=𝔼[V(x0)]+𝔼[∫t0t‍ℒσst(1)V(x(s))ds]≤𝔼[V(x0)]-∫t0t‍κ(t)𝔼[V(x(s))]ds
and by Gronwall’s inequality
(13)E[V(x)]≤E[V(x0)]exp(-∫t0t‍κ(t)ds).
Now from (10) and (13), we obtain the following inequality:
(14)E[|x(t)|p]≤CmaxCminE[|x0|p]exp(-∫t0t‍κ(t)ds),p>0.
Hence, the thesis follows.Notice that functionV 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 Ωl and l∈S, which can be separated into two disjoint subregions: a stable subregion Ωls and an unstable region Ωlus, l∈S (the upper scripts “s” and “us” denote stable and unstable regions, resp.) that is,
(15)Ωl=Ωls∪Ωlus,Ωls∩Ωlus=∅.Let us denote by𝕋s the sum of time intervals of the residence of the system (1) in the regions Ωls, l∈S, and by 𝕋us the sum of time intervals of the residence of the system (1) in the regions Ωlus,  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 functionV(x)∈C2(ℛn;ℛ+) is called a single Lyapunov-like function if there exist positive constants κs and κus such that
(16)ℒσ(t)(1)V(x)≤{-κsV(x)forx∈Ωσ(t),sκusV(x)forx∈Ωσ(t)us,
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 functionV(x)∈C2(ℛn;ℛ+) and positive constants Cmin,Cmax such that
(17)Cmin|x|p≤V(x)≤Cmax|x|p,p>0,(2)
there exist Lebesgue-measurable functionsκs:𝕋→(0,∞) and κus:𝕋→(0,∞) such that partition (15) satisfies the conditions
(18)⋃l∈S‍Ωl=ℛn,whereΩls={x∈ℛn:∀t∈𝕋ℒl(1)V(x)≤-κs(t)V(x)}forl∈S,Ωlus={x∈ℛn:∀t∈𝕋ℒl(1)V(x)≤κus(t)V(x)}forl∈S,(3)
there exists a Lebesguemeasurable functionκ:𝕋→(0,∞) such that 𝒯≠∅, where
(19)𝒯={σ∈𝒮𝒲:∫Tus‍[κus(s)+κ(s)]ds≤∫Ts‍[κs(s)-κ(s)]ds}.Then the null solution x≡0 of (1) is pth mean exponentially stable under the stabilizing switching rule σ¯st∈𝒯.Proof.
From assumptions forσst∈𝒮𝒲, it follows that
(20)𝔼[V(x)]≤𝔼[V(x0)]exp(∫Tusκus(s)ds-∫Ts‍κs(s)ds).
Let us consider the switching strategy σ¯st∈𝒯 described by (19). Then we obtain
(21)𝔼[V(x)]≤𝔼[V(x0)]exp(-∫t0t‍κ(s)ds).
Further proof is similar to the proof of Theorem 6. Hence, the thesis follows.Notice that functionV 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:
(22)dx(t)=[A(σ)x(t)+f(x,σ)]dt+G(σ)dw(t),(σ(t0),x(t0))=(σ0,x0)∈S×ℛn,
where
(23)A(1)=[2.182-4-2.82],A(2)=[-1.9054-21.095],f(x,1)=[0f2(x2,1)],f(x,2)=[0f2(x2,2)],f2(x2,1)={0.8x2-0.2forx2<-1,x2forx2∈[-1,1],0.8x2+0.2forx2>1,f2(x2,2)={0.9x2-0.1forx2<-1,x2forx2∈[-1,1],0.9x2+0.1forx2>1,G(1)=[0.2000.2],G(2)=[0.1000.1].
Let us consider the Lyapunov functionV(x)=xTx. Since 0≤f2(x2,l)x2≤x22, l=1,2, then
(24)ℒ1(22)V(x)=2.2x12-2x1x2-2.8x22+f(x,1)Tx≤-0.8(x12+x22)forΩ1s={x∈ℛ2:(x2+3x1)(x1-x2)≤0},ℒ2(22)V(x)=-1.9x12+2x1x2+1.1x22+f(x,2)Tx≤-0.8(x12+x22)forΩ2s={x∈ℛ2:-1.1x12+2x1x2+2.9x22≤0},ℒ2(22)V(x)=-1.9x12+2x1x2+1.1x22+f(x,2)Tx≤1.1(x12+x22)forΩ2us={x∈ℛ2:(x2+3x1)(x2-x1)≤0}∖(Ω1s∪Ω2s).
Note that ⋃l∈S‍Ωl=ℛ2 and functions κs and κus are constant and are given as follows κs=0.8, κus=1.1.
Condition(25)𝕋us𝕋s≤κs-κκus+κ≈0.61
is satisfied for κ=0.08. Exemplary simulations are shown in Figures 1, 2, and 3.Trajectories of the subsystems of (22).
(a)
Unstable path of the system (22) for l=1
(b)
Unstable path of the system (22) for l=2Trajectories of the hybrid system (22).
(a)
Stable path of the hybrid system (22) with stabilizing switching σ¯st given by  (26)
(b)
Lyapunov functionV=xTx for the hybrid system  (22)Stable path of the hybrid system (22) with stabilizing switching σ¯st given by (26).
(a)
(b)From Theorem8, it follows that V(x)=xTx 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
(26)σ¯st(x,t)={1ifx(t)∈Ω1,2ifx(t)∈Ω2.


## 3.1. Single Lyapunov Functions

We formulate a theorem which establishes sufficient conditions for thepth mean exponential stability of the nonlinear hybrid system (1).Theorem 6.
If the following conditions hold:(1)
there exist a Lyapunov functionV:ℛn→ℛ+ and positive constants Cmin,Cmax such that
(8)Cmin|x|p≤V(x)≤Cmax|x|p,p>0,(2)
there exists a Lebesgue-measurable functionκ:𝕋→(0,∞) such that
(9)⋃l∈S‍Ωl=ℛn,whereΩl={x∈ℛn:∀t∈𝕋ℒl(1)V(x)≤-κ(t)V(x)}forl∈S.Then the null solution x≡0 of the stochastic hybrid system (1) is pth mean exponentially stable under the stabilizing switching rule σst∈𝒮𝒲.Proof.
From assumptions for the Lyapunov functionV:ℛn→ℛ+, we obtain
(10)Cmin|x|p≤V(x)≤Cmax|x|p,p>0,(11)𝔼[ℒσst(1)V(x)]≤-κ(t)𝔼[V(x)].
From (11) and the Itô formula 6, we obtain
(12)𝔼[V(x)]=𝔼[V(x0)]+𝔼[∫t0t‍ℒσst(1)V(x(s))ds]≤𝔼[V(x0)]-∫t0t‍κ(t)𝔼[V(x(s))]ds
and by Gronwall’s inequality
(13)E[V(x)]≤E[V(x0)]exp(-∫t0t‍κ(t)ds).
Now from (10) and (13), we obtain the following inequality:
(14)E[|x(t)|p]≤CmaxCminE[|x0|p]exp(-∫t0t‍κ(t)ds),p>0.
Hence, the thesis follows.Notice that functionV 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 Ωl and l∈S, which can be separated into two disjoint subregions: a stable subregion Ωls and an unstable region Ωlus, l∈S (the upper scripts “s” and “us” denote stable and unstable regions, resp.) that is,
(15)Ωl=Ωls∪Ωlus,Ωls∩Ωlus=∅.Let us denote by𝕋s the sum of time intervals of the residence of the system (1) in the regions Ωls, l∈S, and by 𝕋us the sum of time intervals of the residence of the system (1) in the regions Ωlus,  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 functionV(x)∈C2(ℛn;ℛ+) is called a single Lyapunov-like function if there exist positive constants κs and κus such that
(16)ℒσ(t)(1)V(x)≤{-κsV(x)forx∈Ωσ(t),sκusV(x)forx∈Ωσ(t)us,
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 functionV(x)∈C2(ℛn;ℛ+) and positive constants Cmin,Cmax such that
(17)Cmin|x|p≤V(x)≤Cmax|x|p,p>0,(2)
there exist Lebesgue-measurable functionsκs:𝕋→(0,∞) and κus:𝕋→(0,∞) such that partition (15) satisfies the conditions
(18)⋃l∈S‍Ωl=ℛn,whereΩls={x∈ℛn:∀t∈𝕋ℒl(1)V(x)≤-κs(t)V(x)}forl∈S,Ωlus={x∈ℛn:∀t∈𝕋ℒl(1)V(x)≤κus(t)V(x)}forl∈S,(3)
there exists a Lebesguemeasurable functionκ:𝕋→(0,∞) such that 𝒯≠∅, where
(19)𝒯={σ∈𝒮𝒲:∫Tus‍[κus(s)+κ(s)]ds≤∫Ts‍[κs(s)-κ(s)]ds}.Then the null solution x≡0 of (1) is pth mean exponentially stable under the stabilizing switching rule σ¯st∈𝒯.Proof.
From assumptions forσst∈𝒮𝒲, it follows that
(20)𝔼[V(x)]≤𝔼[V(x0)]exp(∫Tusκus(s)ds-∫Ts‍κs(s)ds).
Let us consider the switching strategy σ¯st∈𝒯 described by (19). Then we obtain
(21)𝔼[V(x)]≤𝔼[V(x0)]exp(-∫t0t‍κ(s)ds).
Further proof is similar to the proof of Theorem 6. Hence, the thesis follows.Notice that functionV 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:
(22)dx(t)=[A(σ)x(t)+f(x,σ)]dt+G(σ)dw(t),(σ(t0),x(t0))=(σ0,x0)∈S×ℛn,
where
(23)A(1)=[2.182-4-2.82],A(2)=[-1.9054-21.095],f(x,1)=[0f2(x2,1)],f(x,2)=[0f2(x2,2)],f2(x2,1)={0.8x2-0.2forx2<-1,x2forx2∈[-1,1],0.8x2+0.2forx2>1,f2(x2,2)={0.9x2-0.1forx2<-1,x2forx2∈[-1,1],0.9x2+0.1forx2>1,G(1)=[0.2000.2],G(2)=[0.1000.1].
Let us consider the Lyapunov functionV(x)=xTx. Since 0≤f2(x2,l)x2≤x22, l=1,2, then
(24)ℒ1(22)V(x)=2.2x12-2x1x2-2.8x22+f(x,1)Tx≤-0.8(x12+x22)forΩ1s={x∈ℛ2:(x2+3x1)(x1-x2)≤0},ℒ2(22)V(x)=-1.9x12+2x1x2+1.1x22+f(x,2)Tx≤-0.8(x12+x22)forΩ2s={x∈ℛ2:-1.1x12+2x1x2+2.9x22≤0},ℒ2(22)V(x)=-1.9x12+2x1x2+1.1x22+f(x,2)Tx≤1.1(x12+x22)forΩ2us={x∈ℛ2:(x2+3x1)(x2-x1)≤0}∖(Ω1s∪Ω2s).
Note that ⋃l∈S‍Ωl=ℛ2 and functions κs and κus are constant and are given as follows κs=0.8, κus=1.1.
Condition(25)𝕋us𝕋s≤κs-κκus+κ≈0.61
is satisfied for κ=0.08. Exemplary simulations are shown in Figures 1, 2, and 3.Trajectories of the subsystems of (22).
(a)
Unstable path of the system (22) for l=1
(b)
Unstable path of the system (22) for l=2Trajectories of the hybrid system (22).
(a)
Stable path of the hybrid system (22) with stabilizing switching σ¯st given by  (26)
(b)
Lyapunov functionV=xTx for the hybrid system  (22)Stable path of the hybrid system (22) with stabilizing switching σ¯st given by (26).
(a)
(b)From Theorem8, it follows that V(x)=xTx 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
(26)σ¯st(x,t)={1ifx(t)∈Ω1,2ifx(t)∈Ω2.


## 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 thepth 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(27)dx(t)=f0(x(t),t,σ)dt+∑i=1p‍ui(x,t,σ)fi(x(t),t,σ)+∑k=1M‍gk(x(t),t,σ)dwk(t),(σ(t0),x(t0))=(σ0,x0),
where x∈ℛn is the state vector, u=[u1,…,um]T is a measurable ℛm—a real-valued control vector law, (σ0,x0)∈S×ℛn is an initial condition, and t∈𝕋. Functions fi:ℛn×𝕋×S→ℛn and gk:ℛn×𝕋×S→ℛn are the locally Lipschitz, fi(0,t,l)=gk(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 ruleσ 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 solutionx≡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 operatorsLl and Lli,  l∈S,  i=1,…,m, for ϕ(x,t,l)∈C2,1(ℛn×𝕋×S;ℛ+):
(28)Llϕ(x,t,l)=∂ϕ(x,t,l)∂t+∑μ=1n‍f0μ(x,t,l)∂ϕ(x,t,l)∂xμ+12∑r,s=1n‍∑k=1M‍gkr(x,t,l)gks(x,t,l)∂2ϕ(x,t,l)∂xr∂xs,Lliϕ(x,t,l)=∑μ=1n‍fiμ(x,t,l)∂ϕ(x,t,l)∂xμ,i=1,…,m,l∈S.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 functionV(x)∈C2(ℛn;ℛ+) and positive constants Cmin,Cmax such that
(29)Cmin|x|p≤V(x)≤Cmax|x|p,p>0,(2)
there exists a Lebesgue-measurable functionκ:𝕋→(0,∞) such that partition determined by (7) satisfies conditions(30)⋃l∈S‍Ωl=ℛn,whereΩl={x∈ℛn:∀t∈𝕋LlV(x)≤-κ(t)V(x)}forl∈S.Then the control law u:ℛn×𝕋×S→ℛm given as follows:
(31)ui(x,t,l)=-LliV(x),i=1,2,…,m,l∈S
together with the stabilizing switching rule σst∈𝒮𝒲 renders the null solution of the stochastic hybrid system (27) the pth mean exponentially stable.Proof.
Applying the infinitesimal operatorℒ defined by (2) to the hybrid system (27), we find that
(32)∀l∈Sℒl(27)V(x)=LlV(x)-∑i=1m‍(LliV(x))2≤-κ(t)V(x),forx∈Ωl.
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 functionV(x)∈C2(ℛn;ℛ+) and positive constants Cmin,Cmax such that
(33)Cmin|x|p≤V(x)≤Cmax|x|p,p>0,(2)
there exist Lebesgue-measurable functionsκs:𝕋→(0,∞) and κus:𝕋→(0,∞) such that partition (15) satisfies conditions
(34)⋃l∈S‍Ωl=ℛn,whereΩls={x∈ℛn:∀t∈𝕋LlV(x)≤-κs(t)V(x)}forl∈S,Ωlus={x∈ℛn:∀t∈𝕋LlV(x)≤κus(t)V(x)}forl∈S,(3)
there exists a Lebesgue-measurable functionκ:𝕋→ℛ+ such that 𝒯≠∅, where
(35)𝒯={σ∈𝒮𝒲:∫Tus‍[κus(s)+κ(s)]ds≤∫Ts‍[κs(s)-κ(s)]ds}.Then the control law u:ℛn×𝕋×S→ℛm given as follows
(36)ui(x,t,l)=-LliV(x),i=1,2,…,m
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 Theorem8.


### 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:
(37)dx(t)=[A(σ(t))x(t)+∑i=1M‍ui(x(t),σ(t))Ci(σ(t))x(t)]dt+∑k=1M‍Bk(σ(t))x(t)dwk(t),(σ(t0),x(t0))=(σ0,x0),
where x∈ℛn is the state vector, u=[u1,…,um]T∈ℛm,ui:ℛn×S→ℛ, is the control vector, t∈𝕋, (σ0,x0)∈S×ℛn is an initial condition, and A(l), Ci(l), Bk(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:
(38)Llϕ(x,l)=xTAT(l)∇ϕ(x,l)+12tr(Bk(l)xxTBk(l)T∇2ϕ(x,l)),l∈S,Lli(x,l)=xTCiT(l)∇ϕ(x,l),i=1,…,m,l∈S,
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 matrixH, constant α1, and positive constants α2, α3 such that the following conditions are satisfied:
(39)⋃l∈S‍Ωl=ℛn,whereΩl={x∈ℛn:xTAT(l)Hx+12∑k=1M‍xTBkT(l)HBk(l)x≤α1xTHx∧α2xTHx≤|xTH∑k=1M‍Bk(l)x|≤α3xTHx}l∈S.
Then control u:ℛn×S→ℛm of a form
(40)ui(x,l)=-p(xTHx)(p/2)-1xTCiT(l)Hx,i=1,2,…,m,l∈S
together with the stabilizing switching rule σst∈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 Theorem10. Let us choose a Lyapunov function of a form
(41)V(x)=(xTHx)(p/2),p>0.
Notice that V satisfies assumption (1) of Theorem 10 for Cmin=λmin(p/2)(H) and Cmax=λmax(p/2)(H). Then using assumptions (39), we obtain
(42)∀l∈SLlV(x)≤-κV(x)κ>0forx∈Ωl,
where
(43)κ={p(|α1|-(p2-1)α32)V(x),for2≤p<2+2|α1|(α3)2,α1<0,p(-α1-(p2-1)α22)V(x),for0<p<2-2α1(α2)2,0≤α1<(α2)2.
From Theorem 10, it follows that the control chosen as follows:
(44)ui(x,l)=-LliV(x)=-p(xTHx)(p/2)-1xTCiT(l)Hx,i=1,2,…,m,l∈S
together with the stabilizing switching rule σst∈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 functionV(x)=(xTHx)(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 ofp=2, we obtain the following criterion.Criterion 14.
Suppose that there exists a symmetric positive definite matrixH such that
(45)⋃l∈S‍Ωl=ℛn,whereΩl={x∈ℛn:xTAT(l)Hx+12∑k=1M‍xTBkT(l)HBk(l)x<0},l∈S.
Then the control u:ℛn×S→ℛm of a form
(46)ui(x,l)=-2xTCiT(l)Hx,i=1,2,…,m,l∈S
together with the stabilizing switching rule σst∈𝒮𝒲 exponentially in mean-square stabilizes the bilinear hybrid system (37).We formulate now a more general theorem which formulates sufficient conditions ofpth 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.Simulation of the hybrid system (53) with stabilizing switching σ=σst for p=1.
(a)
Trajectory of the hybrid system
(b)
Trajectory of the Lyapunov functionFigure 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 matrixH, constants α1s,α1us, and positive constants α2,α3 such that
(47)⋃l∈S‍Ωl=ℛn,whereΩls={x∈ℛn:xTHA(l)x+12∑k=1M‍xTBkT(l)HBk(l)x≤α1sxTHx,α1s<(α2)2∧α2xTHx≤|xTH∑k=1M‍Bk(l)x|≤α3xTHx}l∈S,Ωlus={|xTH∑k=1M‍Bk(l)x|x∈ℛn:α1sxTHx<xTHA(l)x+12∑k=1M‍xTBkT(l)HBk(l)x≤α1usxTHx,∧α2xTHx≤|xTH∑k=1M‍Bk(l)x|≤α3xTHx}l∈S,(2)
there exists constantκ>0 such that 𝒯≠∅, where
(48)𝒯={σ∈𝒮𝒲:𝕋us𝕋s≤κs-κκus+κ},κs={p(|α1s|-(p2-1)α32)dla2≤p<2+2|α1s|(α3)2,α1<0,p(-α1s-(p2-1)α22)dla0<p<2-2α1s(α2)2,0≤α1s<(α2)2,κus={p(α1us+(p2-1)α32)dlap≥2,α1us≥0,p(α1us+(p2-1)α22)dla0<p<2,α1us≥(α2)2.Then the control u:ℛn×S→ℛm of a form
(49)ui(x,l)=-p(xTHx)(p/2)-1xTCiT(l)Hx,i=1,2,…,m,l∈S
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|α1s|/(α3)2 if α1s<0,(b)
0<p<2-2α1s/(α2)2 if 0≤α1s<α22.Proof.
The thesis follows from Theorem11. Let us choose a Lyapunov function as follows:
(50)V(x)=(xTHx)(p/2),p>0.
Function V satisfies assumption (1) of Theorem 11 for Cmin=λmin(p/2)(H) and Cmax=λmax(p/2)(H).
Furthermore,(51)LlV(x)≤{-κsV(x),forx∈Ωls,κusV(x),forx∈Ωlus.
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:ℛn×S→ℛm can be chosen as follows:
(52)ui(x,l)=-LliV(x)=-p(xTHx)(p/2)-1xTCiT(l)Hx,i=1,2,…,m,l∈S.
Hence, the thesis follows.Notice that functionV(x)=(xTHx)(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:
(53)dx(t)=[A(σ(t))x(t)+u1(σ(t),x(t))C1(σ(t))x(t)+u2(σ(t),x(t))C2(σ(t))x(t)]dt+B(σ(t))x(t)dw(t),(σ(0),x(0))=(σ0,x0),
where
(54)A(1)=[-0.024-2-0.02],C1(1)=[02-10],C2(1)=[10-20],B(1)=[0.2000.2],A(2)=[-0.0052-4-0.005],C1(2)=[-1001],C2(2)=[0001],B(2)=[0.1000.1].
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
(55)u:S×ℛ2→ℛ2,u=[u1,u2]T.
Let us choose the Lyapunov function of a form V(x)=xTx. Then regions Ωl,  l∈S, are given as follows:
(56)Ω1={x∈ℛ2:xT[A(1)+12BT(1)B(1)]x≤0}={x∈ℛ2:x1x2≤0},Ω2={x∈ℛ2:xT[A(2)+12BT(2)B(2)]x≤0}={x∈ℛ2:x1x2≥0}Ω1∪Ω2=ℛ2.
Condition (39) of Theorem 12 is satisfied for α1=0. Function V(x)=xTx is a single Lyapunov function for the system (53). From Theorem 12, it follows that the control
(57)u(1,x)=[-p(xTHx)(p/2)-1x1x2,-p(xTHx)(p/2)-1(x12-2x1x2)(xTHx)(p/2)-1]u(2,x)=[-p(xTHx)(p/2)-1(x22-x12),-p(xTHx)(p/2)-1x22]
together with stabilizing switching rule σst given by
(58)σst(t)={1gdyx(t)∈Ω1,2gdyx(t)∈Ω2,
exponentially pth mean, stabilizes the system (53) for p∈(0,2).


## 4.1. Stabilizability of Nonlinear Stochastic Hybrid Systems

Let us consider the stochastic control hybrid system described by the vector Itô differential equations(27)dx(t)=f0(x(t),t,σ)dt+∑i=1p‍ui(x,t,σ)fi(x(t),t,σ)+∑k=1M‍gk(x(t),t,σ)dwk(t),(σ(t0),x(t0))=(σ0,x0),
where x∈ℛn is the state vector, u=[u1,…,um]T is a measurable ℛm—a real-valued control vector law, (σ0,x0)∈S×ℛn is an initial condition, and t∈𝕋. Functions fi:ℛn×𝕋×S→ℛn and gk:ℛn×𝕋×S→ℛn are the locally Lipschitz, fi(0,t,l)=gk(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 ruleσ 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 solutionx≡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 operatorsLl and Lli,  l∈S,  i=1,…,m, for ϕ(x,t,l)∈C2,1(ℛn×𝕋×S;ℛ+):
(28)Llϕ(x,t,l)=∂ϕ(x,t,l)∂t+∑μ=1n‍f0μ(x,t,l)∂ϕ(x,t,l)∂xμ+12∑r,s=1n‍∑k=1M‍gkr(x,t,l)gks(x,t,l)∂2ϕ(x,t,l)∂xr∂xs,Lliϕ(x,t,l)=∑μ=1n‍fiμ(x,t,l)∂ϕ(x,t,l)∂xμ,i=1,…,m,l∈S.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 functionV(x)∈C2(ℛn;ℛ+) and positive constants Cmin,Cmax such that
(29)Cmin|x|p≤V(x)≤Cmax|x|p,p>0,(2)
there exists a Lebesgue-measurable functionκ:𝕋→(0,∞) such that partition determined by (7) satisfies conditions(30)⋃l∈S‍Ωl=ℛn,whereΩl={x∈ℛn:∀t∈𝕋LlV(x)≤-κ(t)V(x)}forl∈S.Then the control law u:ℛn×𝕋×S→ℛm given as follows:
(31)ui(x,t,l)=-LliV(x),i=1,2,…,m,l∈S
together with the stabilizing switching rule σst∈𝒮𝒲 renders the null solution of the stochastic hybrid system (27) the pth mean exponentially stable.Proof.
Applying the infinitesimal operatorℒ defined by (2) to the hybrid system (27), we find that
(32)∀l∈Sℒl(27)V(x)=LlV(x)-∑i=1m‍(LliV(x))2≤-κ(t)V(x),forx∈Ωl.
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 functionV(x)∈C2(ℛn;ℛ+) and positive constants Cmin,Cmax such that
(33)Cmin|x|p≤V(x)≤Cmax|x|p,p>0,(2)
there exist Lebesgue-measurable functionsκs:𝕋→(0,∞) and κus:𝕋→(0,∞) such that partition (15) satisfies conditions
(34)⋃l∈S‍Ωl=ℛn,whereΩls={x∈ℛn:∀t∈𝕋LlV(x)≤-κs(t)V(x)}forl∈S,Ωlus={x∈ℛn:∀t∈𝕋LlV(x)≤κus(t)V(x)}forl∈S,(3)
there exists a Lebesgue-measurable functionκ:𝕋→ℛ+ such that 𝒯≠∅, where
(35)𝒯={σ∈𝒮𝒲:∫Tus‍[κus(s)+κ(s)]ds≤∫Ts‍[κs(s)-κ(s)]ds}.Then the control law u:ℛn×𝕋×S→ℛm given as follows
(36)ui(x,t,l)=-LliV(x),i=1,2,…,m
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 Theorem8.


## 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:
(37)dx(t)=[A(σ(t))x(t)+∑i=1M‍ui(x(t),σ(t))Ci(σ(t))x(t)]dt+∑k=1M‍Bk(σ(t))x(t)dwk(t),(σ(t0),x(t0))=(σ0,x0),
where x∈ℛn is the state vector, u=[u1,…,um]T∈ℛm,ui:ℛn×S→ℛ, is the control vector, t∈𝕋, (σ0,x0)∈S×ℛn is an initial condition, and A(l), Ci(l), Bk(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:
(38)Llϕ(x,l)=xTAT(l)∇ϕ(x,l)+12tr(Bk(l)xxTBk(l)T∇2ϕ(x,l)),l∈S,Lli(x,l)=xTCiT(l)∇ϕ(x,l),i=1,…,m,l∈S,
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 matrixH, constant α1, and positive constants α2, α3 such that the following conditions are satisfied:
(39)⋃l∈S‍Ωl=ℛn,whereΩl={x∈ℛn:xTAT(l)Hx+12∑k=1M‍xTBkT(l)HBk(l)x≤α1xTHx∧α2xTHx≤|xTH∑k=1M‍Bk(l)x|≤α3xTHx}l∈S.
Then control u:ℛn×S→ℛm of a form
(40)ui(x,l)=-p(xTHx)(p/2)-1xTCiT(l)Hx,i=1,2,…,m,l∈S
together with the stabilizing switching rule σst∈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 Theorem10. Let us choose a Lyapunov function of a form
(41)V(x)=(xTHx)(p/2),p>0.
Notice that V satisfies assumption (1) of Theorem 10 for Cmin=λmin(p/2)(H) and Cmax=λmax(p/2)(H). Then using assumptions (39), we obtain
(42)∀l∈SLlV(x)≤-κV(x)κ>0forx∈Ωl,
where
(43)κ={p(|α1|-(p2-1)α32)V(x),for2≤p<2+2|α1|(α3)2,α1<0,p(-α1-(p2-1)α22)V(x),for0<p<2-2α1(α2)2,0≤α1<(α2)2.
From Theorem 10, it follows that the control chosen as follows:
(44)ui(x,l)=-LliV(x)=-p(xTHx)(p/2)-1xTCiT(l)Hx,i=1,2,…,m,l∈S
together with the stabilizing switching rule σst∈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 functionV(x)=(xTHx)(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 ofp=2, we obtain the following criterion.Criterion 14.
Suppose that there exists a symmetric positive definite matrixH such that
(45)⋃l∈S‍Ωl=ℛn,whereΩl={x∈ℛn:xTAT(l)Hx+12∑k=1M‍xTBkT(l)HBk(l)x<0},l∈S.
Then the control u:ℛn×S→ℛm of a form
(46)ui(x,l)=-2xTCiT(l)Hx,i=1,2,…,m,l∈S
together with the stabilizing switching rule σst∈𝒮𝒲 exponentially in mean-square stabilizes the bilinear hybrid system (37).We formulate now a more general theorem which formulates sufficient conditions ofpth 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.Simulation of the hybrid system (53) with stabilizing switching σ=σst for p=1.
(a)
Trajectory of the hybrid system
(b)
Trajectory of the Lyapunov functionFigure 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 matrixH, constants α1s,α1us, and positive constants α2,α3 such that
(47)⋃l∈S‍Ωl=ℛn,whereΩls={x∈ℛn:xTHA(l)x+12∑k=1M‍xTBkT(l)HBk(l)x≤α1sxTHx,α1s<(α2)2∧α2xTHx≤|xTH∑k=1M‍Bk(l)x|≤α3xTHx}l∈S,Ωlus={|xTH∑k=1M‍Bk(l)x|x∈ℛn:α1sxTHx<xTHA(l)x+12∑k=1M‍xTBkT(l)HBk(l)x≤α1usxTHx,∧α2xTHx≤|xTH∑k=1M‍Bk(l)x|≤α3xTHx}l∈S,(2)
there exists constantκ>0 such that 𝒯≠∅, where
(48)𝒯={σ∈𝒮𝒲:𝕋us𝕋s≤κs-κκus+κ},κs={p(|α1s|-(p2-1)α32)dla2≤p<2+2|α1s|(α3)2,α1<0,p(-α1s-(p2-1)α22)dla0<p<2-2α1s(α2)2,0≤α1s<(α2)2,κus={p(α1us+(p2-1)α32)dlap≥2,α1us≥0,p(α1us+(p2-1)α22)dla0<p<2,α1us≥(α2)2.Then the control u:ℛn×S→ℛm of a form
(49)ui(x,l)=-p(xTHx)(p/2)-1xTCiT(l)Hx,i=1,2,…,m,l∈S
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|α1s|/(α3)2 if α1s<0,(b)
0<p<2-2α1s/(α2)2 if 0≤α1s<α22.Proof.
The thesis follows from Theorem11. Let us choose a Lyapunov function as follows:
(50)V(x)=(xTHx)(p/2),p>0.
Function V satisfies assumption (1) of Theorem 11 for Cmin=λmin(p/2)(H) and Cmax=λmax(p/2)(H).
Furthermore,(51)LlV(x)≤{-κsV(x),forx∈Ωls,κusV(x),forx∈Ωlus.
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:ℛn×S→ℛm can be chosen as follows:
(52)ui(x,l)=-LliV(x)=-p(xTHx)(p/2)-1xTCiT(l)Hx,i=1,2,…,m,l∈S.
Hence, the thesis follows.Notice that functionV(x)=(xTHx)(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:
(53)dx(t)=[A(σ(t))x(t)+u1(σ(t),x(t))C1(σ(t))x(t)+u2(σ(t),x(t))C2(σ(t))x(t)]dt+B(σ(t))x(t)dw(t),(σ(0),x(0))=(σ0,x0),
where
(54)A(1)=[-0.024-2-0.02],C1(1)=[02-10],C2(1)=[10-20],B(1)=[0.2000.2],A(2)=[-0.0052-4-0.005],C1(2)=[-1001],C2(2)=[0001],B(2)=[0.1000.1].
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
(55)u:S×ℛ2→ℛ2,u=[u1,u2]T.
Let us choose the Lyapunov function of a form V(x)=xTx. Then regions Ωl,  l∈S, are given as follows:
(56)Ω1={x∈ℛ2:xT[A(1)+12BT(1)B(1)]x≤0}={x∈ℛ2:x1x2≤0},Ω2={x∈ℛ2:xT[A(2)+12BT(2)B(2)]x≤0}={x∈ℛ2:x1x2≥0}Ω1∪Ω2=ℛ2.
Condition (39) of Theorem 12 is satisfied for α1=0. Function V(x)=xTx is a single Lyapunov function for the system (53). From Theorem 12, it follows that the control
(57)u(1,x)=[-p(xTHx)(p/2)-1x1x2,-p(xTHx)(p/2)-1(x12-2x1x2)(xTHx)(p/2)-1]u(2,x)=[-p(xTHx)(p/2)-1(x22-x12),-p(xTHx)(p/2)-1x22]
together with stabilizing switching rule σst given by
(58)σst(t)={1gdyx(t)∈Ω1,2gdyx(t)∈Ω2,
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 thepth 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 thepth 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 Table1.Table 1
The relationship between the theorems proposed in the paper.
p-stability of non-linear hybrid systems
p-stabilizability of non-linear hybrid systems
p-stabilizability of bilinear hybrid systems
Single Lyapunov function
Theorem6
⇐ Theorem 10⇒
Theorem12
Single Lyapunov-like function
Theorem8
⇐ Theorem 11⇒
Theorem15



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*Source: 206190-2013-03-18.xml*