Hindawi Publishing Corporation
Journal of Applied Mathematics
Volume 2013, Article ID 206190, 10 pages
http://dx.doi.org/10.1155/2013/206190
Research Article
𝑝-Stability and 𝑝-Stabilizability of Stochastic Nonlinear and
Bilinear Hybrid Systems under Stabilizing Switching Rules
Ewelina Seroka and LesBaw Socha
Faculty of Mathematics and Natural Sciences, College of Sciences, Cardinal Stefan Wyszyński University in Warsaw,
Dewajtis Street 5, 01-815 Warsaw, Poland
Correspondence should be addressed to Ewelina Seroka; ewelina.seroka@gmail.com
Received 9 November 2012; Accepted 14 January 2013
Academic Editor: Piyapong Niamsup
Copyright © 2013 E. Seroka and L. Socha. This is an open access article distributed under the Creative Commons Attribution
License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly
cited.
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 Lyapunovlike 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 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.
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Journal of Applied Mathematics
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
𝑀
𝑑x (𝑡) = f (x (𝑡) , 𝑡, 𝜎) 𝑑𝑡 + ∑ g𝑘 (x (𝑡) , 𝑡, 𝜎) 𝑑𝑤𝑘 (𝑡) ,
𝑘=1
(1)
(𝜎 (𝑡0 ) , x (𝑡0 )) = (𝜎0 , x0 ) ,
𝑛
where x ∈ R 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 to x ∈ 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 generator L(1)
𝑙 (the Itô operator for the 𝑙th subsystem of the
system (1)) given in every structure by
L(1)
𝑙 𝜙 (x, 𝑡, 𝑙) =
𝑛
𝜕𝜙 (x, 𝑡, 𝑙)
𝜕𝜙 (x, 𝑡, 𝑙)
+ ∑ 𝑓𝜇 (x, 𝑡, 𝑙)
𝜕𝑡
𝜕𝑥𝜇
𝜇=1
+
𝜕2 𝜙 (x, 𝑡, 𝑙)
1 𝑛 𝑀 𝑟
,
∑ ∑ 𝑔𝑘 (x, 𝑡, 𝑙) 𝑔𝑘𝑠 (x, 𝑡, 𝑙)
2 𝑟,𝑠=1 𝑘=1
𝜕𝑥𝑟 𝜕𝑥𝑠
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 ) ∈ R𝑛 × [0, ∞) ,
󵄨
󵄨𝑝
󵄨 󵄨𝑝
E [󵄨󵄨󵄨x (𝑡, x0 , 𝑡0 )󵄨󵄨󵄨 ] ≤ 𝑐E [󵄨󵄨󵄨x0 󵄨󵄨󵄨 ] exp {−𝜅 (𝑡 − 𝑡0 )} ,
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 (R𝑛 ; R+ )
satisfying
∀x ≠ 0 L(1)
𝜎(𝑡) 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) ∈ 𝐶2,1 (R𝑛 × T ×
S; [0, ∞)), then for any switching times 0 ≤ 𝜏1 ≤ 𝜏2 < ∞,
∀𝑙 ∈ S E [𝜙 (x (𝜏2 ) , 𝜏2 , 𝑙)] = E [𝜙 (x (𝜏1 ) , 𝜏1 , 𝑙)]
𝜏2
+ E [∫ L𝑙 𝜙 (x (𝑠) , 𝑠, 𝑙) 𝑑𝑠]
𝜏1
(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 Ω𝑙 , 𝑙 ∈ S, and ⋃𝑙∈S Ω𝑙 = R𝑛 .
We consider a special class of switching rule SW given by
SW = {𝜎 : (𝜎 (𝑡) = 𝑙) 󳨐⇒ (x (𝑡) ∈ Ω𝑙 ) , 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 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
𝑡 ≥ 𝑡0 ,
(3)
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.
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.
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).
Definition 3. A Lyapunov function 𝑉(x) ∈ 𝐶2 (R𝑛 ; R+ )
satisfying
(1) there exist a Lyapunov function 𝑉 : R𝑛 → R+ and
positive constants 𝐶min , 𝐶max such that
where 𝜅 is called the decay rate.
∀𝑙 ∈ S, ∀x ≠ 0 L(1)
𝑙 𝑉 (x) < 0
(4)
Theorem 6. If the following conditions hold:
𝐶min |x|𝑝 ≤ 𝑉 (x) ≤ 𝐶max |x|𝑝 ,
𝑝 > 0,
(8)
Journal of Applied Mathematics
3
(2) there exists a Lebesgue-measurable function 𝜅 : T →
(0, ∞) such that
⋃ Ω𝑙 = R𝑛 , where
𝑙∈S
Ω𝑙 = {x ∈ R𝑛 : ∀𝑡 ∈ T L(1)
𝑙 𝑉 (x) ≤ −𝜅 (𝑡) 𝑉 (x)}
for 𝑙 ∈ S.
(9)
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 function 𝑉 :
R𝑛 → R+ , we obtain
𝐶min |x|𝑝 ≤ 𝑉 (x) ≤ 𝐶max |x|𝑝 ,
𝑝 > 0,
E [L(1)
𝜎st 𝑉 (x)] ≤ −𝜅 (𝑡) E [𝑉 (x)] .
(10)
(11)
From (11) and the Itô formula 6, we obtain
L(1)
𝜎(𝑡) 𝑉 (x) ≤ {
E [𝑉 (x)] = E [𝑉 (x0 )] + E [∫ L(1)
𝜎st 𝑉 (x (𝑠)) 𝑑𝑠]
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 (R𝑛 ; R+ )
and positive constants 𝐶min , 𝐶max such that
𝐶min |x|𝑝 ≤ 𝑉 (x) ≤ 𝐶max |x|𝑝 ,
Ωs𝑙 = {x ∈ R𝑛 : ∀𝑡 ∈ T L(1)
𝑙 𝑉 (x) ≤ −𝜅s (𝑡) 𝑉 (x)}
and by Gronwall’s inequality
𝑡
𝑡0
(13)
Now from (10) and (13), we obtain the following inequality:
𝑡
𝐶
󵄨 󵄨𝑝
𝐸 [|x (𝑡)| ] ≤ max 𝐸 [󵄨󵄨󵄨x0 󵄨󵄨󵄨 ] exp (− ∫ 𝜅 (𝑡) 𝑑𝑠) ,
𝐶min
𝑡0
𝑝
(17)
𝑙∈S
𝑡0
𝑝 > 0.
(14)
Hence, the thesis follows.
𝑛
(1)
Ωus
𝑙 = {x ∈ R : ∀𝑡 ∈ T L𝑙 𝑉 (x) ≤ 𝜅us (𝑡) 𝑉 (x)}
for 𝑙 ∈ S,
for 𝑙 ∈ S,
(18)
(3) there exists a Lebesguemeasurable function 𝜅 : T →
(0, ∞) such that T ≠ 0, where
T = {𝜎 ∈ SW : ∫ [𝜅us (𝑠)+𝜅 (𝑠)] 𝑑𝑠 ≤ ∫ [𝜅s (𝑠)−𝜅 (𝑠)] 𝑑𝑠} .
𝑇us
𝑇s
(19)
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 Ω𝑙 and 𝑙 ∈ S,
which can be separated into two disjoint subregions: a stable
subregion Ωs𝑙 and an unstable region Ωus
𝑙 , 𝑙 ∈ S (the upper
scripts “s” and “us” denote stable and unstable regions, resp.)
that is,
Ω𝑙 = Ωs𝑙 ∪ Ωus
𝑙 ,
𝑝 > 0,
⋃ Ω𝑙 = R𝑛 , where
≤ E [𝑉 (x0 )] − ∫ 𝜅 (𝑡) E [𝑉 (x (𝑠))] 𝑑𝑠
𝐸 [𝑉 (x)] ≤ 𝐸 [𝑉 (x0 )] exp (− ∫ 𝜅 (𝑡) 𝑑𝑠) .
(16)
for some switching rule 𝜎.
(12)
𝑡
−𝜅s 𝑉 (x) for x ∈ Ωs𝜎(𝑡),
𝜅us 𝑉 (x) for x ∈ Ωus
𝜎(𝑡) ,
(2) there exist Lebesgue-measurable functions 𝜅s : T →
(0, ∞) and 𝜅us : T → (0, ∞) such that partition (15)
satisfies the conditions
𝑡
𝑡0
Definition 7. A Lyapunov function 𝑉(x) ∈ 𝐶2 (R𝑛 ; R+ ) is
called a single Lyapunov-like function if there exist positive
constants 𝜅s and 𝜅us such that
Ωs𝑙 ∩ Ωus
𝑙 = 0.
(15)
Let us denote by Ts the sum of time intervals of the
residence of the system (1) in the regions Ωs𝑙 , 𝑙 ∈ S, and by
Tus the sum of time intervals of the residence of the system
(1) in the regions Ωus
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.
Then the null solution x ≡ 0 of (1) is 𝑝th mean exponentially
stable under the stabilizing switching rule 𝜎st ∈ T.
Proof. From assumptions for 𝜎st ∈ SW, it follows that
E [𝑉 (x)] ≤ E [𝑉 (x0 )] exp (∫ 𝜅us (𝑠) 𝑑𝑠 − ∫ 𝜅s (𝑠) 𝑑𝑠) .
𝑇us
𝑇s
(20)
Let us consider the switching strategy 𝜎st ∈ T described by
(19). Then we obtain
𝑡
E [𝑉 (x)] ≤ E [𝑉 (x0 )] exp (− ∫ 𝜅 (𝑠) 𝑑𝑠) .
𝑡0
(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).
4
Journal of Applied Mathematics
Example 9. Let us consider a special case of the system (1)
with the two subsystems (S = {1, 2}) given as follows:
𝑑x (𝑡) = [A (𝜎) x (𝑡) + f (x, 𝜎)] 𝑑𝑡 + G (𝜎) 𝑑𝑤 (𝑡) ,
𝑛
(𝜎 (𝑡0 ) , x (𝑡0 )) = (𝜎0 , x0 ) ∈ S × R ,
(22)
where
exponentially stable in mean square with a decay rate 𝜅 = 0.08
for the switching strategy 𝜎st ∈ T given by
𝜎st (x, 𝑡) = {
1 if x (𝑡) ∈ Ω1 ,
2 if x (𝑡) ∈ Ω2 .
(26)
4. Stabilizability of Stochastic Hybrid Systems
2.18 2
A (1) = [
],
−4 −2.82
f (x, 1) = [
0
],
𝑓2 (𝑥2 , 1)
−1.905 4
A (2) = [
],
−2 1.095
f (x, 2) = [
0
],
𝑓2 (𝑥2 , 2)
0.8𝑥2 − 0.2
{
{
𝑓2 (𝑥2 , 1) = {𝑥2
{
{0.8𝑥2 + 0.2
for 𝑥2 < −1,
for 𝑥2 ∈ [−1, 1] ,
for 𝑥2 > 1,
0.9𝑥2 − 0.1
{
{
𝑓2 (𝑥2 , 2) = {𝑥2
{
{0.9𝑥2 + 0.1
for 𝑥2 < −1,
for 𝑥2 ∈ [−1, 1] ,
for 𝑥2 > 1,
G (1) = [
0.2 0
],
0 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
𝑝
𝑑x (𝑡) = f0 (x (𝑡) , 𝑡, 𝜎) 𝑑𝑡 + ∑𝑢𝑖 (x, 𝑡, 𝜎) f𝑖 (x (𝑡) , 𝑡, 𝜎)
𝑖=1
𝑀
0.1 0
G (2) = [
].
0 0.1
(27)
+ ∑ g𝑘 (x (𝑡) , 𝑡, 𝜎) 𝑑𝑤𝑘 (𝑡) ,
𝑘=1
(23)
Let us consider the Lyapunov function 𝑉(x) = x𝑇 x. Since
0 ≤ 𝑓2 (𝑥2 , 𝑙)𝑥2 ≤ 𝑥22 , 𝑙 = 1, 2, then
2
2
𝑇
L(22)
1 𝑉 (x) = 2.2𝑥1 − 2𝑥1 𝑥2 − 2.8𝑥2 + 𝑓(x, 1) x
≤ − 0.8 (𝑥12 + 𝑥22 ) for
Ωs1 = {x ∈ R2 : (𝑥2 + 3𝑥1 ) (𝑥1 − 𝑥2 ) ≤ 0} ,
2
2
𝑇
L(22)
2 𝑉 (x) = −1.9𝑥1 + 2𝑥1 𝑥2 + 1.1𝑥2 + 𝑓(x, 2) x
≤ −0.8 (𝑥12 + 𝑥22 ) for
Ωs2 = {x ∈ R2 : −1.1𝑥12 + 2𝑥1 𝑥2 + 2.9𝑥22 ≤ 0} ,
2
2
𝑇
L(22)
2 𝑉 (x) = −1.9𝑥1 + 2𝑥1 𝑥2 + 1.1𝑥2 + 𝑓(x, 2) x
≤ 1.1 (𝑥12 + 𝑥22 ) for
2
s
s
Ωus
2 = {x ∈ R : (𝑥2 + 3𝑥1 ) (𝑥2 − 𝑥1 ) ≤ 0} \ (Ω1 ∪ Ω2 ) .
(24)
Note that ⋃𝑙∈S Ω𝑙 = R2 and functions 𝜅s and 𝜅us are constant
and are given as follows 𝜅s = 0.8, 𝜅us = 1.1.
Condition
𝜅 −𝜅
Tus
≤ s
≈ 0.61
Ts
𝜅us + 𝜅
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.
(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 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
𝐿𝑖𝑙 , 𝑙 ∈ S, 𝑖 = 1, . . . , 𝑚, for 𝜙(x, 𝑡, 𝑙) ∈ 𝐶2,1 (R𝑛 × T × S; R+ ):
𝐿 𝑙 𝜙 (x, 𝑡, 𝑙) =
+
(25)
is satisfied for 𝜅 = 0.08. Exemplary simulations are shown in
Figures 1, 2, and 3.
From Theorem 8, it follows that 𝑉(x) = x𝑇 x is the
single Lyapunov-like function for the system (22), and it is
𝑛
𝜕𝜙 (x, 𝑡, 𝑙)
𝜕𝜙 (x, 𝑡, 𝑙)
𝜇
+ ∑ 𝑓0 (x, 𝑡, 𝑙)
𝜕𝑡
𝜕𝑥𝜇
𝜇=1
𝑛
𝜕2 𝜙 (x, 𝑡, 𝑙)
1 𝑛 𝑀 𝑟
,
∑ ∑ 𝑔𝑘 (x, 𝑡, 𝑙) 𝑔𝑘s (x, 𝑡, 𝑙)
2 𝑟,s=1 𝑘=1
𝜕𝑥𝑟 𝜕𝑥s
𝜇
𝐿𝑖𝑙 𝜙 (x, 𝑡, 𝑙) = ∑ 𝑓𝑖 (x, 𝑡, 𝑙)
𝜇=1
𝜕𝜙 (x, 𝑡, 𝑙)
,
𝜕𝑥𝜇
𝑖 = 1, . . . , 𝑚, 𝑙 ∈ S.
(28)
Journal of Applied Mathematics
5
𝑥2
𝑥2
4
20
2
𝑥0
−4
𝑥1
2
−2
4
𝑥0
𝑥1
20
−20
−2
40
−4
(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).
𝑥2
1.5
Ω1
1
𝑥0
𝑉(𝑥(𝑡))
3
0.5
Ω2
𝑥1
−1
0.5
−0.5
−0.5
2
1
Ω2
Ω1
1
0
𝑡
0
1
2
3
4
5
(b) Lyapunov function 𝑉 = x𝑇 x for the hybrid system (22)
(a) Stable path of the hybrid system (22) with stabilizing
switching 𝜎st given by (26)
Figure 2: Trajectories of the hybrid system (22).
Then, the following stabilization result for the control
hybrid system (27) holds.
Theorem 10. Suppose that the following conditions hold:
(2) there exists a Lebesgue-measurable function 𝜅 : T →
(0, ∞) such that partition determined by (7) satisfies
conditions
⋃ Ω𝑙 = R𝑛 , where
𝑙∈S
(1) there exist a Lyapunov function 𝑉(x) ∈ 𝐶2 (R𝑛 ; R+ )
and positive constants 𝐶min , 𝐶max such that
𝐶min |x|𝑝 ≤ 𝑉 (x) ≤ 𝐶max |x|𝑝 ,
𝑝 > 0,
(29)
𝑛
Ω𝑙 = {x ∈ R : ∀𝑡 ∈ T 𝐿 𝑙 𝑉 (x) ≤ −𝜅 (𝑡) 𝑉 (x)}
for 𝑙 ∈ S.
(30)
Then the control law u : R𝑛 × T × S → R𝑚 given as follows:
(31)
𝑢𝑖 (x, 𝑡, 𝑙) = −𝐿𝑖𝑙 𝑉 (x) , 𝑖 = 1, 2, . . . , 𝑚, 𝑙 ∈ S
6
Journal of Applied Mathematics
𝑥(𝑡)
𝑞(𝑡)
𝑥2 (𝑡)
1
2
𝑥1 (𝑡)
𝑡
2
1
3
4
5
1
−1
𝑡
2
1
(a)
3
4
5
(b)
Figure 3: Stable path of the hybrid system (22) with stabilizing switching 𝜎st given by (26).
together with the stabilizing switching rule 𝜎st ∈ SW renders
the null solution of the stochastic hybrid system (27) the 𝑝th
mean exponentially stable.
together with the stabilizing switching rule 𝜎st ∈ T 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
Proof. The thesis follows from Theorem 8.
𝑚
𝑖
∀𝑙 ∈ S L(27)
𝑙 𝑉 (x) = 𝐿 𝑙 𝑉 (x) − ∑(𝐿 𝑙 𝑉 (x))
2
(32)
𝑖=1
≤ − 𝜅 (𝑡) 𝑉 (x) ,
for x ∈ Ω𝑙 .
𝑀
𝑑x (𝑡) =[A (𝜎 (𝑡)) x (𝑡)+ ∑𝑢𝑖 (x (𝑡) , 𝜎 (𝑡)) C𝑖 (𝜎 (𝑡)) x (𝑡)] 𝑑𝑡
Now the thesis follows from Theorem 6.
𝑖=1
We can formulate a more general theorem in a case when
a Lyapunov-like function exists as follows.
(1) there exist a Lyapunov function 𝑉(x) ∈ 𝐶2 (R𝑛 ; R+ )
and positive constants 𝐶min , 𝐶max such that
𝐶min |x|𝑝 ≤ 𝑉 (x) ≤ 𝐶max |x|𝑝 ,
𝑝 > 0,
(33)
(2) there exist Lebesgue-measurable functions 𝜅s : T →
(0, ∞) and 𝜅us : T → (0, ∞) such that partition (15)
satisfies conditions
⋃ Ω𝑙 = R𝑛 , where
𝑙∈S
Ωs𝑙 = {x ∈ R𝑛 : ∀𝑡 ∈ T 𝐿 𝑙 𝑉 (x) ≤ −𝜅s (𝑡) 𝑉 (x)}
for 𝑙 ∈ S,
𝑛
= {x ∈ R : ∀𝑡 ∈ T 𝐿 𝑙 𝑉 (x) ≤ 𝜅us (𝑡) 𝑉 (x)}
for 𝑙 ∈ S,
(34)
(3) there exists a Lebesgue-measurable function 𝜅 : T →
R+ such that T ≠ 0, where
T = {𝜎 ∈ SW : ∫ [𝜅us (𝑠)+𝜅 (𝑠)] 𝑑𝑠 ≤ ∫ [𝜅s (𝑠)−𝜅 (𝑠)] 𝑑𝑠} .
𝑇us
𝑀
+ ∑ B𝑘 (𝜎 (𝑡)) x (𝑡) 𝑑𝑤𝑘 (𝑡) ,
𝑘=1
Theorem 11. Suppose that the following conditions hold:
Ωus
𝑙
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:
(𝜎 (𝑡0 ) , x (𝑡0 )) = (𝜎0 , x0 ) ,
(37)
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:
𝐿 𝑙 𝜙 (x, 𝑙) = x𝑇 A𝑇 (𝑙) ∇𝜙 (x, 𝑙)
+
1
tr (B𝑘 (𝑙) xx𝑇 B𝑘 (𝑙)𝑇 ∇2 𝜙 (x, 𝑙)) ,
2
(35)
𝑛
𝑚
Then the control law u : R × T × S → R given as follows
𝑢𝑖 (x, 𝑡, 𝑙) = −𝐿𝑖𝑙 𝑉 (x) ,
𝑖 = 1, 2, . . . , 𝑚
(36)
(38)
𝑙 ∈ S,
𝑇s
𝐿𝑖𝑙 (x, 𝑙) = x𝑇 C𝑇𝑖 (𝑙) ∇𝜙 (x, 𝑙) ,
𝑖 = 1, . . . , 𝑚, 𝑙 ∈ S,
where ∇𝜙(⋅, 𝑞) and ∇2 𝜙(⋅, 𝑞) denote gradient and Hessian of
the function 𝜙(⋅, 𝑞), respectively.
Journal of Applied Mathematics
7
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:
⋃ Ω𝑙 = R𝑛 , where
𝑙∈S
1𝑀 𝑇 𝑇
∑ x B𝑘 (𝑙) HB𝑘 (𝑙) x
2 𝑘=1
Ω𝑙 = {x ∈ R𝑛 : x𝑇 A𝑇 (𝑙) Hx +
≤ 𝛼1 x𝑇Hx ∧ 𝛼2 x𝑇 Hx
󵄨󵄨
󵄨󵄨
󵄨󵄨 𝑇 𝑀
󵄨󵄨
󵄨
≤ 󵄨󵄨x H ∑ B𝑘 (𝑙) x󵄨󵄨󵄨 ≤ 𝛼3 x𝑇 Hx}
󵄨󵄨
󵄨󵄨
𝑘=1
󵄨
󵄨
(𝑝/2)−1 𝑇
x C𝑇𝑖 (𝑙) Hx,
⋃ Ω𝑙 = R𝑛 , where
1𝑀
Ω𝑙 = {x ∈ R𝑛 : x𝑇 A𝑇 (𝑙) Hx+ ∑ x𝑇 B𝑇𝑘 (𝑙) HB𝑘 (𝑙) x < 0} ,
2 𝑘=1
𝑖 = 1, 2, . . . , 𝑚, 𝑙 ∈ S
(40)
(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
,
𝑝 > 0.
(41)
Notice that 𝑉 satisfies assumption (1) of Theorem 10 for
(𝑝/2)
𝐶min = 𝜆 min (H) and 𝐶max = 𝜆(𝑝/2)
max (H). Then using
assumptions (39), we obtain
∀𝑙 ∈ S 𝐿 𝑙 𝑉 (x) ≤ −𝜅𝑉 (x)
𝑙∈S
(39)
(a) 2 ≤ 𝑝 < 2 + 2|𝛼1 |/(𝛼3 )2 if 𝛼1 < 0,
(𝑝/2)
Remark 13. In a particular case of 𝑝 = 2, we obtain the
following criterion.
𝑙 ∈ S.
together with the stabilizing switching rule 𝜎st ∈ SW makes
the hybrid system (37) the 𝑝th mean exponentially stable for
𝑉 (x) = (x𝑇 Hx)
Notice that function 𝑉(x) = (x𝑇 Hx)(𝑝/2) is a single
Lyapunov function for the hybrid system (37). See [16] for the
details of the proof.
Criterion 14. Suppose that there exists a symmetric positive
definite matrix H such that
Then control u : R𝑛 × S → R𝑚 of a form
𝑢𝑖 (x, 𝑙) = −𝑝(x𝑇 Hx)
Hence, the thesis follows.
𝜅 > 0 for x ∈ Ω𝑙 ,
(42)
where
󵄨 󵄨
2 󵄨󵄨󵄨𝛼1 󵄨󵄨󵄨
󵄨󵄨 󵄨󵄨 𝑝
2
{
{
−(
)
𝑉
,
for
2
≤
𝑝
<
2+
,
−1)
𝛼
𝛼
𝑝
(
(x)
󵄨
󵄨
{
3
󵄨 1󵄨 2
2
{
{
(𝛼
)
{
3
{
{
{
𝛼1 < 0,
𝜅={
𝑝
2𝛼1
{
2
{
𝑝 (−𝛼1 −( −1) 𝛼2 ) 𝑉 (x) , for 0 < 𝑝 < 2 −
,
{
{
2
{
2
(𝛼2 )
{
{
{
2
0 ≤ 𝛼1 < (𝛼2 ) .
{
(43)
From Theorem 10, it follows that the control chosen as
follows:
(𝑝/2)−1 𝑇
𝑢𝑖 (x, 𝑙) = −𝐿𝑖𝑙 𝑉 (x) = −𝑝(x𝑇 Hx)
x C𝑇𝑖 (𝑙) Hx,
(44)
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 .
𝑙 ∈ S.
(45)
Then the control u : R𝑛 × S → R𝑚 of a form
𝑢𝑖 (x, 𝑙) = −2x𝑇 C𝑇𝑖 (𝑙) Hx,
𝑖 = 1, 2, . . . , 𝑚, 𝑙 ∈ S
(46)
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 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.
Theorem 15. Suppose that the following conditions are satisfied:
(1) there exist symmetric positive definite matrix H, constants 𝛼1s , 𝛼1us , and positive constants 𝛼2 , 𝛼3 such that
⋃ Ω𝑙 = R𝑛 , where
𝑙∈S
Ωs𝑙 = {x ∈ R𝑛 : x𝑇 HA (𝑙) x +
1𝑀 𝑇 𝑇
∑ x B𝑘 (𝑙) HB𝑘 (𝑙) x
2 𝑘=1
2
≤ 𝛼1s x𝑇 Hx, 𝛼1s < (𝛼2 ) ∧ 𝛼2 x𝑇 Hx
󵄨󵄨
󵄨󵄨
𝑀
󵄨󵄨
󵄨󵄨
≤ 󵄨󵄨󵄨x𝑇 H ∑ B𝑘 (𝑙) x󵄨󵄨󵄨 ≤ 𝛼3 x𝑇 Hx} 𝑙 ∈ S,
󵄨󵄨
󵄨󵄨
𝑘=1
󵄨
󵄨
𝑛
s 𝑇
𝑇
Ωus
𝑙 = {x ∈ R : 𝛼1 x Hx < x HA (𝑙) x
1𝑀 𝑇 𝑇
∑ x B𝑘 (𝑙) HB𝑘 (𝑙) x ≤ 𝛼1us x𝑇 Hx, ∧𝛼2 x𝑇 Hx
2 𝑘=1
󵄨󵄨
󵄨󵄨
󵄨󵄨 𝑇 𝑀
󵄨󵄨
󵄨
≤ 󵄨󵄨x H ∑ B𝑘 (𝑙) x󵄨󵄨󵄨 ≤ 𝛼3 x𝑇 Hx} 𝑙 ∈ S,
󵄨󵄨
󵄨󵄨
𝑘=1
󵄨
󵄨
(47)
+
8
Journal of Applied Mathematics
𝑉(𝑥(𝑡))
0.2 𝑥2
0.4
𝑥1
−0.2
−0.4
0.2
0.4
−0.2
0.2
−0.4
𝑥0
𝑡
−0.6
0.5
(a) Trajectory of the hybrid system
1
1.5
2
2.5
3
3.5
(b) Trajectory of the Lyapunov function
Figure 4: Simulation of the hybrid system (53) with stabilizing switching 𝜎 = 𝜎st for 𝑝 = 1.
Then the control u : R𝑛 × S → R𝑚 of a form
(𝑞(𝑡), 𝑥(𝑡))
𝑢𝑖 (x, 𝑙) = −𝑝(x𝑇 Hx)
2
(𝑝/2)−1 𝑇 𝑇
x 𝐶𝑖 (𝑙) Hx,
𝑖 = 1, 2, . . . , 𝑚, 𝑙 ∈ S
(49)
𝑞(𝑡)
together with the stabilizing switching rule 𝜎st ∈ T makes the
hybrid system (37) the 𝑝th mean exponentially stable for
1
𝑥1 (𝑡)
0.5
(a) 2 ≤ 𝑝 < 2 + 2|𝛼1s |/(𝛼3 )2 if 𝛼1s < 0,
𝑡
1
1.5
2
2.5
3
(b) 0 < 𝑝 < 2 − 2𝛼1s /(𝛼2 )2 if 0 ≤ 𝛼1s < 𝛼22 .
3.5
Proof. The thesis follows from Theorem 11. Let us choose a
Lyapunov function as follows:
𝑥2 (𝑡)
(𝑝/2)
Figure 5: Simulation of the hybrid system (53) with stabilizing
switching 𝜎 = 𝜎st for 𝑝 = 1.
(2) there exists constant 𝜅 > 0 such that T ≠ 0, where
T = {𝜎 ∈ SW :
𝑝
us
2
{
{𝑝 (𝛼1 +( − 1) 𝛼3 )
2
𝜅us = {
{𝑝 (𝛼us +( 𝑝 − 1) 𝛼2 )
1
2
2
{
󵄨󵄨 s 󵄨󵄨
󵄨𝛼 󵄨
dla 2 ≤ p < 2 + 2 󵄨 1 󵄨2 ,
(𝛼3 )
𝛼1 < 0,
𝛼s
dla 0 < p < 2 − 2 1 2 ,
(𝛼2 )
2
0 ≤ 𝛼1s < (𝛼2 ) ,
dla p ≥ 2, 𝛼1us ≥ 0,
,
𝑝 > 0.
(50)
Function 𝑉 satisfies assumption (1) of Theorem 11 for 𝐶min =
(𝑝/2)
𝜆 min (H) and 𝐶max = 𝜆(𝑝/2)
max (H).
Furthermore,
−𝜅 𝑉 (x) ,
𝐿 𝑙 𝑉 (x) ≤ { s
𝜅us 𝑉 (x) ,
Tus
𝜅 −𝜅
≤ s
},
Ts
𝜅us + 𝜅
𝑝
󵄨 󵄨
{
{
𝑝 (󵄨󵄨󵄨𝛼1s 󵄨󵄨󵄨 − ( − 1) 𝛼32 )
{
{
2
{
{
{
{
{
𝜅s = {
𝑝
{
{
{
𝑝 (−𝛼1s − ( − 1) 𝛼22 )
{
{
2
{
{
{
{
𝑉 (x) = (x𝑇 Hx)
for x ∈ Ωs𝑙 ,
for x ∈ Ωus
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 : R𝑛 × S →
R𝑚 can be chosen as follows:
(𝑝/2)−1 𝑇
𝑢𝑖 (x, 𝑙) = −𝐿𝑖𝑙 𝑉 (x) = −𝑝(x𝑇 Hx)
x C𝑇𝑖 (𝑙) Hx,
(52)
𝑖 = 1, 2, . . . , 𝑚, 𝑙 ∈ S.
Hence, the thesis follows.
2
dla 0 < p < 2, 𝛼1us ≥ (𝛼2 ) .
(48)
Notice that function 𝑉(x) = (x𝑇 Hx)(𝑝/2) is a single
Lyapunov-like function for the hybrid system (37). See [16]
for the details of the proof.
Journal of Applied Mathematics
9
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
Theorem 6
Theorem 8
⇐ Theorem 10⇒
⇐ Theorem 11⇒
Theorem 12
Theorem 15
Single Lyapunov function
Single Lyapunov-like function
Example 16. Let us consider a special case of the hybrid
system (37) given as follows:
together with stabilizing switching rule 𝜎st given by
1
𝜎st (𝑡) = {
2
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
(𝜎 (0) , x (0)) = (𝜎0 , x0 ) ,
(53)
where
C2 (1) = [
0 2
C1 (1) = [
],
−1 0
1 0
],
−2 0
0.2 0
B (1) = [
],
0 0.2
−1 0
],
C1 (2) = [
0 1
−0.005
2
A (2) = [
],
−4 −0.005
C2 (2) = [
0 0
],
0 1
B (2) = [
(54)
0.1 0
].
0 0.1
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
u : S × R 2 → R2 ,
𝑇
u = [𝑢1 , 𝑢2 ] .
(55)
Let us choose the Lyapunov function of a form 𝑉(x) = x𝑇 x.
Then regions Ω𝑙 , 𝑙 ∈ S, are given as follows:
1
Ω1 = {x ∈ R2 : x𝑇 [A (1) + B𝑇 (1) B (1)] x ≤ 0}
2
= {x ∈ R2 : 𝑥1 𝑥2 ≤ 0} ,
1
Ω2 = {x ∈ R2 : x𝑇 [A (2) + B𝑇 (2) B (2)] x ≤ 0}
2
(56)
= {x ∈ R2 : 𝑥1 𝑥2 ≥ 0}
Ω1 ∪ Ω2 = R2 .
Condition (39) of Theorem 12 is satisfied for 𝛼1 = 0. Function
𝑉(x) = x𝑇 x is a single Lyapunov function for the system (53).
From Theorem 12, it follows that the control
𝑇
(𝑝/2)−1
u (1, x) = [−𝑝(x Hx)
𝑥1 𝑥2 ,
(𝑝/2)−1
−𝑝(x𝑇 Hx)
(𝑝/2)−1
u (2, x) = [−𝑝(x𝑇 Hx)
(58)
exponentially 𝑝th mean, stabilizes the system (53) for 𝑝 ∈
(0, 2).
+ B (𝜎 (𝑡)) x (𝑡) 𝑑𝑤 (𝑡) ,
−0.02 4
A (1) = [
],
−2 −0.02
gdy x (𝑡) ∈ Ω1 ,
gdy x (𝑡) ∈ Ω2 ,
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 nonGaussian 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.
(𝑥12 − 2𝑥1 𝑥2 ) ]
(𝑝/2)−1 2
𝑥2 ]
(𝑥22 − 𝑥12 ) , −𝑝(x𝑇 Hx)
(57)
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10
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