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On almost sure stability conditions of linear switching stochastic differential systems
Nonlinear Analysis: Hybrid Systems 22 (2016) 108–115
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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
On almost sure stability conditions of linear switching stochastic differential systems Shen Cong School of Mechanical & Electrical Engineering, Heilongjiang University, Harbin 150080, China
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Article history: Received 29 September 2014 Accepted 29 March 2016 Keywords: Switching systems Stochastic systems Almost sure stability
abstract In modeling practical systems, it can be efficient to apply Poisson process and Wiener process to represent the abrupt changes and the environmental noise, respectively. Therefore, we consider the systems affected by these random processes and investigate their joint effects on stability. In order to apply Lyapunov stability method, we formulate the action of the infinitesimal generator corresponding to such a system. Then, we derive the almost sure stability conditions by using some fundamental convergence theorem. To illustrate the theoretical results, we construct an example to show that it is possible to achieve stabilization by using random perturbations. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Switching systems can be used to describe some practical systems that undergo abrupt changes. Mathematically, such a system is composed of several subsystems with a switching signal to orchestrate among them. Therefore, one of the fundamental problems in the research area is to characterize the mechanism triggering switching and study its effect on stability. One way to deal with switching signal is to assume it to evolve according to the law of continuous-time Markov chain. In fact, this pure assumption implies that the occurrence of switching is a Poisson event; see, e.g., [1]. From this perspective, roughly speaking, the stability relies on two aspects, namely, the embedded Poisson distribution and the embedded discrete-time Markov chain; see, e.g., [2–4]. Indeed, the embedded Poisson distribution represents the density of switching points, while the embedded Markov chain indicates the likelihood of a subsystem to be activated at a switching point. In this sense, focusing on the aspect how the varying rate of switching signal influences stability, we might simply suppose the evolution of switching signal over time to obey Poisson distribution; see, e.g., [5]. Motivated by the above observation, we shall consider the systems affected by two basic kinds of Lévy processes, namely, Poisson processes and Wiener Processes. Thereby, we can in a unified stochastic framework account for the abrupt changes and the environment noise, which are commonly encountered in modeling practical systems. Recently, Chatterjee and Liberzon [6,7] proved that the solutions of such systems, if exist, have some nice statistic properties. In this paper, we make an attempt to investigate the joint effects of these random processes on the stability in almost sure sense. Within this context, as pointed out in [8], Wiener process can be used to stabilize a given unstable system or to make a system more stable even when it is already stable. Also, from [9,10] we learn that modulating random switching signal can influence the dynamical behavior and make it very different. In this paper, we observe that it is possible to achieve stabilization by letting Poisson and Wiener processes work together. Thus, we may gain a more comprehensive insight into the randomly perturbed stability problem.
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.nahs.2016.03.010 1751-570X/© 2016 Elsevier Ltd. All rights reserved.
S. Cong / Nonlinear Analysis: Hybrid Systems 22 (2016) 108–115
109
In parallel with the stochastic framework to account for switching, it is commonly used to classify switching signals in terms of average dwell-time. And it provides a conventional way to address the stability problem of switching systems by using multiple Lyapunov functions approach; see, e.g., [11] and the references therein. In this paper, we shall explain the connection between the deterministic and stochastic frameworks to deal with switching. In fact, we pose an analogy to the multiple Lyapunov functions approach. It requires us to formulate the infinitesimal generator of the systems under investigation, which plays a basic role for capturing the stochastic dynamics. Then, we derive the stability conditions by using some fundamental convergence theorem. The remainder of the paper is organized as follows. In Section 2 we describe the system under consideration and formulate the problem. In Section 3, we present the stability conditions and interpret the construction of the switching signal that obeys Poisson distribution. In Section 4, a simulation example is worked out to illustrate the results. Finally, the paper is briefly summarized in Section 5. 2. Problem description Consider the linear switching stochastic system described as follows dx(t ) = Aσ (t ) x(t )dt + Gσ (t ) x(t )dw(t ),
t ≥ t0 = 0,
(1)
where x(t ) ∈ R is state vector and w(t ) stands for the normalized one-dimensional Wiener Process, which is defined on the filtered complete probability space {Ω , F , Ft , P }. We suppose {Ft }t ≥0 to satisfy the usual conditions, namely, each Ft is P -complete and Ft = ∩s>t Fs for every t. Besides, write E for the mathematical expectation operator. System (1) is generated by the switching signal σ (t ) that orchestrates among the following subsystems n
dx(t ) = Ai x(t )dt + Gi x(t )dw(t ),
i ∈ {1, . . . , N }.
According to the evolution of switching signal over time, it can be expanded into the following sequential form
{(σ (t0 ), t0 ), (σ (t1 ), t1 ), . . . , (σ (tk ), tk ), . . .}.
(2)
It means that the σ (tk )th subsystem is activated during the interval [tk , tk+1 ). We now suppose the switching signal in (2) to obey Poisson distribution and to be independent of w(t ). That is, for any ∆ > 0 we have
P {tk+1 − tk ≥ ∆} = e−λ∆ ,
k = 0, 1, 2, . . . ,
(3)
where λ > 0 is referred to as Poisson exponent. Therefore, the switching points t0 < t1 < · · · < tk < · · · turn out to constitute a sequence of stopping times tending to infinity. Equivalently, it reads that
P {σ (t + ∆) ̸= σ (t )} = 1 − e−λ∆ .
(4)
As is well known, the Poisson exponent λ defines the mean value of the number of the switching points distributed within the time interval of unit length. Then it allows us to make a sense of the varying rate of switching signal. Accordingly, we categorize switching signals by means of λ. Namely, we write Sλ for the set of all the switching signals obeying Poisson distribution exactly with the exponent λ. For given switching signal σ , we denote by x(t ; x0 , σ ) the corresponding motion of system (1) at time t starting from x0 at initial time t0 . Definition 1. System (1) is said to be almost surely exponentially stable, if
P lim sup t →∞
ln |x(t ; x0 , σ )| t
<0 =1
for all x0 ∈ Rn . 3. Main results We aim at establishing the conditions that guarantee system (1) to be almost surely exponentially stable with respect to certain Sλ . The starting point is to formulate the action of the infinitesimal generator for it corresponds to the one-parameter semigroup of operators, which in turn determine the state-transition. In particular, the infinitesimal generator plays a fundamental role for addressing the stochastic stability problem due to the fact that, in general, we are not be able to express the state-transition in an analytical closed form (cf. [12]). For the sake of generality, we shall first consider the switching stochastic differential system as follows dx(t ) = fσ (t ) (x(t ))dt + gσ (t ) (x(t ))dw(t ),
(5)
where the switching signal obeys the Poisson distribution as in (4). We suppose the drift coefficients fi (x) and the diffusion coefficients gi (x) to satisfy certain conditions so that we can guarantee the solution of (5) to exist on [0, ∞). To each subsystem dx(t ) = fi (x(t ))dt + gi (x(t ))dw(t ), assign a positive-definite function Vi (x), which has the partial derivatives
110
S. Cong / Nonlinear Analysis: Hybrid Systems 22 (2016) 108–115
at least to second order in x. Let L be the infinitesimal generator corresponding to the stochastic differential system in (5). For V (x(t ), σ (t ) = i) = Vi (x(t )), we compute the action of L on it as 1
LV (x(t ), σ (t ) = i) = lim
E [V (x(t + ∆), σ (t + ∆))|x(t ), σ (t ) = i] − V (x(t ), σ (t ) = i)
∆ (1 − e−λ∆ ) [V (x(t + ∆), σ (t + ∆) ̸= i) − V (x(t ), σ (t ) = i)] = lim ∆ ↓0 ∆ ∆ ↓0
+ lim
e−λ∆
∆
∆ ↓0
[V (x(t + ∆), σ (t + ∆) = i) − V (x(t ), σ (t ) = i)]
2 ∂ Vi (x(t )) 1 ′ ∂ Vi (x(t )) = λ Vj (x(t )) − Vi (x(t )) + fi (x(t )) + gi (x(t )) gi (x(t )), j ̸= i. (6) ∂x 2 ∂ xp ∂ xq n×n ′ ∂V ∂2V ∂U ∂2U ∂V ∂ Vi For the composed function Ui (x(t )) = ln Vi (x(t )), noting that ∂ xi = V1 ∂ xi and ∂ x2i = V1 ∂ x2i − 12 ∂ xi , we compute its ∂x V i i
i
differential along the solution of system (5) as d ln V (x(t ), σ (t ) = i) = L ln V (x(t ), σ (t ) = i)dt 1 ∂ V (x(t ), σ (t ))
+
=
V (x(t ), σ (t ))
σ (t )=i
dw(t )
2 ∂ Vi (x(t )) 1 ∂ Vi (x(t )) fi (x(t )) + gi′ (x(t )) gi (x(t )) Vi (x(t )) Vi (x(t )) ∂x 2 ∂ xp ∂ xq n×n 2 1 ∂ Vi (x(t )) ∂ Vi (x(t )) 1 − 2 gi (x(t )) gi (x(t ))dw(t ), j ̸= i. dt + ∂ x V ( x ( t )) ∂x 2Vi (x(t )) i λ ln
Vj (x(t ))
∂x
gσ (t ) (x(t ))
1
+
(7)
We now are in the position to prove the main result. Theorem 1. Given λ > 0. If there exist a family of positive-definite matrices Pi and a family of numbers χi > 0 and γi such that for all i, j ∈ {1, . . . , N } the following inequalities hold
1 (Ai − γi Gi )′ Pi + Pi (Ai − γi Gi ) + G′i Pi Gi + λ ln χi + γi2 Pi < 0,
(8)
Pj ≤ χi Pi ,
(9)
2
j ̸= i,
then system (1) is almost surely exponentially stable for all switching signals belonging to Sλ . Proof. For system (1), we construct the Lyapunov function as V (x(t ), σ (t )) = Vσ (t ) (x(t ))
(10)
with Vi (x) = x Pi x, i ∈ {1, . . . , N }. Firstly, from (9) it is immediately seen that ′
ln
Vj (x) Vi (x)
≤ ln χi ,
j ̸= i.
Noting this and using (7), we know that the differential of ln V (x(t ), σ (t )) along the solution of system (1) satisfies d ln V (x(t ), σ (t )) ≤
x′ (t )(A′σ (t ) Pσ (t ) + Pσ (t ) Aσ (t ) + G′σ (t ) Pσ (t ) Gσ (t ) )x(t ) dt x′ (t )Pσ (t ) x(t )
+ λ ln χσ (t ) dt − +
1
x′ (t )(G′σ (t ) Pσ (t ) + Pσ (t ) Gσ (t ) )x(t ) x′ (t )Pσ (t ) x(t )
2
x′ (t )(G′σ (t ) Pσ (t ) + Pσ (t ) Gσ (t ) )x(t ) x′ (t )Pσ (t ) x(t )
2 dt
dw(t ).
(11)
In particular, we have
−
1 2
x′ (t )(G′σ (t ) Pσ (t ) + Pσ (t ) Gσ (t ) )x(t ) x′ (t )Pσ (t ) x(t )
2 =−
1
x′ (t )(G′σ (t ) Pσ (t ) + Pσ (t ) Gσ (t ) − γσ (t ) Pσ (t ) )x(t )
x′ (t )Pσ (t ) x(t ) ′ x (t )(Gσ (t ) Pσ (t ) + Pσ (t ) Gσ (t ) )x(t ) 1 + γσ2(t ) − γσ (t ) x′ (t )Pσ (t ) x(t ) 2 ′ ′ x (t )(Gσ (t ) Pσ (t ) + Pσ (t ) Gσ (t ) )x(t ) 1 −γσ (t ) + γσ2(t ) . x′ (t )Pσ (t ) x(t ) 2
2
2
′
≤
(12)
S. Cong / Nonlinear Analysis: Hybrid Systems 22 (2016) 108–115
111
For any t ≥ t0 , we then have ln V (x(t ), σ (t )) ≤ ln V (x(t0 ), σ (t0 )) t ′ x (s)(A′σ (s) Pσ (s) + Pσ (s) Aσ (s) + G′σ (s) Pσ (s) Gσ (s) + (λ ln χσ (s) + 21 γσ2(s) )Pσ (s) )x(s) + ds x′ (s)Pσ (s) x(s) t0
t
γσ (s)
−
x′ (s)(G′σ (s) Pσ (s) + Pσ (s) Gσ (s) )x(s) x′ (s)Pσ (s) x(s)
t0
t
ds +
x′ (s)(G′σ (s) Pσ (s) + Pσ (s) Gσ (s) )x(s) x′ (s)Pσ (s) x(s)
t0
dw(s).
(13)
Since the right-hand side of (8) is negative-definite, there must be a sufficiently small positive number ϵ such that
1 (Ai − γi Gi )′ Pi + Pi (Ai − γi Gi ) + G′i Pi Gi + λ ln χi + γi2 Pi < −ϵ Pi 2
holds true uniformly for all i ∈ {1, . . . , N }. Taking this into account, from (13) we arrive at ln V (x(t ), σ (t )) < ln V (x(t0 ), σ (t0 )) − ϵ t + M (t ),
(14)
where M (t ) :=
t
x′ (s)(G′σ (s) Pσ (s) + Pσ (s) Gσ (s) )x(s) x′ (s)Pσ (s) x(s)
t0
dw(s).
It can be seen that M (t ) is a continuous local martingale with respect to {Ft }t ≥t0 and vanishes at t0 . Moreover, the quadratic variation of M (t ) (for the definition of quadratic variation please refer to p. 12 in [12]) satisfies
t ′ x (s)(G′σ (s) Pσ (s) + Pσ (s) Gσ (s) )x(s) 2 ds ⟨M (t ), M (t )⟩ = x′ (s)Pσ (s) x(s) t0 ≤ ρ 2 (t − t0 ).
(15)
where
ρ := max
max{|λmin (G′i Pi + Pi Gi )|, |λmax (G′i Pi + Pi Gi )|}
λmin (Pi )
i=1,...,N
.
Then, by the strong law of large numbers (cf. [12]), there is probability 1 that lim
M (t )
t →∞
t
= 0.
From (14) it then follows that, with probability 1, lim sup t →∞
ln |x(t )| t
=
1 2
lim sup
The proof is thus completed.
ln V (x(t ), σ (t )) t
t →∞
=−
ϵ 2
< 0.
Remark 1. A discussion on the parameters γi and χi , which should be properly chosen so that the derived stability conditions are feasible, would be helpful for us to make a sense of the stability mechanism. As shown in (12), the effect of the part of the system polluted by the environment noise on stability is partly represented through γi . At the same time, χi renders the intensity parameter λ available to represent the role that the switching signal plays in preserving stability. Therefore, adjusting χi and γi to test the feasibility of the stability conditions brings an insight into the way of the randomly varying switching signal and the environment noise to influence stability. We now turn to take a look at the construction of the switching signals that obey Poisson distribution. Given a switching signal σ (t ), let
Π = {t0 = 0, t1 , . . . , tk , . . . : lim tk = ∞} k→∞
be the collection of its switching points. Meanwhile, let N ([s1 , s2 ]) be counter of the switching points of σ (t ) distributed within the interval [s1 , s2 ]. We then have
N ([t0 , s]) = sup m :
m
(tk+1 − tk ) ≤ s .
k=0
That is to say, the number of switching points is depending on the length of the interval of time and, therefore, is invariant for the shift transformation. As we know, the Poisson exponent λ indicates the density of the switching points distributed within an interval of unit length. That is, fixing an interval of time [s1 , s2 ], one has that
E {N ([s1 , s2 ])} = λ(s2 − s1 ).
(16)
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S. Cong / Nonlinear Analysis: Hybrid Systems 22 (2016) 108–115
At the same time, merging switching signals provides an alternative way to observe the linear relationship between the intensity parameter λ and the density of switching points. To be precise, let Π1 and Π2 be the collections of the switching points of σ1 (t ) and σ2 (t ), respectively. The switching signal, which is denoted by σ1 (t ) ⊕ σ2 (t ), is generated by merging σ1 (t ) and σ2 (t ) in such a way that its switching points correspond to the set Π1 ∪ Π2 ; please refer to [11] for more details. Assume σ1 (t ) ∈ Sλ1 , σ2 (t ) ∈ Sλ2 , and they are independent of each other, then
σ1 (t ) ⊕ σ2 (t ) ∈ Sλ1 +λ2 .
(17)
Next, we consider the special case that only noise-free subsystems are involved. Namely, the system under consideration is x˙ (t ) = Aσ (t ) x(t ),
t ≥ t0 = 0,
(18)
which is generated by a switching signal obeying the Poisson distribution in (4) and orchestrating among several noise-free linear subsystems. Corollary 1. Given λ > 0. If there exist a family of numbers χi and βi , and a family of positive-definite matrices Pi such that A′i Pi + Pi Ai − βi Pi ≤ 0,
(19)
Pj ≤ χi Pi ,
(20)
j ̸= i,
and
λ ln χi + βi < 0.
(21)
then system (18) is almost surely exponentially stable for all switching signals belonging to Sλ . Obviously, Corollary 1 can be deduced directly from Theorem 1. However, assume there exist χ and β such that (19)–(21) hold true uniformly for all subsystems, namely, they become A′i Pi + Pi Ai − β Pi ≤ 0,
(22)
Pj ≤ χ Pi ,
(23)
j ̸= i, λ ln χ + β < 0,
(24)
respectively. Then, we can provide an alternative proof by using the law of large numbers of Poisson processes. Proof. Construct a Lyapunov function as V (x(t ), σ (t ) = i) = x′ (t )Pi x(t ).
(25)
For any t ≥ t0 , let N ([t0 , t ]) be equal to an integer, say, k. At the same time, let the following sequence t0 < t1 < t2 < · · · < tk ≤ t denote the corresponding switching points. Indeed, (22) implies that β(ti −ti−1 ) V (x(ti ), σ (ti+ V (x(ti−1 ), σ (ti+ −1 )) ≤ e −1 )),
1 ≤ i ≤ k.
This together with (23) yields β(ti −ti−1 ) β(ti −ti−2 ) V (x(ti ), σ (ti+ V (x(ti−1 ), σ (ti+ V (x(ti−2 ), σ (ti+ −1 )) ≤ χ e −2 )) ≤ χ e −2 )).
Iterating the preceding inequalities from tk to t0 , we obtain V (x(t ), σ (tk+ )) ≤ χ k eβ(t −t0 ) V (x(t0 ), σ (t0 ))
= exp[N ([t0 , t ]) ln χ + β(t − t0 )]V (x(t0 ), σ (t0 )). It then follows that lim sup
ln |x(t ; x0 , σ )| t
t →∞
≤
1 2
lim
t →∞
N ([t0 , t ]) ln χ + β(t − t0 ) t
.
Furthermore, by the law of large numbers of Poisson processes (cf. [13]), there is probability 1 that lim
t →∞
N ([t0 , t ]) t
= λ.
Hence, in view of (24), we arrive at
P lim sup t →∞
ln |x(t ; x0 , σ )| t
The proof is thus completed.
≤
1 2
(λ ln χ + β) < 0 = 1.
(26)
S. Cong / Nonlinear Analysis: Hybrid Systems 22 (2016) 108–115
113
In parallel with (24), the following inequality
λ(χ − 1) + β < 0
(27)
can guarantee the system to be almost surely asymptotically stable as well provided the conditions in (22) and (23) remain true; please refer to [5]. Though the inequality in (27) is more conservative than that in (24) for there must be χ > 1, we now prove that it makes exp[N ([t0 , t ]) ln χ + β(t − t0 )] become a continuous-time supermartingale. In fact, for any ∆ > 0, by the independent increments property of Poisson processes we can calculate that
E exp[N ([t0 , t + ∆]) ln χ]N ([t0 , t ])
= E exp[N ([t0 , t ]) ln χ + (N ([t0 , t + ∆]) − N ([t0 , t ])) ln χ]N ([t0 , t ]) = exp[N ([t0 , t ]) ln χ]E exp[(N ([t0 , t + ∆]) − N ([t0 , t ])) ln χ] = exp[N ([t0 , t ]) ln χ]E χ N ([t0 ,t +∆])−N ([t0 ,t ]) ∞ (λ∆)k = exp[N ([t0 , t ]) ln χ] e−λ∆ χk k! k=0 = exp[N ([t0 , t ]) ln χ + λ(χ − 1)∆]. Multiplying both sides of the above equality by exp[β(t + ∆ − t0 )] and recalling (27) results in
E exp[N ([t0 , t + ∆]) ln χ + β(t + ∆ − t0 )]N ([t0 , t ])
= exp[N ([t0 , t ]) ln χ + λ(χ − 1)∆ + β(t + ∆ − t0 )] = exp[(λ(χ − 1) + β)∆] exp[N ([t0 , t ]) ln χ + β(t − t0 )] < exp[N ([t0 , t ]) ln χ + β(t − t0 )]. It implies that the expression on the right-hand side of (26) constitutes a continuous-time supermartingale. Therefore, one can show the system in (18) is almost surely asymptotically stable via some typical approach; see, e.g., [14]. 4. An illustrative example Consider three linear unstable subsystems as x˙ (t ) =
−0.35 0.5
−0.28 0.32
0.6 ˙ (t ) −0.4 x(t ), x
=
0.31 −0.5
0.74 ˙ (t ) −0.12 x(t ), and x
=
0.37 −0.3 x(t ). In order to demonstrate the effectiveness of the results, we shall show that it is possible to achieve
stabilization by simultaneously introducing environment noise and modulating switching signal. As a matter of fact, for the following stochastic subsystems dx(t ) = dx(t ) =
−0.28 0.32
0.31 −0.5
0.6 0.46 x(t )dt + −0.4 0.25
0.74 0.3 x(t )dt + −0.12 0
0.43 x(t )dw(t ), 0.5
0 x(t )dw(t ), 0.24
and dx(t ) =
−0.35 0.5
0.37 0.5 x(t )dt + −0.3 0.28
0.47 x(t )dw(t ), 0.56
by fixing λ = 1.0 and setting χ1 = 1.15, χ2 = 0.87, χ3 = 1.15 and γ1 = 0.84, γ2 = 0.34, γ3 = 0.9, we find the stability conditions in Theorem 1 feasible. The sample path of the switching signal is shown in Fig. 1, which triggers the change of subsystems by its edges. The corresponding state-response is depicted in Fig. 2. A simulation exposes that the sample paths of the solutions of both the second and third stochastically perturbed subsystems diverge to infinity. This reveals that introducing Wiener processes alone may be not enough to achieve stabilization, yet letting it work together with Poisson processes can make the overall system exponentially stable. Generally speaking, stability can be achieved if the switching signal is properly modulated so as to retain balance among the stable subsystems and the unstable ones. 5. Conclusion We considered a class of linear switching stochastic systems. By supposing the evolution of switching signal over time to obey Poisson distribution, we set this kind of systems entirely in a stochastic framework and formulated the action of the corresponding infinitesimal generator. Therefore, we characterized the joint effects of Poisson processes and Wiener processes on the sample paths of solution to such a system and derived the almost sure stability conditions. We provided an example to illustrate the theoretical results.
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S. Cong / Nonlinear Analysis: Hybrid Systems 22 (2016) 108–115
Fig. 1. The switching signal corresponding to λ = 1.0.
Fig. 2. The state-response corresponding to Poisson switching signal.
Acknowledgments The author would like to thank the anonymous reviewers for their very helpful comments and suggestions. The work is supported by the National Natural Science Foundation of China (Grant No. 61573131) and the Natural Science Foundation of Heilongjiang Province of China (Grant No. LC201428). References [1] X. Feng, K.A. Loparo, Y. Ji, H.J. Chizeck, Stochastic stability properites of jump linear systems, IEEE Trans. Automat. Control 37 (1) (1992) 38–53. [2] P. Bolzern, P. Colaneri, G. De Nicolao, On almost sure stability of continuous-time Markov jump linear systems, Automatica 42 (6) (2006) 983–988. [3] P. Bolzern, P. Colaneri, G. De Nicolao, Markov jump linear systems with switching transition rates: mean square stability with dwell-time, Automatica 46 (6) (2010) 1081–1088. [4] M. Tanelli, P. Bolzern, P. Colaneri, Almost sure stabilization of uncertain continuous-time Markov jump linear systems, IEEE Trans. Automat. Control 55 (1) (2010) 195–201. [5] A. Cetinkaya, K. Kashima, T. Hayakawa, Stability of stochastic systems with probabilistic mode switchings and state jumps, in: Proc. 2010 American Control Conference, Baltimore, June 30–July 02, 2010, pp. 4046–4051. [6] D. Chatterjee, D. Liberzon, Stability analysis of deterministic and stochastic switched systems via a comprison principle and multiple Lyapunov functions, SIAM J. Control Optim. 45 (1) (2006) 174–206. [7] D. Chatterjee, D. Liberzon, Stabilizing randomly switched systems, SIAM J. Control Optim. 49 (5) (2011) 2008–2031. [8] F. Deng, Q. Luo, X. Mao, S. Pang, Noise suppresses and expresses exponential growth, Systems Control Lett. 57 (3) (2008) 262–270. [9] G. Hu, M. Liu, X. Mao, M. Song, Noise suppresses exponential growth under regime switching, J. Math. Anal. Appl. 355 (2) (2009) 783–795.
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Nonlinear Analysis: Hybrid Systems journal homepage: www.elsevier.com/locate/nahs
On almost sure stability conditions of linear switching stochastic differential systems Shen Cong School of Mechanical & Electrical Engineering, Heilongjiang University, Harbin 150080, China
article
info
Article history: Received 29 September 2014 Accepted 29 March 2016 Keywords: Switching systems Stochastic systems Almost sure stability
abstract In modeling practical systems, it can be efficient to apply Poisson process and Wiener process to represent the abrupt changes and the environmental noise, respectively. Therefore, we consider the systems affected by these random processes and investigate their joint effects on stability. In order to apply Lyapunov stability method, we formulate the action of the infinitesimal generator corresponding to such a system. Then, we derive the almost sure stability conditions by using some fundamental convergence theorem. To illustrate the theoretical results, we construct an example to show that it is possible to achieve stabilization by using random perturbations. © 2016 Elsevier Ltd. All rights reserved.
1. Introduction Switching systems can be used to describe some practical systems that undergo abrupt changes. Mathematically, such a system is composed of several subsystems with a switching signal to orchestrate among them. Therefore, one of the fundamental problems in the research area is to characterize the mechanism triggering switching and study its effect on stability. One way to deal with switching signal is to assume it to evolve according to the law of continuous-time Markov chain. In fact, this pure assumption implies that the occurrence of switching is a Poisson event; see, e.g., [1]. From this perspective, roughly speaking, the stability relies on two aspects, namely, the embedded Poisson distribution and the embedded discrete-time Markov chain; see, e.g., [2–4]. Indeed, the embedded Poisson distribution represents the density of switching points, while the embedded Markov chain indicates the likelihood of a subsystem to be activated at a switching point. In this sense, focusing on the aspect how the varying rate of switching signal influences stability, we might simply suppose the evolution of switching signal over time to obey Poisson distribution; see, e.g., [5]. Motivated by the above observation, we shall consider the systems affected by two basic kinds of Lévy processes, namely, Poisson processes and Wiener Processes. Thereby, we can in a unified stochastic framework account for the abrupt changes and the environment noise, which are commonly encountered in modeling practical systems. Recently, Chatterjee and Liberzon [6,7] proved that the solutions of such systems, if exist, have some nice statistic properties. In this paper, we make an attempt to investigate the joint effects of these random processes on the stability in almost sure sense. Within this context, as pointed out in [8], Wiener process can be used to stabilize a given unstable system or to make a system more stable even when it is already stable. Also, from [9,10] we learn that modulating random switching signal can influence the dynamical behavior and make it very different. In this paper, we observe that it is possible to achieve stabilization by letting Poisson and Wiener processes work together. Thus, we may gain a more comprehensive insight into the randomly perturbed stability problem.
E-mail address: [email protected]. http://dx.doi.org/10.1016/j.nahs.2016.03.010 1751-570X/© 2016 Elsevier Ltd. All rights reserved.
S. Cong / Nonlinear Analysis: Hybrid Systems 22 (2016) 108–115
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In parallel with the stochastic framework to account for switching, it is commonly used to classify switching signals in terms of average dwell-time. And it provides a conventional way to address the stability problem of switching systems by using multiple Lyapunov functions approach; see, e.g., [11] and the references therein. In this paper, we shall explain the connection between the deterministic and stochastic frameworks to deal with switching. In fact, we pose an analogy to the multiple Lyapunov functions approach. It requires us to formulate the infinitesimal generator of the systems under investigation, which plays a basic role for capturing the stochastic dynamics. Then, we derive the stability conditions by using some fundamental convergence theorem. The remainder of the paper is organized as follows. In Section 2 we describe the system under consideration and formulate the problem. In Section 3, we present the stability conditions and interpret the construction of the switching signal that obeys Poisson distribution. In Section 4, a simulation example is worked out to illustrate the results. Finally, the paper is briefly summarized in Section 5. 2. Problem description Consider the linear switching stochastic system described as follows dx(t ) = Aσ (t ) x(t )dt + Gσ (t ) x(t )dw(t ),
t ≥ t0 = 0,
(1)
where x(t ) ∈ R is state vector and w(t ) stands for the normalized one-dimensional Wiener Process, which is defined on the filtered complete probability space {Ω , F , Ft , P }. We suppose {Ft }t ≥0 to satisfy the usual conditions, namely, each Ft is P -complete and Ft = ∩s>t Fs for every t. Besides, write E for the mathematical expectation operator. System (1) is generated by the switching signal σ (t ) that orchestrates among the following subsystems n
dx(t ) = Ai x(t )dt + Gi x(t )dw(t ),
i ∈ {1, . . . , N }.
According to the evolution of switching signal over time, it can be expanded into the following sequential form
{(σ (t0 ), t0 ), (σ (t1 ), t1 ), . . . , (σ (tk ), tk ), . . .}.
(2)
It means that the σ (tk )th subsystem is activated during the interval [tk , tk+1 ). We now suppose the switching signal in (2) to obey Poisson distribution and to be independent of w(t ). That is, for any ∆ > 0 we have
P {tk+1 − tk ≥ ∆} = e−λ∆ ,
k = 0, 1, 2, . . . ,
(3)
where λ > 0 is referred to as Poisson exponent. Therefore, the switching points t0 < t1 < · · · < tk < · · · turn out to constitute a sequence of stopping times tending to infinity. Equivalently, it reads that
P {σ (t + ∆) ̸= σ (t )} = 1 − e−λ∆ .
(4)
As is well known, the Poisson exponent λ defines the mean value of the number of the switching points distributed within the time interval of unit length. Then it allows us to make a sense of the varying rate of switching signal. Accordingly, we categorize switching signals by means of λ. Namely, we write Sλ for the set of all the switching signals obeying Poisson distribution exactly with the exponent λ. For given switching signal σ , we denote by x(t ; x0 , σ ) the corresponding motion of system (1) at time t starting from x0 at initial time t0 . Definition 1. System (1) is said to be almost surely exponentially stable, if
P lim sup t →∞
ln |x(t ; x0 , σ )| t
<0 =1
for all x0 ∈ Rn . 3. Main results We aim at establishing the conditions that guarantee system (1) to be almost surely exponentially stable with respect to certain Sλ . The starting point is to formulate the action of the infinitesimal generator for it corresponds to the one-parameter semigroup of operators, which in turn determine the state-transition. In particular, the infinitesimal generator plays a fundamental role for addressing the stochastic stability problem due to the fact that, in general, we are not be able to express the state-transition in an analytical closed form (cf. [12]). For the sake of generality, we shall first consider the switching stochastic differential system as follows dx(t ) = fσ (t ) (x(t ))dt + gσ (t ) (x(t ))dw(t ),
(5)
where the switching signal obeys the Poisson distribution as in (4). We suppose the drift coefficients fi (x) and the diffusion coefficients gi (x) to satisfy certain conditions so that we can guarantee the solution of (5) to exist on [0, ∞). To each subsystem dx(t ) = fi (x(t ))dt + gi (x(t ))dw(t ), assign a positive-definite function Vi (x), which has the partial derivatives
110
S. Cong / Nonlinear Analysis: Hybrid Systems 22 (2016) 108–115
at least to second order in x. Let L be the infinitesimal generator corresponding to the stochastic differential system in (5). For V (x(t ), σ (t ) = i) = Vi (x(t )), we compute the action of L on it as 1
LV (x(t ), σ (t ) = i) = lim
E [V (x(t + ∆), σ (t + ∆))|x(t ), σ (t ) = i] − V (x(t ), σ (t ) = i)
∆ (1 − e−λ∆ ) [V (x(t + ∆), σ (t + ∆) ̸= i) − V (x(t ), σ (t ) = i)] = lim ∆ ↓0 ∆ ∆ ↓0
+ lim
e−λ∆
∆
∆ ↓0
[V (x(t + ∆), σ (t + ∆) = i) − V (x(t ), σ (t ) = i)]
2 ∂ Vi (x(t )) 1 ′ ∂ Vi (x(t )) = λ Vj (x(t )) − Vi (x(t )) + fi (x(t )) + gi (x(t )) gi (x(t )), j ̸= i. (6) ∂x 2 ∂ xp ∂ xq n×n ′ ∂V ∂2V ∂U ∂2U ∂V ∂ Vi For the composed function Ui (x(t )) = ln Vi (x(t )), noting that ∂ xi = V1 ∂ xi and ∂ x2i = V1 ∂ x2i − 12 ∂ xi , we compute its ∂x V i i
i
differential along the solution of system (5) as d ln V (x(t ), σ (t ) = i) = L ln V (x(t ), σ (t ) = i)dt 1 ∂ V (x(t ), σ (t ))
+
=
V (x(t ), σ (t ))
σ (t )=i
dw(t )
2 ∂ Vi (x(t )) 1 ∂ Vi (x(t )) fi (x(t )) + gi′ (x(t )) gi (x(t )) Vi (x(t )) Vi (x(t )) ∂x 2 ∂ xp ∂ xq n×n 2 1 ∂ Vi (x(t )) ∂ Vi (x(t )) 1 − 2 gi (x(t )) gi (x(t ))dw(t ), j ̸= i. dt + ∂ x V ( x ( t )) ∂x 2Vi (x(t )) i λ ln
Vj (x(t ))
∂x
gσ (t ) (x(t ))
1
+
(7)
We now are in the position to prove the main result. Theorem 1. Given λ > 0. If there exist a family of positive-definite matrices Pi and a family of numbers χi > 0 and γi such that for all i, j ∈ {1, . . . , N } the following inequalities hold
1 (Ai − γi Gi )′ Pi + Pi (Ai − γi Gi ) + G′i Pi Gi + λ ln χi + γi2 Pi < 0,
(8)
Pj ≤ χi Pi ,
(9)
2
j ̸= i,
then system (1) is almost surely exponentially stable for all switching signals belonging to Sλ . Proof. For system (1), we construct the Lyapunov function as V (x(t ), σ (t )) = Vσ (t ) (x(t ))
(10)
with Vi (x) = x Pi x, i ∈ {1, . . . , N }. Firstly, from (9) it is immediately seen that ′
ln
Vj (x) Vi (x)
≤ ln χi ,
j ̸= i.
Noting this and using (7), we know that the differential of ln V (x(t ), σ (t )) along the solution of system (1) satisfies d ln V (x(t ), σ (t )) ≤
x′ (t )(A′σ (t ) Pσ (t ) + Pσ (t ) Aσ (t ) + G′σ (t ) Pσ (t ) Gσ (t ) )x(t ) dt x′ (t )Pσ (t ) x(t )
+ λ ln χσ (t ) dt − +
1
x′ (t )(G′σ (t ) Pσ (t ) + Pσ (t ) Gσ (t ) )x(t ) x′ (t )Pσ (t ) x(t )
2
x′ (t )(G′σ (t ) Pσ (t ) + Pσ (t ) Gσ (t ) )x(t ) x′ (t )Pσ (t ) x(t )
2 dt
dw(t ).
(11)
In particular, we have
−
1 2
x′ (t )(G′σ (t ) Pσ (t ) + Pσ (t ) Gσ (t ) )x(t ) x′ (t )Pσ (t ) x(t )
2 =−
1
x′ (t )(G′σ (t ) Pσ (t ) + Pσ (t ) Gσ (t ) − γσ (t ) Pσ (t ) )x(t )
x′ (t )Pσ (t ) x(t ) ′ x (t )(Gσ (t ) Pσ (t ) + Pσ (t ) Gσ (t ) )x(t ) 1 + γσ2(t ) − γσ (t ) x′ (t )Pσ (t ) x(t ) 2 ′ ′ x (t )(Gσ (t ) Pσ (t ) + Pσ (t ) Gσ (t ) )x(t ) 1 −γσ (t ) + γσ2(t ) . x′ (t )Pσ (t ) x(t ) 2
2
2
′
≤
(12)
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For any t ≥ t0 , we then have ln V (x(t ), σ (t )) ≤ ln V (x(t0 ), σ (t0 )) t ′ x (s)(A′σ (s) Pσ (s) + Pσ (s) Aσ (s) + G′σ (s) Pσ (s) Gσ (s) + (λ ln χσ (s) + 21 γσ2(s) )Pσ (s) )x(s) + ds x′ (s)Pσ (s) x(s) t0
t
γσ (s)
−
x′ (s)(G′σ (s) Pσ (s) + Pσ (s) Gσ (s) )x(s) x′ (s)Pσ (s) x(s)
t0
t
ds +
x′ (s)(G′σ (s) Pσ (s) + Pσ (s) Gσ (s) )x(s) x′ (s)Pσ (s) x(s)
t0
dw(s).
(13)
Since the right-hand side of (8) is negative-definite, there must be a sufficiently small positive number ϵ such that
1 (Ai − γi Gi )′ Pi + Pi (Ai − γi Gi ) + G′i Pi Gi + λ ln χi + γi2 Pi < −ϵ Pi 2
holds true uniformly for all i ∈ {1, . . . , N }. Taking this into account, from (13) we arrive at ln V (x(t ), σ (t )) < ln V (x(t0 ), σ (t0 )) − ϵ t + M (t ),
(14)
where M (t ) :=
t
x′ (s)(G′σ (s) Pσ (s) + Pσ (s) Gσ (s) )x(s) x′ (s)Pσ (s) x(s)
t0
dw(s).
It can be seen that M (t ) is a continuous local martingale with respect to {Ft }t ≥t0 and vanishes at t0 . Moreover, the quadratic variation of M (t ) (for the definition of quadratic variation please refer to p. 12 in [12]) satisfies
t ′ x (s)(G′σ (s) Pσ (s) + Pσ (s) Gσ (s) )x(s) 2 ds ⟨M (t ), M (t )⟩ = x′ (s)Pσ (s) x(s) t0 ≤ ρ 2 (t − t0 ).
(15)
where
ρ := max
max{|λmin (G′i Pi + Pi Gi )|, |λmax (G′i Pi + Pi Gi )|}
λmin (Pi )
i=1,...,N
.
Then, by the strong law of large numbers (cf. [12]), there is probability 1 that lim
M (t )
t →∞
t
= 0.
From (14) it then follows that, with probability 1, lim sup t →∞
ln |x(t )| t
=
1 2
lim sup
The proof is thus completed.
ln V (x(t ), σ (t )) t
t →∞
=−
ϵ 2
< 0.
Remark 1. A discussion on the parameters γi and χi , which should be properly chosen so that the derived stability conditions are feasible, would be helpful for us to make a sense of the stability mechanism. As shown in (12), the effect of the part of the system polluted by the environment noise on stability is partly represented through γi . At the same time, χi renders the intensity parameter λ available to represent the role that the switching signal plays in preserving stability. Therefore, adjusting χi and γi to test the feasibility of the stability conditions brings an insight into the way of the randomly varying switching signal and the environment noise to influence stability. We now turn to take a look at the construction of the switching signals that obey Poisson distribution. Given a switching signal σ (t ), let
Π = {t0 = 0, t1 , . . . , tk , . . . : lim tk = ∞} k→∞
be the collection of its switching points. Meanwhile, let N ([s1 , s2 ]) be counter of the switching points of σ (t ) distributed within the interval [s1 , s2 ]. We then have
N ([t0 , s]) = sup m :
m
(tk+1 − tk ) ≤ s .
k=0
That is to say, the number of switching points is depending on the length of the interval of time and, therefore, is invariant for the shift transformation. As we know, the Poisson exponent λ indicates the density of the switching points distributed within an interval of unit length. That is, fixing an interval of time [s1 , s2 ], one has that
E {N ([s1 , s2 ])} = λ(s2 − s1 ).
(16)
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S. Cong / Nonlinear Analysis: Hybrid Systems 22 (2016) 108–115
At the same time, merging switching signals provides an alternative way to observe the linear relationship between the intensity parameter λ and the density of switching points. To be precise, let Π1 and Π2 be the collections of the switching points of σ1 (t ) and σ2 (t ), respectively. The switching signal, which is denoted by σ1 (t ) ⊕ σ2 (t ), is generated by merging σ1 (t ) and σ2 (t ) in such a way that its switching points correspond to the set Π1 ∪ Π2 ; please refer to [11] for more details. Assume σ1 (t ) ∈ Sλ1 , σ2 (t ) ∈ Sλ2 , and they are independent of each other, then
σ1 (t ) ⊕ σ2 (t ) ∈ Sλ1 +λ2 .
(17)
Next, we consider the special case that only noise-free subsystems are involved. Namely, the system under consideration is x˙ (t ) = Aσ (t ) x(t ),
t ≥ t0 = 0,
(18)
which is generated by a switching signal obeying the Poisson distribution in (4) and orchestrating among several noise-free linear subsystems. Corollary 1. Given λ > 0. If there exist a family of numbers χi and βi , and a family of positive-definite matrices Pi such that A′i Pi + Pi Ai − βi Pi ≤ 0,
(19)
Pj ≤ χi Pi ,
(20)
j ̸= i,
and
λ ln χi + βi < 0.
(21)
then system (18) is almost surely exponentially stable for all switching signals belonging to Sλ . Obviously, Corollary 1 can be deduced directly from Theorem 1. However, assume there exist χ and β such that (19)–(21) hold true uniformly for all subsystems, namely, they become A′i Pi + Pi Ai − β Pi ≤ 0,
(22)
Pj ≤ χ Pi ,
(23)
j ̸= i, λ ln χ + β < 0,
(24)
respectively. Then, we can provide an alternative proof by using the law of large numbers of Poisson processes. Proof. Construct a Lyapunov function as V (x(t ), σ (t ) = i) = x′ (t )Pi x(t ).
(25)
For any t ≥ t0 , let N ([t0 , t ]) be equal to an integer, say, k. At the same time, let the following sequence t0 < t1 < t2 < · · · < tk ≤ t denote the corresponding switching points. Indeed, (22) implies that β(ti −ti−1 ) V (x(ti ), σ (ti+ V (x(ti−1 ), σ (ti+ −1 )) ≤ e −1 )),
1 ≤ i ≤ k.
This together with (23) yields β(ti −ti−1 ) β(ti −ti−2 ) V (x(ti ), σ (ti+ V (x(ti−1 ), σ (ti+ V (x(ti−2 ), σ (ti+ −1 )) ≤ χ e −2 )) ≤ χ e −2 )).
Iterating the preceding inequalities from tk to t0 , we obtain V (x(t ), σ (tk+ )) ≤ χ k eβ(t −t0 ) V (x(t0 ), σ (t0 ))
= exp[N ([t0 , t ]) ln χ + β(t − t0 )]V (x(t0 ), σ (t0 )). It then follows that lim sup
ln |x(t ; x0 , σ )| t
t →∞
≤
1 2
lim
t →∞
N ([t0 , t ]) ln χ + β(t − t0 ) t
.
Furthermore, by the law of large numbers of Poisson processes (cf. [13]), there is probability 1 that lim
t →∞
N ([t0 , t ]) t
= λ.
Hence, in view of (24), we arrive at
P lim sup t →∞
ln |x(t ; x0 , σ )| t
The proof is thus completed.
≤
1 2
(λ ln χ + β) < 0 = 1.
(26)
S. Cong / Nonlinear Analysis: Hybrid Systems 22 (2016) 108–115
113
In parallel with (24), the following inequality
λ(χ − 1) + β < 0
(27)
can guarantee the system to be almost surely asymptotically stable as well provided the conditions in (22) and (23) remain true; please refer to [5]. Though the inequality in (27) is more conservative than that in (24) for there must be χ > 1, we now prove that it makes exp[N ([t0 , t ]) ln χ + β(t − t0 )] become a continuous-time supermartingale. In fact, for any ∆ > 0, by the independent increments property of Poisson processes we can calculate that
E exp[N ([t0 , t + ∆]) ln χ]N ([t0 , t ])
= E exp[N ([t0 , t ]) ln χ + (N ([t0 , t + ∆]) − N ([t0 , t ])) ln χ]N ([t0 , t ]) = exp[N ([t0 , t ]) ln χ]E exp[(N ([t0 , t + ∆]) − N ([t0 , t ])) ln χ] = exp[N ([t0 , t ]) ln χ]E χ N ([t0 ,t +∆])−N ([t0 ,t ]) ∞ (λ∆)k = exp[N ([t0 , t ]) ln χ] e−λ∆ χk k! k=0 = exp[N ([t0 , t ]) ln χ + λ(χ − 1)∆]. Multiplying both sides of the above equality by exp[β(t + ∆ − t0 )] and recalling (27) results in
E exp[N ([t0 , t + ∆]) ln χ + β(t + ∆ − t0 )]N ([t0 , t ])
= exp[N ([t0 , t ]) ln χ + λ(χ − 1)∆ + β(t + ∆ − t0 )] = exp[(λ(χ − 1) + β)∆] exp[N ([t0 , t ]) ln χ + β(t − t0 )] < exp[N ([t0 , t ]) ln χ + β(t − t0 )]. It implies that the expression on the right-hand side of (26) constitutes a continuous-time supermartingale. Therefore, one can show the system in (18) is almost surely asymptotically stable via some typical approach; see, e.g., [14]. 4. An illustrative example Consider three linear unstable subsystems as x˙ (t ) =
−0.35 0.5
−0.28 0.32
0.6 ˙ (t ) −0.4 x(t ), x
=
0.31 −0.5
0.74 ˙ (t ) −0.12 x(t ), and x
=
0.37 −0.3 x(t ). In order to demonstrate the effectiveness of the results, we shall show that it is possible to achieve
stabilization by simultaneously introducing environment noise and modulating switching signal. As a matter of fact, for the following stochastic subsystems dx(t ) = dx(t ) =
−0.28 0.32
0.31 −0.5
0.6 0.46 x(t )dt + −0.4 0.25
0.74 0.3 x(t )dt + −0.12 0
0.43 x(t )dw(t ), 0.5
0 x(t )dw(t ), 0.24
and dx(t ) =
−0.35 0.5
0.37 0.5 x(t )dt + −0.3 0.28
0.47 x(t )dw(t ), 0.56
by fixing λ = 1.0 and setting χ1 = 1.15, χ2 = 0.87, χ3 = 1.15 and γ1 = 0.84, γ2 = 0.34, γ3 = 0.9, we find the stability conditions in Theorem 1 feasible. The sample path of the switching signal is shown in Fig. 1, which triggers the change of subsystems by its edges. The corresponding state-response is depicted in Fig. 2. A simulation exposes that the sample paths of the solutions of both the second and third stochastically perturbed subsystems diverge to infinity. This reveals that introducing Wiener processes alone may be not enough to achieve stabilization, yet letting it work together with Poisson processes can make the overall system exponentially stable. Generally speaking, stability can be achieved if the switching signal is properly modulated so as to retain balance among the stable subsystems and the unstable ones. 5. Conclusion We considered a class of linear switching stochastic systems. By supposing the evolution of switching signal over time to obey Poisson distribution, we set this kind of systems entirely in a stochastic framework and formulated the action of the corresponding infinitesimal generator. Therefore, we characterized the joint effects of Poisson processes and Wiener processes on the sample paths of solution to such a system and derived the almost sure stability conditions. We provided an example to illustrate the theoretical results.
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S. Cong / Nonlinear Analysis: Hybrid Systems 22 (2016) 108–115
Fig. 1. The switching signal corresponding to λ = 1.0.
Fig. 2. The state-response corresponding to Poisson switching signal.
Acknowledgments The author would like to thank the anonymous reviewers for their very helpful comments and suggestions. The work is supported by the National Natural Science Foundation of China (Grant No. 61573131) and the Natural Science Foundation of Heilongjiang Province of China (Grant No. LC201428). References [1] X. Feng, K.A. Loparo, Y. Ji, H.J. Chizeck, Stochastic stability properites of jump linear systems, IEEE Trans. Automat. Control 37 (1) (1992) 38–53. [2] P. Bolzern, P. Colaneri, G. De Nicolao, On almost sure stability of continuous-time Markov jump linear systems, Automatica 42 (6) (2006) 983–988. [3] P. Bolzern, P. Colaneri, G. De Nicolao, Markov jump linear systems with switching transition rates: mean square stability with dwell-time, Automatica 46 (6) (2010) 1081–1088. [4] M. Tanelli, P. Bolzern, P. Colaneri, Almost sure stabilization of uncertain continuous-time Markov jump linear systems, IEEE Trans. Automat. Control 55 (1) (2010) 195–201. [5] A. Cetinkaya, K. Kashima, T. Hayakawa, Stability of stochastic systems with probabilistic mode switchings and state jumps, in: Proc. 2010 American Control Conference, Baltimore, June 30–July 02, 2010, pp. 4046–4051. [6] D. Chatterjee, D. Liberzon, Stability analysis of deterministic and stochastic switched systems via a comprison principle and multiple Lyapunov functions, SIAM J. Control Optim. 45 (1) (2006) 174–206. [7] D. Chatterjee, D. Liberzon, Stabilizing randomly switched systems, SIAM J. Control Optim. 49 (5) (2011) 2008–2031. [8] F. Deng, Q. Luo, X. Mao, S. Pang, Noise suppresses and expresses exponential growth, Systems Control Lett. 57 (3) (2008) 262–270. [9] G. Hu, M. Liu, X. Mao, M. Song, Noise suppresses exponential growth under regime switching, J. Math. Anal. Appl. 355 (2) (2009) 783–795.
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