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Stochastic stabilization and destabilization of nonlinear differential equations
Systems & Control Letters 62 (2013) 163–169
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Stochastic stabilization and destabilization of nonlinear differential equations Lirong Huang ∗ Automatic Control Laboratory, D-ITET, ETH Zurich, Switzerland
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Article history: Received 6 May 2011 Accepted 13 November 2012 Available online 29 December 2012
This paper is concerned with the stochastic stabilization and destabilization of nonlinear differential equations, which, more precisely, further develops the general theory of stochastic stabilization and destabilization that was studied by Mao (1994) [4] and Appleby et al. (2008) [1]. The proposed results are of substantial improvements and reveal the more fundamental principle for stochastic stabilization and destabilization of dynamical systems, which is also verified with examples of application to massaction systems. © 2012 Elsevier B.V. All rights reserved.
Keywords: Almost sure stability Brownian motion Nonlinear dynamical systems Stochastic destabilization Stochastic stabilization Mass-action kinetics
1. Introduction
is almost surely stable (resp., unstable) provided that f satisfies a global linear bound of the form
It is well known that noise can not only be used to destabilize a given stable system but also be used to stabilize a given unstable system or to make a system even more stable. The literature on stabilization and destabilization by noise is extensive (see [1–4] and the reference therein). Stabilization by deterministic periodic ‘‘noise’’ was investigated a great deal in the references, e.g., [5–7]. But the study in the case of random noise, viz, stochastic stabilization, was initiated by Has’minskii who employed two white noise sources to stabilize a system [3, p. 229]. Later, the problems of stabilization and destabilization by random noise were studied in many works, e.g., [8,9,4,10,2,11,1] and the references therein. Particularly, Mao [4] proposed a general theory on stabilization and destabilization by Brownian motion. In [4], it was found that a general n-dimensional ordinary differential equation (ODE)
|f (x)| ≤ K0 |x|, x ∈ Rn (3) for some K0 > 0. Although, under some assumptions, (3) can be
x˙ (t ) = f (x(t )),
t > 0;
x(0) = x0 ∈ Rn
(1)
could be stabilized (resp., destabilized) by an m-dimensional Brownian motion w(t ) = [w1 (t ) . . . wm (t )]T . More precisely, matrices Bk ∈ Rn×n , k = 1, . . . , m, can be found such that the equilibrium solution of the stochastic differential equation (SDE) dx(t ) = f (x(t ))dt +
m
Bk x(t )dwk (t ),
t > 0;
k=1
x(0) = x0 ∈ Rn
∗
Tel.: +41 44 633 8281. E-mail addresses: [email protected], [email protected].
0167-6911/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2012.11.008
(2)
weakened to a one-sided growth condition (see, e.g., [12]), many nonlinear systems with equilibrium at 0 are still excluded. Recently, these results have been significantly improved by Appleby et al. [1], which includes a much more general class of nonlinear dynamical systems. Suppose that f and g in SDE (7) satisfy conditions (8) and (9). It has been shown in [1] that a function g can be appropriately chosen such that the equilibrium solution of SDE (7) is (i) almost surely stable provided that there exists α ∈ (0, 1) such that
|x|2 (2xT f (x) + |g (x)|2F ) − (2 − α)|xT g (x)|2 ≤ 0
(4)
and g (L) := min |xT g (x)| > 0 |x|=L
(5)
for every L > 0; (ii) almost surely unstable provided that there exists α ∈ (0, 1) such that
|x|2 (2xT f (x) + |g (x)|2F ) − (2 + α)|xT g (x)|2 ≥ 0.
(6)
This work aims to further develop the general theory of stochastic stabilization and destabilization of nonlinear dynamical systems that was studied in [4,1]. The proposed results exploit the system structure and therefore are more applicable, e.g., to massaction dynamical systems (see, e.g., Examples 5.1 and 5.2). As will be shown, our proposed results reveal the fundamental principle for stochastic stabilization and destabilization of dynamical systems (see Remarks 3.1 and 4.1).
164
L. Huang / Systems & Control Letters 62 (2013) 163–169
2. Preliminaries Throughout this paper, unless otherwise specified, we shall employ the following notation. Let (Ω , F , {Ft }t ≥0 , P) be a complete probability space with a filtration {Ft }t ≥0 satisfying the usual conditions (i.e. it is right continuous and F0 contains all Pnull sets) and E[·] be the expectation operator with respect to the probability measure. Let w(t ) = (w1 (t ), . . . , wm (t ))T be an m-dimensional Brownian motion defined on the probability space. If x, y are real numbers, then x ∨ y denotes the maximum of x and y, and x ∧ y stands for the minimum of x and y. Let | · | denote the Euclidean norm of a vector and its induced norm of a matrix while | · |F the Frobenius norm of a matrix. Let C (Rn ; R+ ) the family of nonnegative continuous functions µ : Rn → R+ and CB (Rn ; R) be the class of upper-bounded continuous functions µ : Rn → R, i.e., there is a constant Hµ for µ ∈ CB (Rn ; R) such that µ(x) ≤ Hµ for all x ∈ Rn . Denote by CK (Rn ; R+ ) the family of nonnegative continuous functions µ : Rn → R+ such that µ(x) > 0 for all x ̸= 0, 0 ≤ µ(0) < ∞, and, moreover, lim inf|x|→∞ µ(x) = ∞ if µ(0) = 0. It is observed that µ ∈ CK (Rn ; R+ ) if µ(x) = c0 +c1 µ1 (x) for all x ∈ Rn with c0 ≥ 0, c1 ≥ 0, c0 + c1 > 0 and a pair of functions ν1 and ν2 of K∞ class such that ν1 (|x|) ≤ µ1 (x) ≤ ν2 (|x|) for all x ∈ Rn , where K∞ is a family of functions λ ∈ K with λ(r ) → ∞ as r → ∞ while K denotes the class of continuous strictly increasing functions λ from R+ to R+ with λ(0) = 0. Let us consider the following n-dimensional SDE dx(t ) = f (x(t ))dt + g (x(t ))dw(t )
(7)
Lemma 2.1 ([14, Theorem 7, p. 139]). Let A(t ) and U (t ) be two continuous adapted increasing processes on t ≥ 0 with A(0) = U (0) = 0 a.s. Let M (t ) be a real-valued continuous local martingale with M (0) = 0 a.s. Let X0 be a nonnegative F0 -measurable random variable such that EX0 < ∞. Define X (t ) = X0 + A(t ) − U (t ) + M (t ) for all t ≥ 0. If X (t ) is nonnegative, then
lim A(t ) < ∞ ⊂
t →∞
lim U (t ) < ∞ ∩
t →∞
lim X (t ) < ∞
t →∞
a.s.
3. Stochastic stabilization of nonlinear systems In this section, we wish to give conditions on function g such that solutions of SDE (7) will tend to zero almost surely. Before we discuss and prove the asymptotic stability, let us consider the existence and uniqueness of global solutions of SDE (7). Assume that the following condition about f and g is satisfied: Assumption 3.1. There exists a function λ ∈ CB (Rn ; R) such that 1 4
|x|
|x|2 (2xT f (x) + |g (x)|2F ) − 2|xT g (x)|2 ≤ λ(x).
(14)
So we have the following result on the solutions of SDE (7). Proposition 3.1. Suppose that Assumption 3.1 holds. Then there exists a unique continuous adapted process x, which is a solution of (7) such that τe = θ0 = ∞ a.s.
(8)
The proof and application of this result are given in Appendices A and B, respectively. This statement says that the solution does not explode, but also may not end up at the equilibrium 0. Let us proceed to consider the asymptotic stability of SDE (7) with the following assumption.
and satisfy the local Lipschitz continuous condition, that is, for any integer k ≥ 1, there is Lk > 0 such that
Assumption 3.2. There exist functions µ ∈ CK (Rn ; R+ ) and ν ∈ C (Rn ; R+ ) such that
on t ≥ 0 with initial data x(0) = x0 ∈ R \ {0}, where both f : Rn → Rn and g : Rn → Rn×m obey n
f (0) = 0
and g (0) = 0
|f (x) − f (y)| ∨ |g (x) − g (y)| ≤ Lk (|x − y|)
(9)
for all (x, y) ∈ R × R with |x|∨|y| ≤ k. Obviously, both (1) and (7) admit the equilibrium solution x(t ) = 0 for all t ≥ 0 when initial condition x0 = 0. Therefore, the noise perturbation preserves the equilibrium of the system (1). By virtue of the local Lipschitz continuous condition (9), we know that there is a unique continuous adapted process x (see, e.g., [12,13]) such that n
x(t ∧ τk ) = x0 +
n
t ∧τk
f (x(s))ds +
0
t ∧τk
4
|x|
(10)
where τk = inf{t > 0 : |x(t ; x0 )| ≥ k}. We set inf ∅ = ∞ as usual. Eq. (7) has a global solution if the explosion time τe defined by k→∞
(11)
obeys τe = ∞ a.s. In this work, it is important to show that solutions to Eq. (7) cannot reach zero in finite time, i.e., P ({θ0 < ∞}) = 0, where the stopping time θ0 is defined by x
θ0 = θ0 0 = inf{t > 0 : |x(t ; x0 )| = 0}.
(15)
(16)
and inf
µ(x) >0 ν(x)
(17)
for every β > 0.
g (x(s))dw(s),
0
τe = τex0 = lim τk = inf{t > 0 : |x(t ; x0 )| ̸∈ [0, ∞)}
|x|2 (2xT f (x) + |g (x)|2F ) − 2|xT g (x)|2 ≤ −µ(x),
|xT g (x)|2 ≤ ν(x) |x|4
|x|≥β
t ≥ 0, a.s.
(12)
According to the existing results (see, e.g., [1,13]), we have
τe ≤ θ0 and θ0 = ∞ a.s.
1
(13)
The purpose of our work is to further develop this theory of stochastic stabilization and destabilization of nonlinear systems so that it reveals the more fundamental principle and hence are more applicable. Let us begin with the following well-known lemma.
With Assumption 3.2, we are guaranteed the almost sure asymptotic stability of the solution of SDE (7). Theorem 3.1. Suppose that f and g obey Assumption 3.2. Then there exists a unique continuous adapted process x, which is a global solution of (7) and which obeys lim x(t ; x0 ) = 0 a.s.
t →∞
Proof. By Proposition 3.1, we know that SDE (7) has a unique global continuous solution denoted by x(t ) = x(t ; x0 ) and hence |x(t )| is well-defined and continuous for all t ≥ 0. We decompose the sample space into three mutually exclusive events as follows:
ω : lim sup |x(t )| ≥ lim inf |x(t )| > 0 , t →∞ t →∞ E2 = ω : lim sup |x(t )| > 0 and lim inf |x(t )| = 0 ,
E1 =
t →∞
t →∞
E3 = ω : lim |x(t )| = 0 . t →∞
(18)
L. Huang / Systems & Control Letters 62 (2013) 163–169
Applying Itô’s formula, we have d[|x(t )| ] = (2x f (x(t )) + |g (x(t ))| )dt 2
T
2 F
+ 2xT (t )g (x(t ))dw(t ).
(19)
Since θ0 = ∞ a.s., we may apply the Itô formula to the positive process |x|2 to obtain that log |x(t )|2 = log |x0 |2 +
t
1
≤ log |x0 |2 −
t
µ(x(s))ds + M (t )
(20)
p(x) = |x| (2x f (x) + |g (x)| ) − 2|x g (x)| M (t ) =
t
2 F
2x (s)g (x(s)) T
|x(s)|2
0
T
2
and
dw(s).
By (16), we have
⟨M (t )⟩ =
t
0
|xT (s)g (x(s))|2 ds ≤ |x(s)|4
t
ν(x(s))ds,
(21)
0
where ⟨M (t )⟩ is the quadratic variation of the continuous local martingale M (t ). Alternatively, we decompose the sample space into the following two mutually exclusive events:
A1 =
ω:
∞
µ(x(s))ds < ∞
and
0
A2 =
ω:
∞
µ(x(s))ds = ∞ .
(22)
0
First, we show that A1 ⊂ E3 a.s. (that is, P(A1 ∩ E3c ) = 0 with E3c = Ω \ E3 ), which is obvious if P(A1 ) = 0. Suppose P(A1 ) > 0. Since µ ∈ CK (Rn ; R+ ), we immediately have P(A1 ∩ E1 ) = 0. Moreover, µ(0) = 0, lim inf|x|→∞ µ(x) = ∞ and hence sup0≤t <∞ |x(t )| < ∞ a.s. on A1 . We claim
P(A1 ∩ E2 ) = 0,
(23)
which is shown in Appendix C with the techniques proposed in [15] (see also [16]). But this implies A1 ⊂ E3 a.s. Next, we need to show A2 ⊂ E3 a.s. when P(A2 ) > 0. Obviously, A2 = A21 ∪ A22 and A21 ∩ A22 = ∅, where A21 = {ω ∈ A2 : limt →∞ ⟨M (t )⟩ < ∞} and A22 = {ω ∈ A2 : limt →∞ ⟨M (t )⟩ = ∞}. ¯ (t ) + ⟨M (t )⟩ is nonnegative and M ¯ (t ) = M 2 (t ) − Since M 2 (t ) = M ⟨M (t )⟩ is a continuous local martingale, by Lemma 2.1, we have limt →∞ M 2 (t ) < ∞ and, therefore, limt →∞ M (t ) exists and is finite a.s. on A21 . But this yields log |x(t )|2 lim sup t ≤ −1 t →∞ µ(x(s))ds 0
a.s.
and hence limt →∞ |x(t )| = 0 a.s. on A21 . So A21 ⊂ E3 a.s. Now we proceed to prove A22 ⊂ E3 a.s. This will be true if we show that
P(A22 ∩ E1 ) = P(A22 ∩ E2 ) = 0.
(24)
lim inf t →∞
0
µ(x(s))ds > 0 a.s. ⟨M (t )⟩
(25)
and, by the strong law of large numbers, therefore lim sup t →∞
log |x(t )|2
⟨M (t )⟩
≥ ε0 ,
(27)
j = 1, 2, 3, . . .
[(ξ2j ∧ t ) − (ξ2j−1 ∧ t )] j≥1 Aε = ω ∈ Aσ : lim inf >0 , t →∞ t
(28)
ξ1 = 0, ξ2j = inf{t ≥ ξ2j−1 : |x(t )| ≤ ε}, ξ2j+1 = inf{t ≥ ξ2j : |x(t )| > ε}, j = 1, 2, 3, . . . . Observe that P(Aε ) → P(Aσ ) ≥ ε0 > 0 as ε → 0 since Aε ⊆ Aσ , P(Aσ ) ≥ ε0 > 0, θ0 = ∞ a.s. and x(t ) is continuous on t ≥ 0 a.s.. So there exists ε > 0 sufficiently small for P(Aε ) > 0. But, by (17), (21) and (29), we have lim sup t →∞
log |x(t )|2
⟨M (t )⟩
< 0 a.s. on Aε
(30)
and hence limt →∞ |x(t )| = 0 a.s. on Aε , which contradicts Aε ⊆ Aσ ⊆ A22 ∩ E2 . So we must have P(A22 ∩ E2 ) = 0. The proof is complete. Remark 3.1. The inequalities (15)–(17) exploit the system structure and yield more general results. It can be observed that Theorem 8 in [1] is a special case of Theorem 3.1 when µ(x)/ν(x) > c > 0 for all x ∈ Rn while Theorem 3.4 in [4] can be considered as a version with µ(x) = a > 0 and b2 ≥ ν(x) ≥ b1 > 0 for all x ∈ Rn . 4. Stochastic destabilization of nonlinear systems In this section, we wish to give conditions on g such that the solutions of (7) tend to explosions with probability one, that is, lim supt →τe |x(t )| = ∞ a.s. Let us consider the following conditions. Assumption 4.1. There exist functions µ ∈ CK (Rn ; R+ ) and ν ∈ C (Rn ; R+ ) such that 1 4
| x|
|x|2 (2xT f (x) + |g (x)|2F ) − 2|xT g (x)|2 ≥ µ(x),
(31)
(32)
and inf
(26)
(29)
where {ξj }j≥1 is a sequence of stopping times defined by
|x|≤β
< 0 a.s.,
σ2j+1 = inf{t ≥ σ2j : |x(t )| ≥ 2ε1 },
|xT g (x)|2 ≤ ν(x) |x|4
By inequalities (17) and (21), on A22 ∩ E1 , we have
t
and Z is a set of natural numbers that includes infinitely many elements. Obviously, Aσ := ω ∈ A22 : σj < ∞, j ∈ Z ⊆ A22 ∩ E2 and there are infinitely many even numbers in Z . Given any ε ∈ [0, ε1 ∧ |x0 |), we define Aε ⊆ Aσ by
for t ≥ 0, where T
P ω ∈ A22 : σj < ∞, j ∈ Z
σ1 = inf{t ≥ 0 : |x(t )| ≥ 2ε1 }, σ2j = inf{t ≥ σ2j−1 : |x(t )| ≤ ε1 },
0
2
which implies limt →∞ |x(t )| = 0 a.s. on A22 ∩ E1 . Obviously, this is a contradiction. So we must have P(A22 ∩ E1 ) = 0. Suppose that P(A22 ∩ E2 ) > 0. Then there exist constants ε0 > 0 and ε1 > 0 such that
where {σj }j≥1 is a sequence of stopping times defined by p(x(s))ds + M (t ) 4
|x(s)|
0
165
µ(x) >0 ν(x)
for every β > 0.
(33)
166
L. Huang / Systems & Control Letters 62 (2013) 163–169
Theorem 4.1. Suppose that f and g obey Assumption 4.1. Let x be the unique continuous adapted process which is a solution of (7) on [0, τe ), where τe is defined by (11). Then, x obeys lim sup |x(t ; x0 )| = ∞ a.s. t →τe
Proof. We know that a solution of SDE (7) is well-defined and continuous on t ∈ [0, τe ). We decompose the sample space into two mutually exclusive events as follows:
¯E1 = ω : lim sup |x(t )| < ∞ and t →τe E¯ 2 = ω : lim sup |x(t )| = ∞ .
(34)
By (20) and (31), we have log |x(t )|2 = log |x0 |2 +
t
|x(s)|4
0 2
1
p(x(s))ds + M (t )
µ(x(s))ds + M (t )
≥ log |x0 | +
k1
aX (a + b)X (35)
for t ≥ 0. By (32), we have t
0
|xT (s)g (x(s))|2 ds ≤ |x(s)|4
t
ν(x(s))ds.
(36)
0
Recall that events A1 and A2 are defined by (22). Clearly, we shall have the required result if we show A1 ⊂ E¯ 2 and A2 ⊂ E¯ 2 a.s. First, we show P(A1 ∩ E¯ 1 ) = 0, which gives A1 ⊂ E¯ 2 a.s. This is obvious when P(A1 ) = 0. Suppose P(A1 ) > 0. Similarly, since µ ∈ CK (Rn ; R+ ), we immediately have µ(0) = 0, lim inf|x|→∞ µ(x) = ∞, sup0≤t <∞ |x(t )| < ∞ a.s. on A1 and hence A1 ⊆ E¯ 1 a.s., which implies P(A1 ∩ E¯ 1 ) = P(A1 ) > 0. But, using the techniques presented in [15] (see Appendix C), we can prove limt →∞ |x(t )| = 0 a.s. on A1 . Given h > 0, define a subset Bh of A1 as
Bh =
ω ∈ A1 : sup |x(t )| ≤ h .
(37)
t ≥0
Since P(A1 ) > 0, there exists h > 0 sufficiently large for P(Bh ) > 0. By (33) and (36), limt →∞ ⟨M (t )⟩ < ∞ a.s. on Bh . By Lemma 2.1, this gives limt →∞ M 2 (t ) and hence limt →∞ M (t ) exists and is finite a.s. on Bh . But, by (35), we see lim inft →∞ log |x(t )|2 > −∞ a.s. on Bh , which contradicts Bh ⊂ A1 . So we must have P(A1 ) = 0. As above, we divide A2 into two disjoint parts A2 = A21 ∪ A22 . Since limt →∞ M (t ) exists and is finite a.s. on A21 , so, by (35), we have log |x(t )|
2
lim inf t t →∞
0
µ(x(s))ds
≥ 1 a.s.
and hence lim inft →∞ |x(t )| = ∞ a.s. on A21 . So A21 ⊂ E¯ 2 a.s. Suppose P(A22 ∩ E¯ 1 ) > 0. Then there exists sufficiently large h > 0 such that P(Ch ) > 0, where
¯ Ch = ω ∈ (A22 ∩ E1 ) : sup |x(t )| ≤ h .
(41)
k2
0
⟨M (t )⟩ =
In this section, application examples are given to verify that our proposed results reveal the more general principle and therefore are more applicable. Example 5.1. Let us consider the following one-species reversible mass-action system (see, e.g., [17,18])
t
Remark 4.1. The inequalities (31)–(33) give more applicable results by exploiting the system structure. As is observed, Theorem 12 in [1] is a special case of Theorem 4.1 when µ(x)/ν(x) > c > 0 for all x ∈ Rn while Theorem 4.3 in [4] can be considered as a version with µ(x) = a > 0 and b2 ≥ ν(x) ≥ b1 > 0 for all x ∈ Rn . 5. Examples
t →τe
which implies lim inft →∞ |x(t )| = ∞ a.s. on Ch . But this contradicts Ch ⊂ A22 ∩ E¯ 2 . So we must have P(A22 ∩ E¯ 1 ) = 0 and hence A22 ⊂ E¯ 2 a.s., which completes the proof.
(38)
with positive initial condition, where a and b are positive integers while k1 and k2 are positive real numbers. The population of the species is described by the scalar ODE (see, e.g., [17]) x˙ (t ) = xa (t ) k1 − k2 xb (t ) .
(42)
According to [17, Theorem 6.4], mass-action system (41) is permanent, that is, there is a positive constant c such that the solution of ODE (42) obeys c < inf x(t ) ≤ sup x(t ) < t ≥0
t ≥0
1 c
.
It is easy to see that a special case of (42) with a = b = 1 is the well-known logistic population model (see, e.g., [19]). It is well known that the behavior of a system may change significantly if environmental noise is taken into account (see, e.g., [20,21]). For example, let w(t ) be a scalar standard Brownian motion. Then, ∀β ∈ R, there exists a unique continuous positive solution x(t ) = x(t ; x0 ) to SDE dx(t ) = xa (t ) k1 − k2 xb (t ) dt + β x
a+1 2
(t )dw(t )
(43)
on t ≥ 0 with x(0) = x0 > 0, which has the property (13) with τe = ∞ a.s. It is noticed the results in [22,23] do not apply to the cases of (43) with a > 1. For any pair of positive integers a and b, the solution of SDE (43) obeys limt →∞ x(t ) = 0 a.s. if, by (4),
β 2 > 2k1 ,
(44)
which means the population of stochastic mass-action model (43) will become extinct with probability one. Note that the results in [1] do not apply to system (43) with β 2 = 2k1 . But, by Theorem 3.1, it is easy to see that the solution of stochastic population system (43) will become extinct asymptotically almost surely, i.e., limt →∞ x(t ) = 0 a.s., if
β 2 ≥ 2k1 .
(45)
t ≥0
Example 5.2. Let us consider a 3-dimensional version of the logistic population models as follows:
By (33) and (36), on Ch , we have
t lim inf t →∞
0
µ(x(s))ds > 0 a.s. ⟨M (t )⟩
lim inf t →∞
log |x(t )|
⟨M (t )⟩
> 0 a.s.,
x˙ 2 (t ) = x2 (t ) a2 − b21 x1 (t ) − b22 x2 (t ) − b23 x3 (t )
x˙ 3 (t ) = x3 (t ) a3 − b31 x1 (t ) − b32 x2 (t ) − b33 x3 (t )
(39)
and, by the strong law of large numbers, therefore 2
x˙ 1 (t ) = x1 (t ) a1 − b11 x1 (t ) − b12 x2 (t ) − b13 x3 (t )
(40)
(46)
on t ≥ 0 with initial data x1 (0) > 0, x2 (0) > 0 and x3 (0) > 0. Assume that ai and bij , 1 ≤ i, j ≤ 3, are positive constants such
L. Huang / Systems & Control Letters 62 (2013) 163–169
condition, that is, there are nonnegative constants α , κ and γ such that
that ai >
bij aj
(47)
bjj
j̸=i
for 1 ≤ i ≤ 3, see, e.g., ai = bii = 3c for 1 ≤ i ≤ 3 and bij = c for i ̸= j with c > 0. According to the results in [19], we see 0 < inf xi (t ) ≤ sup xi (t ) < ∞ t ≥0
for 1 ≤ i ≤ 3. Clearly, system (45) is a specific case of (1) with
x=
x1 (a1 − b11 x1 − b12 x2 − b13 x3 ) x2 (a2 − b21 x1 − b22 x2 − b23 x3 ) . x3 (a3 − b31 x1 − b32 x2 − b33 x3 )
and f (x) =
xT f (x) ≤ κ|x|α+2 + γ |x|2 .
(53)
It is also assumed that f satisfies conditions (8) and (9). This problem was investigated in [24] with two Brownian motion sources w1 (t ) and w2 (t ) in the form of dx(t ) = f (x(t ))dt + qx(t )dw1 (t ) + σ |x(t )|β x(t )dw2 (t ).
t ≥0
x1 x2 x3
167
According to [24, Theorem 4.1], the solution of SDE (54) is almost surely stable if
2β > α,
Can this population system be destabilized by noise so that its solution goes to explosion with probability one, i.e., lim supt →τe |x(t )| = ∞ a.s.? It is noticed that the results in [1] are not prepared for such a purpose though [1, Theorem 12] states that, for an appropriate choice of g, the solution of SDE (7) obeys lim inft →τe |x(t )| > 0 a.s. Now let us turn to Theorem 4.1 with a design method of g derived from the examples in [4,1]. Let γ : R3 → R+ be a function as follows:
3 3 6 x2 b x ij j i=1 i j=1 γ (x) = . 3 2
(48)
xi
i =1
σ ̸= 0 and
σ2 > max − y2β + κ yα + γ . (55)
q2 2
2
y≥0
But this can be easily obtained from Theorem 3.1. It is observed that system (54) is indeed a special case of SDE (7) with g (x) = [qx σ |x|β x] and w(t ) = [w1 (t ) w2 (t )]T . Obviously, g is locally Lipschitz continuous with
|g (x)|2F = q2 |x|2 + σ 2 |x|2β+2 and
(56)
|xT g (x)|2 = q2 |x|4 + σ 2 |x|2β+4 . By inequalities (53) and (56), we have
1 2 |x| (2xT f (x) + |g (x)|2F ) − 2|xT g (x)|2 |x|4 ≤ 2κ|x|α + 2γ + q2 + σ 2 |x|2β − 2(q2 + σ 2 |x|2β ) = 2κ|x|α + 2γ − (q2 + σ 2 |x|2β ) =: −µ(x).
So we have
γ 2 (x)|x|2 + 6xT f (x) ≥ 6
ai x2i ≥ 6a|x|2 ,
x ∈ R3
(49)
i =1
with a = min1≤i≤3 ai > 0. Using the convention that x = [x1 x2 · · · xn ]T and xn+1 = x1 with n = 3 in this case, define g : R3 → R3×3 by g (x) = γ (x) diag{x2 , x3 , x1 }. Then g is locally Lipschitz continuous with
|g (x)|2F = γ 2 (x)|x|2 and |xT g (x)|2 = γ 2 (x)
3
x2i x2i+1
(57) 2β
Note that |x g (x)| /|x| = q + σ |x| = ν(x). It is observed that Assumption 3.2 holds if one of the following conditions is satisfied: T
3
(54)
2
4
2
2
(i) 2β > α > 0, σ 2 > 0 and q2 > maxy≥0 −σ 2 y2β + 2κ yα + 2γ ; (ii) 2β = α > 0, σ 2 > 2κ and q2 ≥ 2γ ; (iii) 2β = α = 0 and σ 2 + q2 > 2(k + γ ).
Therefore, by Theorem 3.1, the solution of (54) is almost surely stable in these cases. Alternatively, it is easy to observe that the sufficient condition (55) can be obtained by applying the existing result (4)–(5), i.e., Theorem 8 in [1] to system (54).
(50) 6. Conclusion
i=1 3
for all x ∈ R . Note that
|x|4 =
3 2
x2i
i=1
3
=
3
x4i + 2
i=1
3
x2i x2i+1 ≤ 3
i =1
x4i .
(51)
i =1
Substitution of (49)–(51) into (31) gives 1
|x|4 =
|x|2 (2xT f (x) + |g (x)|2F ) − 2|xT g (x)|2 1
|x|4
2|x|2 xT f (x) + γ 2 (x) |x|4 − 2
≥
1 1 2|x|2 xT f (x) + γ 2 (x)|x|4 4 |x| 3
=
1 T 6x f (x) + γ 2 (x)|x|2 ≥ 2a, 2 3|x|
3
x2i x2i+1
This paper has presented new results on stochastic stabilization and destabilization of nonlinear differential equations. It has been shown (see Remarks 3.1 and 4.1) and illustrated with examples that our proposed results reveal the more fundamental principle and are more applicable. It is noticed that the theorems in [24] can be easily obtained by applying our results or those in [1]. Recently, the techniques proposed in [24] have been extended to study stochastic stabilization of functional differential equations (see [25] and also [26]) x˙ (t ) = f (xt )
i=1
(58)
by introducing two Brownian motions in the form of dx(t ) = f (xt )dt + qx(t )dw1 (t ) + σ |x(t )|β x(t )dw2 (t ), (52)
which implies Assumption 4.1 holds. By Theorem 4.1, it follows the desired result lim supt →τe |x(t )| = ∞ a.s.
(59)
where f satisfies the local Lipschitz condition and, moreover, there are some nonnegative constants α , κ , κ¯ , γ and a probability measure µ on [−τ , 0] such that
|f (φ)| = |φ(0)| κ|φ(0)|α + κ¯
0
|φ(θ )|α dµ(θ ) + γ
(60)
−τ
Example 5.3. Let us study the problem of stochastic stabilization of ODE (1) with the so-called one-sided polynomial growth
for any (φ, t ) ∈ C ([−τ , 0]; Rn ) × R+ (see Assumptions 1.2 and 4.1 in [25]). Obviously, (58) is a special class of functional differential
168
L. Huang / Systems & Control Letters 62 (2013) 163–169
equations where its right-hand side vanishes whenever the current state is zero. Therefore, it shares some properties with differential equations (1), e.g., the current state φ(0) plays a dominant role in the system dynamics which can be verified, say, in the proof of θ0 = ∞a.s. if x(0) ̸= 0. It appears that the techniques proposed in [1] and our results could be also extended to such a class of functional differential equations.
(i) 2β > α > 0 and σ ̸= 0; (ii) 2β = α > 0 and σ 2 ≥ 2κ ; (iii) 2β ≥ α = 0 and σ ∈ R. So, by Proposition 3.1, there exists a unique continuous adapted process x, which is a solution of (63) such that τe = θ0 = ∞a.s. Appendix C. Proof of claim (23)
Acknowledgments The author gratefully acknowledges the reviewer’s comments. The author would like to thank Prof. G. Craciun for his lectures at ETH Zurich and thank Prof. H. Koeppel for the discussion, which helped with the Example 5.1 of mass-action kinetics.
We prove P(A1 ∩ E2 ) = 0 by contradiction. Suppose that P(A1 ∩ E2 ) > 0. There exist α0 > 0 and α1 > 0 such that P ω ∈ A1 : there are infinitely many j such that ρj < ∞ = P ω ∈ A1 : σj < ∞, j ∈ Z¯ ≥ α0 , (65)
Appendix A. Proof of Proposition 3.1
where {ρj }j≥1 are a sequence of stopping times defined by
By (13), we only need to prove τe = ∞ a.s. Since θ0 = ∞ a.s., by Itô’s formula, we obtain
ρ1 = inf{t ≥ 0 : µ(x(t )) ≥ 2α1 }, ρ2j = inf{t ≥ σ2j−1 : µ(x(t )) ≤ α1 }, ρ2j+1 = inf{t ≥ σ2j : µ(x(t )) ≥ 2α1 },
log |x(t )|2 = log |x0 |2 +
t
|x(s)|
4
0
≤ log |x0 |2 +
1
p(x(s))ds + M (t )
t
λ(x(s))ds + M (t )
(61)
0
on t ≥ 0, where p(x) = |x|2 (2xT f (x) + |g (x)|2F ) − 2|xT g (x)|2 M (t ) =
t
2xT (s)g (x(s))
0
|x(s)|
2
for all |x| ≤ k. Let τ (t ) = (ρj + t ) ∧ τk for t ≥ 0 and IA be the indicator of set A. For any j ∈ Z¯ , by Hölder’s inequality and Doob’s martingale inequality (see, e.g., [13]), we compute
E log |x(t ∧ τk )| ≤ E log |x0 | + E 2
t ∧τk
λ(x(s))ds
Appendix B. An application example of Proposition 3.1 Over the past few years, there have been many works dedicated to the study that environmental noise can be used to suppress explosions (in finite time) of differential equations (see, e.g., [27,22,24] and the references therein). For example, in [24], a scalar Brownian motion w(t ) is introduced into ODE (1) satisfying conditions (8), (9) and (53) in the form of dx(t ) = f (x(t ))dt + σ |x(t )| x(t )dw(t ),
(63)
which guarantees the existence and uniqueness of its global solutions. As an example of applications, we apply Proposition 3.1 to this case and figure out the sufficient conditions for existence and uniqueness of solutions to (63). Obviously, in system (63), g (x) = σ |x|β x is locally Lipschitz continuous with
|g (x)|2F = σ 2 |x|2β+2 and |xT g (x)|2 = σ 2 |x|2β+4 . Therefore, we have 1 2 |x| (2xT f (x) + |g (x)|2F ) − 2|xT g (x)|2 4 |x| 2β
=: λ(x).
j 2 τ j (t ) τ k (t ) k = E IA1 ∩{ρj <τk } sup g (x(s))dB(s) f (x(s))ds + 0≤t ≤T ρj σj j 2 τ k (t ) ≤ 2E IA1 ∩{ρj <τk } sup f (x(s))ds 0≤t ≤T ρj j 2 τ k (t ) + 2E IA1 ∩{ρj <τk } sup g (x(s))dB(s) 0≤t ≤T ρj j
τ k (T )
≤ 2E IA1 ∩{ρj <τk }
ρj
+ 8E IA1 ∩{ρj <τk }
|f (x(s))|2 ds
j
τ k (T ) ρj
|g (x(s))| ds 2
≤ 2Kk2 T (T + 4),
(67)
where T is some positive constant. Since µ(·) is continuous in Rn , it must be uniformly continuous in the closed ball S¯k = {x ∈ Rn : |x| ≤ k}. For any given b > 0, we can choose cb > 0 such that |µ(x) − µ(y)| < b whenever x, y ∈ S¯k and |x − y| < cb . Let us choose
ε=
α0 3
,
k ≥ kε
and b = α1 .
(68)
By (67) and Chebyshev’s inequality (see, e.g., [13]), we have
2β
≤ 2κ|x| + 2γ + σ |x| − 2σ |x| = 2κ|x|α + 2γ − σ 2 |x|2β 2
0≤t ≤T
≤ E log |x0 |2 + Hλ t =: H¯ t . (62) ¯ t is independent of k and log |x(τk )|2 = 2 log k. So Note that H ¯t. 2 log kP({τk ≤ t }) ≤ E log |x(t ∧ τk )|2 ≤ H Letting k → ∞ and then t → ∞, we obtain P({τe < ∞}) = 0, that is, τe = ∞a.s., which completes the proof.
β
j
E IA1 ∩{ρj <τk } sup |x(τk (t )) − x(ρj )|2
0
2
(66) j k
But this yields
α
and Z¯ is a set of natural numbers that includes infinitely many elements. Since x(t ) and hence µ(x(t )) are continuous on t ≥ 0, we see that ρj → ∞ a.s. as j → ∞. By local Lipschitz condition (9), for any given k > 0, there exists Kk > 0 such that
|f (x)| ∨ |g (x)| ≤ Kk
and
dw(s).
2
j = 1, 2, 3, . . . ;
P ω ∈ A1 : τk ≤ ρj + P ω ∈ A1 : ρj < τk and (64)
It is easy to see that λ ∈ CB (Rn ; R) and hence Assumption 3.1 holds if one of the following conditions is satisfied:
sup |µ(x(ρj + t )) − µ(x(ρj ))| ≥ α1
0≤t ≤T
≤ P ω ∈ A1 : τk ≤ ρj + P ω ∈ A1 : ρj < τk ≤ ρj + T
L. Huang / Systems & Control Letters 62 (2013) 163–169
+ P ω ∈ A1 : ρj + T < τk and
References
sup |µ(x(ρj + t )) − µ(x(ρj ))| ≥ α1 0 ≤t ≤T
≤ P ω ∈ A1 : τk ≤ ρj + T + P ω ∈ A1 : ρj + T < τk and sup x(ρj + t ) − x(ρj ) ≥ cα1 0 ≤t ≤T
≤
2Kk2 T (T + 4) cα21
+ (1 − 2ε) .
(69)
We furthermore choose T = T (ε, α1 , k) > 0 sufficiently small for 2Kk2 T (T + 4) cα21
≤ ε.
(70)
Inequalities (69) and (70) yield
P ω ∈ A1 : ρj < τk and sup |µ(x(ρj + t )) − µ(x(ρj ))| < α1
≥ ε.
(71)
0≤t ≤T
According to (65), j − 1 ∈ Z¯ whenever j ∈ Z¯ and j ≥ 2, which implies there are infinitely many even numbers in Z¯ . By (22) and (71), we have ∞
∞ > E IA1 0
≥
≥
≥
µ(x(t ))dt ρ2j
E IA1 ∩{ρ2j−1 <τk }
2j∈Z¯
ρ2j−1
µ(x(t ))dt
α1 E IA1 ∩{ρ2j−1 <τk } (ρ2j − ρ2j−1 )
2j∈Z¯
T α1 P ω ∈ A1 : ρ2j−1 < τk and
2j∈Z¯
sup |µ(x(ρ2j−1 + t )) − µ(x(ρ2j−1 ))| < α1
0 ≤t ≤T
≥
2j∈Z¯
T α1 ε =
1 3
T α0 α1 = ∞,
169
(72)
2j∈Z¯
which is a contradiction. So we must have P(A1 ∩ E2 ) = 0. The proof is complete.
[1] J.A.D. Appleby, X. Mao, A. Rodkina, Stabilization and destabilization of nonlinear differential equations by noise, IEEE Trans. Automat. Control 53 (2008) 683–691. [2] T. Caraballo, M. Garrido-Atienza, J. Real, Stochastic stabilisation of differential systems with general decay rate, Systems Control Lett. 48 (2003) 397–406. [3] R.Z. Has’minskii, Stochastic Stability of Differential Equations, Sithoff & Noordhoff, Alphen aan den Rijn, Netherlands, 1980. [4] X. Mao, Stochastic stabilisation and destabilisation, Systems Control Lett. 23 (1994) 279–290. [5] R. Bellman, J. Bentsman, S. Meerkov, Stability of fast periodic systems, IEEE Trans. Automat. Control 30 (1985) 289–291. [6] H.J. Kushner, On the stability of processes defined by stochastic differencedifferential equations, J. Differ. Equ. 4 (1968) 424–443. [7] S. Meerkov, Condition of vibrational stabilizability of a class of non-linear systems, IEEE Trans. Automat. Control 27 (1982) 485–487. [8] L. Arnold, H. Crauel, V. Wihstutz, Stabilization of linear systems by noise, SIAM J. Control Optim. 21 (1983) 451–461. [9] M. Scheutzow, Stabilization and destabilization by noise in the plane, Stoch. Anal. Appl. 11 (1993) 97–113. [10] X. Mao, Stability of stochastic differential equations with Markovian switching, Stochastic Process. Appl. 79 (1999) 45–67. [11] R.Z. Khasminskii, C. Zhu, G. Yin, Stability of regime-switching diffusions, Stochastic Process. Appl. 117 (2007) 1037–1051. [12] X. Mao, G.G. Yin, C. Yuan, Stabilisation and destabilisation of hybrid systems of stochastic differential equations, Automatica 43 (2007) 264–273. [13] X. Mao, Stochastic Differential Equations and Applications, second ed., Horwood Publishing, Chichester, UK, 2007. [14] R.Sh. Lipster, A.N. Shiryayev, Theory of Martingales, Kluwer Academic, Dordrecht, Netherlands, 1989. [15] L. Huang, X. Mao, On almost sure stability of hybrid stochastic systems with mode-dependent interval delays, IEEE Trans. Automat. Control 55 (2010) 1946–1952. [16] X. Mao, A note on the LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl. 268 (2002) 125–142. [17] G. Craciun, F. Nazarov, C. Pantea, Persistence and permanence of mass-action and power-law dynamical systems, SIAM J. Appl. Math. (in press). [18] D.J. Wilkinson, Stochastic Modelling for Systems Biology, Chapman & Hall/CRC, London, UK, 2006. [19] S. Ahmad, A.C. Lazer, Average conditions for global asymptotic stability in a nonautonomous Lotka–Volterra system, Nonlinear Anal. 40 (2000) 37–49. [20] X. Mao, G. Marion, E. Renshaw, Environmental noise suppresses explosion in population dynamics, Stoch. Process. Appl. 97 (2002) 95–110. [21] X. Mao, S. Sabanis, E. Renshaw, Asymptotic behavior of the stochastic Lotka–Volterra model, J. Math. Anal. Appl. 287 (2003) 141–156. [22] Q. Luo, X. Mao, Stochastic population dynamics under regime switching II, J. Math. Anal. Appl. 355 (2009) 577–593. [23] X. Li, A. Gray, D. Jiang, X. Mao, Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, J. Math. Anal. Appl. 376 (2011) 11–28. [24] F. Wu, S. Hu, Suppression and stabilisation of noise, Int. J. Control 82 (2009) 2150–2157. [25] F. Wu, X. Mao, S. Hu, Stochastic suppression and stabilization of functional differential equations, Systems & Control Letters 59 (2010) 745–753. [26] F. Wu, S. Hu, Stochastic suppression and stabilization of delay differential systems, Int. J. Robust Nonlinear Control 21 (2010) 488–500. [27] F. Deng, Q. Luo, X. Mao, S. Peng, Noise suppresses or expresses exponential growth, Systems & Control Letters 57 (2008) 262–270.
Contents lists available at SciVerse ScienceDirect
Systems & Control Letters journal homepage: www.elsevier.com/locate/sysconle
Stochastic stabilization and destabilization of nonlinear differential equations Lirong Huang ∗ Automatic Control Laboratory, D-ITET, ETH Zurich, Switzerland
article
abstract
info
Article history: Received 6 May 2011 Accepted 13 November 2012 Available online 29 December 2012
This paper is concerned with the stochastic stabilization and destabilization of nonlinear differential equations, which, more precisely, further develops the general theory of stochastic stabilization and destabilization that was studied by Mao (1994) [4] and Appleby et al. (2008) [1]. The proposed results are of substantial improvements and reveal the more fundamental principle for stochastic stabilization and destabilization of dynamical systems, which is also verified with examples of application to massaction systems. © 2012 Elsevier B.V. All rights reserved.
Keywords: Almost sure stability Brownian motion Nonlinear dynamical systems Stochastic destabilization Stochastic stabilization Mass-action kinetics
1. Introduction
is almost surely stable (resp., unstable) provided that f satisfies a global linear bound of the form
It is well known that noise can not only be used to destabilize a given stable system but also be used to stabilize a given unstable system or to make a system even more stable. The literature on stabilization and destabilization by noise is extensive (see [1–4] and the reference therein). Stabilization by deterministic periodic ‘‘noise’’ was investigated a great deal in the references, e.g., [5–7]. But the study in the case of random noise, viz, stochastic stabilization, was initiated by Has’minskii who employed two white noise sources to stabilize a system [3, p. 229]. Later, the problems of stabilization and destabilization by random noise were studied in many works, e.g., [8,9,4,10,2,11,1] and the references therein. Particularly, Mao [4] proposed a general theory on stabilization and destabilization by Brownian motion. In [4], it was found that a general n-dimensional ordinary differential equation (ODE)
|f (x)| ≤ K0 |x|, x ∈ Rn (3) for some K0 > 0. Although, under some assumptions, (3) can be
x˙ (t ) = f (x(t )),
t > 0;
x(0) = x0 ∈ Rn
(1)
could be stabilized (resp., destabilized) by an m-dimensional Brownian motion w(t ) = [w1 (t ) . . . wm (t )]T . More precisely, matrices Bk ∈ Rn×n , k = 1, . . . , m, can be found such that the equilibrium solution of the stochastic differential equation (SDE) dx(t ) = f (x(t ))dt +
m
Bk x(t )dwk (t ),
t > 0;
k=1
x(0) = x0 ∈ Rn
∗
Tel.: +41 44 633 8281. E-mail addresses: [email protected], [email protected].
0167-6911/$ – see front matter © 2012 Elsevier B.V. All rights reserved. doi:10.1016/j.sysconle.2012.11.008
(2)
weakened to a one-sided growth condition (see, e.g., [12]), many nonlinear systems with equilibrium at 0 are still excluded. Recently, these results have been significantly improved by Appleby et al. [1], which includes a much more general class of nonlinear dynamical systems. Suppose that f and g in SDE (7) satisfy conditions (8) and (9). It has been shown in [1] that a function g can be appropriately chosen such that the equilibrium solution of SDE (7) is (i) almost surely stable provided that there exists α ∈ (0, 1) such that
|x|2 (2xT f (x) + |g (x)|2F ) − (2 − α)|xT g (x)|2 ≤ 0
(4)
and g (L) := min |xT g (x)| > 0 |x|=L
(5)
for every L > 0; (ii) almost surely unstable provided that there exists α ∈ (0, 1) such that
|x|2 (2xT f (x) + |g (x)|2F ) − (2 + α)|xT g (x)|2 ≥ 0.
(6)
This work aims to further develop the general theory of stochastic stabilization and destabilization of nonlinear dynamical systems that was studied in [4,1]. The proposed results exploit the system structure and therefore are more applicable, e.g., to massaction dynamical systems (see, e.g., Examples 5.1 and 5.2). As will be shown, our proposed results reveal the fundamental principle for stochastic stabilization and destabilization of dynamical systems (see Remarks 3.1 and 4.1).
164
L. Huang / Systems & Control Letters 62 (2013) 163–169
2. Preliminaries Throughout this paper, unless otherwise specified, we shall employ the following notation. Let (Ω , F , {Ft }t ≥0 , P) be a complete probability space with a filtration {Ft }t ≥0 satisfying the usual conditions (i.e. it is right continuous and F0 contains all Pnull sets) and E[·] be the expectation operator with respect to the probability measure. Let w(t ) = (w1 (t ), . . . , wm (t ))T be an m-dimensional Brownian motion defined on the probability space. If x, y are real numbers, then x ∨ y denotes the maximum of x and y, and x ∧ y stands for the minimum of x and y. Let | · | denote the Euclidean norm of a vector and its induced norm of a matrix while | · |F the Frobenius norm of a matrix. Let C (Rn ; R+ ) the family of nonnegative continuous functions µ : Rn → R+ and CB (Rn ; R) be the class of upper-bounded continuous functions µ : Rn → R, i.e., there is a constant Hµ for µ ∈ CB (Rn ; R) such that µ(x) ≤ Hµ for all x ∈ Rn . Denote by CK (Rn ; R+ ) the family of nonnegative continuous functions µ : Rn → R+ such that µ(x) > 0 for all x ̸= 0, 0 ≤ µ(0) < ∞, and, moreover, lim inf|x|→∞ µ(x) = ∞ if µ(0) = 0. It is observed that µ ∈ CK (Rn ; R+ ) if µ(x) = c0 +c1 µ1 (x) for all x ∈ Rn with c0 ≥ 0, c1 ≥ 0, c0 + c1 > 0 and a pair of functions ν1 and ν2 of K∞ class such that ν1 (|x|) ≤ µ1 (x) ≤ ν2 (|x|) for all x ∈ Rn , where K∞ is a family of functions λ ∈ K with λ(r ) → ∞ as r → ∞ while K denotes the class of continuous strictly increasing functions λ from R+ to R+ with λ(0) = 0. Let us consider the following n-dimensional SDE dx(t ) = f (x(t ))dt + g (x(t ))dw(t )
(7)
Lemma 2.1 ([14, Theorem 7, p. 139]). Let A(t ) and U (t ) be two continuous adapted increasing processes on t ≥ 0 with A(0) = U (0) = 0 a.s. Let M (t ) be a real-valued continuous local martingale with M (0) = 0 a.s. Let X0 be a nonnegative F0 -measurable random variable such that EX0 < ∞. Define X (t ) = X0 + A(t ) − U (t ) + M (t ) for all t ≥ 0. If X (t ) is nonnegative, then
lim A(t ) < ∞ ⊂
t →∞
lim U (t ) < ∞ ∩
t →∞
lim X (t ) < ∞
t →∞
a.s.
3. Stochastic stabilization of nonlinear systems In this section, we wish to give conditions on function g such that solutions of SDE (7) will tend to zero almost surely. Before we discuss and prove the asymptotic stability, let us consider the existence and uniqueness of global solutions of SDE (7). Assume that the following condition about f and g is satisfied: Assumption 3.1. There exists a function λ ∈ CB (Rn ; R) such that 1 4
|x|
|x|2 (2xT f (x) + |g (x)|2F ) − 2|xT g (x)|2 ≤ λ(x).
(14)
So we have the following result on the solutions of SDE (7). Proposition 3.1. Suppose that Assumption 3.1 holds. Then there exists a unique continuous adapted process x, which is a solution of (7) such that τe = θ0 = ∞ a.s.
(8)
The proof and application of this result are given in Appendices A and B, respectively. This statement says that the solution does not explode, but also may not end up at the equilibrium 0. Let us proceed to consider the asymptotic stability of SDE (7) with the following assumption.
and satisfy the local Lipschitz continuous condition, that is, for any integer k ≥ 1, there is Lk > 0 such that
Assumption 3.2. There exist functions µ ∈ CK (Rn ; R+ ) and ν ∈ C (Rn ; R+ ) such that
on t ≥ 0 with initial data x(0) = x0 ∈ R \ {0}, where both f : Rn → Rn and g : Rn → Rn×m obey n
f (0) = 0
and g (0) = 0
|f (x) − f (y)| ∨ |g (x) − g (y)| ≤ Lk (|x − y|)
(9)
for all (x, y) ∈ R × R with |x|∨|y| ≤ k. Obviously, both (1) and (7) admit the equilibrium solution x(t ) = 0 for all t ≥ 0 when initial condition x0 = 0. Therefore, the noise perturbation preserves the equilibrium of the system (1). By virtue of the local Lipschitz continuous condition (9), we know that there is a unique continuous adapted process x (see, e.g., [12,13]) such that n
x(t ∧ τk ) = x0 +
n
t ∧τk
f (x(s))ds +
0
t ∧τk
4
|x|
(10)
where τk = inf{t > 0 : |x(t ; x0 )| ≥ k}. We set inf ∅ = ∞ as usual. Eq. (7) has a global solution if the explosion time τe defined by k→∞
(11)
obeys τe = ∞ a.s. In this work, it is important to show that solutions to Eq. (7) cannot reach zero in finite time, i.e., P ({θ0 < ∞}) = 0, where the stopping time θ0 is defined by x
θ0 = θ0 0 = inf{t > 0 : |x(t ; x0 )| = 0}.
(15)
(16)
and inf
µ(x) >0 ν(x)
(17)
for every β > 0.
g (x(s))dw(s),
0
τe = τex0 = lim τk = inf{t > 0 : |x(t ; x0 )| ̸∈ [0, ∞)}
|x|2 (2xT f (x) + |g (x)|2F ) − 2|xT g (x)|2 ≤ −µ(x),
|xT g (x)|2 ≤ ν(x) |x|4
|x|≥β
t ≥ 0, a.s.
(12)
According to the existing results (see, e.g., [1,13]), we have
τe ≤ θ0 and θ0 = ∞ a.s.
1
(13)
The purpose of our work is to further develop this theory of stochastic stabilization and destabilization of nonlinear systems so that it reveals the more fundamental principle and hence are more applicable. Let us begin with the following well-known lemma.
With Assumption 3.2, we are guaranteed the almost sure asymptotic stability of the solution of SDE (7). Theorem 3.1. Suppose that f and g obey Assumption 3.2. Then there exists a unique continuous adapted process x, which is a global solution of (7) and which obeys lim x(t ; x0 ) = 0 a.s.
t →∞
Proof. By Proposition 3.1, we know that SDE (7) has a unique global continuous solution denoted by x(t ) = x(t ; x0 ) and hence |x(t )| is well-defined and continuous for all t ≥ 0. We decompose the sample space into three mutually exclusive events as follows:
ω : lim sup |x(t )| ≥ lim inf |x(t )| > 0 , t →∞ t →∞ E2 = ω : lim sup |x(t )| > 0 and lim inf |x(t )| = 0 ,
E1 =
t →∞
t →∞
E3 = ω : lim |x(t )| = 0 . t →∞
(18)
L. Huang / Systems & Control Letters 62 (2013) 163–169
Applying Itô’s formula, we have d[|x(t )| ] = (2x f (x(t )) + |g (x(t ))| )dt 2
T
2 F
+ 2xT (t )g (x(t ))dw(t ).
(19)
Since θ0 = ∞ a.s., we may apply the Itô formula to the positive process |x|2 to obtain that log |x(t )|2 = log |x0 |2 +
t
1
≤ log |x0 |2 −
t
µ(x(s))ds + M (t )
(20)
p(x) = |x| (2x f (x) + |g (x)| ) − 2|x g (x)| M (t ) =
t
2 F
2x (s)g (x(s)) T
|x(s)|2
0
T
2
and
dw(s).
By (16), we have
⟨M (t )⟩ =
t
0
|xT (s)g (x(s))|2 ds ≤ |x(s)|4
t
ν(x(s))ds,
(21)
0
where ⟨M (t )⟩ is the quadratic variation of the continuous local martingale M (t ). Alternatively, we decompose the sample space into the following two mutually exclusive events:
A1 =
ω:
∞
µ(x(s))ds < ∞
and
0
A2 =
ω:
∞
µ(x(s))ds = ∞ .
(22)
0
First, we show that A1 ⊂ E3 a.s. (that is, P(A1 ∩ E3c ) = 0 with E3c = Ω \ E3 ), which is obvious if P(A1 ) = 0. Suppose P(A1 ) > 0. Since µ ∈ CK (Rn ; R+ ), we immediately have P(A1 ∩ E1 ) = 0. Moreover, µ(0) = 0, lim inf|x|→∞ µ(x) = ∞ and hence sup0≤t <∞ |x(t )| < ∞ a.s. on A1 . We claim
P(A1 ∩ E2 ) = 0,
(23)
which is shown in Appendix C with the techniques proposed in [15] (see also [16]). But this implies A1 ⊂ E3 a.s. Next, we need to show A2 ⊂ E3 a.s. when P(A2 ) > 0. Obviously, A2 = A21 ∪ A22 and A21 ∩ A22 = ∅, where A21 = {ω ∈ A2 : limt →∞ ⟨M (t )⟩ < ∞} and A22 = {ω ∈ A2 : limt →∞ ⟨M (t )⟩ = ∞}. ¯ (t ) + ⟨M (t )⟩ is nonnegative and M ¯ (t ) = M 2 (t ) − Since M 2 (t ) = M ⟨M (t )⟩ is a continuous local martingale, by Lemma 2.1, we have limt →∞ M 2 (t ) < ∞ and, therefore, limt →∞ M (t ) exists and is finite a.s. on A21 . But this yields log |x(t )|2 lim sup t ≤ −1 t →∞ µ(x(s))ds 0
a.s.
and hence limt →∞ |x(t )| = 0 a.s. on A21 . So A21 ⊂ E3 a.s. Now we proceed to prove A22 ⊂ E3 a.s. This will be true if we show that
P(A22 ∩ E1 ) = P(A22 ∩ E2 ) = 0.
(24)
lim inf t →∞
0
µ(x(s))ds > 0 a.s. ⟨M (t )⟩
(25)
and, by the strong law of large numbers, therefore lim sup t →∞
log |x(t )|2
⟨M (t )⟩
≥ ε0 ,
(27)
j = 1, 2, 3, . . .
[(ξ2j ∧ t ) − (ξ2j−1 ∧ t )] j≥1 Aε = ω ∈ Aσ : lim inf >0 , t →∞ t
(28)
ξ1 = 0, ξ2j = inf{t ≥ ξ2j−1 : |x(t )| ≤ ε}, ξ2j+1 = inf{t ≥ ξ2j : |x(t )| > ε}, j = 1, 2, 3, . . . . Observe that P(Aε ) → P(Aσ ) ≥ ε0 > 0 as ε → 0 since Aε ⊆ Aσ , P(Aσ ) ≥ ε0 > 0, θ0 = ∞ a.s. and x(t ) is continuous on t ≥ 0 a.s.. So there exists ε > 0 sufficiently small for P(Aε ) > 0. But, by (17), (21) and (29), we have lim sup t →∞
log |x(t )|2
⟨M (t )⟩
< 0 a.s. on Aε
(30)
and hence limt →∞ |x(t )| = 0 a.s. on Aε , which contradicts Aε ⊆ Aσ ⊆ A22 ∩ E2 . So we must have P(A22 ∩ E2 ) = 0. The proof is complete. Remark 3.1. The inequalities (15)–(17) exploit the system structure and yield more general results. It can be observed that Theorem 8 in [1] is a special case of Theorem 3.1 when µ(x)/ν(x) > c > 0 for all x ∈ Rn while Theorem 3.4 in [4] can be considered as a version with µ(x) = a > 0 and b2 ≥ ν(x) ≥ b1 > 0 for all x ∈ Rn . 4. Stochastic destabilization of nonlinear systems In this section, we wish to give conditions on g such that the solutions of (7) tend to explosions with probability one, that is, lim supt →τe |x(t )| = ∞ a.s. Let us consider the following conditions. Assumption 4.1. There exist functions µ ∈ CK (Rn ; R+ ) and ν ∈ C (Rn ; R+ ) such that 1 4
| x|
|x|2 (2xT f (x) + |g (x)|2F ) − 2|xT g (x)|2 ≥ µ(x),
(31)
(32)
and inf
(26)
(29)
where {ξj }j≥1 is a sequence of stopping times defined by
|x|≤β
< 0 a.s.,
σ2j+1 = inf{t ≥ σ2j : |x(t )| ≥ 2ε1 },
|xT g (x)|2 ≤ ν(x) |x|4
By inequalities (17) and (21), on A22 ∩ E1 , we have
t
and Z is a set of natural numbers that includes infinitely many elements. Obviously, Aσ := ω ∈ A22 : σj < ∞, j ∈ Z ⊆ A22 ∩ E2 and there are infinitely many even numbers in Z . Given any ε ∈ [0, ε1 ∧ |x0 |), we define Aε ⊆ Aσ by
for t ≥ 0, where T
P ω ∈ A22 : σj < ∞, j ∈ Z
σ1 = inf{t ≥ 0 : |x(t )| ≥ 2ε1 }, σ2j = inf{t ≥ σ2j−1 : |x(t )| ≤ ε1 },
0
2
which implies limt →∞ |x(t )| = 0 a.s. on A22 ∩ E1 . Obviously, this is a contradiction. So we must have P(A22 ∩ E1 ) = 0. Suppose that P(A22 ∩ E2 ) > 0. Then there exist constants ε0 > 0 and ε1 > 0 such that
where {σj }j≥1 is a sequence of stopping times defined by p(x(s))ds + M (t ) 4
|x(s)|
0
165
µ(x) >0 ν(x)
for every β > 0.
(33)
166
L. Huang / Systems & Control Letters 62 (2013) 163–169
Theorem 4.1. Suppose that f and g obey Assumption 4.1. Let x be the unique continuous adapted process which is a solution of (7) on [0, τe ), where τe is defined by (11). Then, x obeys lim sup |x(t ; x0 )| = ∞ a.s. t →τe
Proof. We know that a solution of SDE (7) is well-defined and continuous on t ∈ [0, τe ). We decompose the sample space into two mutually exclusive events as follows:
¯E1 = ω : lim sup |x(t )| < ∞ and t →τe E¯ 2 = ω : lim sup |x(t )| = ∞ .
(34)
By (20) and (31), we have log |x(t )|2 = log |x0 |2 +
t
|x(s)|4
0 2
1
p(x(s))ds + M (t )
µ(x(s))ds + M (t )
≥ log |x0 | +
k1
aX (a + b)X (35)
for t ≥ 0. By (32), we have t
0
|xT (s)g (x(s))|2 ds ≤ |x(s)|4
t
ν(x(s))ds.
(36)
0
Recall that events A1 and A2 are defined by (22). Clearly, we shall have the required result if we show A1 ⊂ E¯ 2 and A2 ⊂ E¯ 2 a.s. First, we show P(A1 ∩ E¯ 1 ) = 0, which gives A1 ⊂ E¯ 2 a.s. This is obvious when P(A1 ) = 0. Suppose P(A1 ) > 0. Similarly, since µ ∈ CK (Rn ; R+ ), we immediately have µ(0) = 0, lim inf|x|→∞ µ(x) = ∞, sup0≤t <∞ |x(t )| < ∞ a.s. on A1 and hence A1 ⊆ E¯ 1 a.s., which implies P(A1 ∩ E¯ 1 ) = P(A1 ) > 0. But, using the techniques presented in [15] (see Appendix C), we can prove limt →∞ |x(t )| = 0 a.s. on A1 . Given h > 0, define a subset Bh of A1 as
Bh =
ω ∈ A1 : sup |x(t )| ≤ h .
(37)
t ≥0
Since P(A1 ) > 0, there exists h > 0 sufficiently large for P(Bh ) > 0. By (33) and (36), limt →∞ ⟨M (t )⟩ < ∞ a.s. on Bh . By Lemma 2.1, this gives limt →∞ M 2 (t ) and hence limt →∞ M (t ) exists and is finite a.s. on Bh . But, by (35), we see lim inft →∞ log |x(t )|2 > −∞ a.s. on Bh , which contradicts Bh ⊂ A1 . So we must have P(A1 ) = 0. As above, we divide A2 into two disjoint parts A2 = A21 ∪ A22 . Since limt →∞ M (t ) exists and is finite a.s. on A21 , so, by (35), we have log |x(t )|
2
lim inf t t →∞
0
µ(x(s))ds
≥ 1 a.s.
and hence lim inft →∞ |x(t )| = ∞ a.s. on A21 . So A21 ⊂ E¯ 2 a.s. Suppose P(A22 ∩ E¯ 1 ) > 0. Then there exists sufficiently large h > 0 such that P(Ch ) > 0, where
¯ Ch = ω ∈ (A22 ∩ E1 ) : sup |x(t )| ≤ h .
(41)
k2
0
⟨M (t )⟩ =
In this section, application examples are given to verify that our proposed results reveal the more general principle and therefore are more applicable. Example 5.1. Let us consider the following one-species reversible mass-action system (see, e.g., [17,18])
t
Remark 4.1. The inequalities (31)–(33) give more applicable results by exploiting the system structure. As is observed, Theorem 12 in [1] is a special case of Theorem 4.1 when µ(x)/ν(x) > c > 0 for all x ∈ Rn while Theorem 4.3 in [4] can be considered as a version with µ(x) = a > 0 and b2 ≥ ν(x) ≥ b1 > 0 for all x ∈ Rn . 5. Examples
t →τe
which implies lim inft →∞ |x(t )| = ∞ a.s. on Ch . But this contradicts Ch ⊂ A22 ∩ E¯ 2 . So we must have P(A22 ∩ E¯ 1 ) = 0 and hence A22 ⊂ E¯ 2 a.s., which completes the proof.
(38)
with positive initial condition, where a and b are positive integers while k1 and k2 are positive real numbers. The population of the species is described by the scalar ODE (see, e.g., [17]) x˙ (t ) = xa (t ) k1 − k2 xb (t ) .
(42)
According to [17, Theorem 6.4], mass-action system (41) is permanent, that is, there is a positive constant c such that the solution of ODE (42) obeys c < inf x(t ) ≤ sup x(t ) < t ≥0
t ≥0
1 c
.
It is easy to see that a special case of (42) with a = b = 1 is the well-known logistic population model (see, e.g., [19]). It is well known that the behavior of a system may change significantly if environmental noise is taken into account (see, e.g., [20,21]). For example, let w(t ) be a scalar standard Brownian motion. Then, ∀β ∈ R, there exists a unique continuous positive solution x(t ) = x(t ; x0 ) to SDE dx(t ) = xa (t ) k1 − k2 xb (t ) dt + β x
a+1 2
(t )dw(t )
(43)
on t ≥ 0 with x(0) = x0 > 0, which has the property (13) with τe = ∞ a.s. It is noticed the results in [22,23] do not apply to the cases of (43) with a > 1. For any pair of positive integers a and b, the solution of SDE (43) obeys limt →∞ x(t ) = 0 a.s. if, by (4),
β 2 > 2k1 ,
(44)
which means the population of stochastic mass-action model (43) will become extinct with probability one. Note that the results in [1] do not apply to system (43) with β 2 = 2k1 . But, by Theorem 3.1, it is easy to see that the solution of stochastic population system (43) will become extinct asymptotically almost surely, i.e., limt →∞ x(t ) = 0 a.s., if
β 2 ≥ 2k1 .
(45)
t ≥0
Example 5.2. Let us consider a 3-dimensional version of the logistic population models as follows:
By (33) and (36), on Ch , we have
t lim inf t →∞
0
µ(x(s))ds > 0 a.s. ⟨M (t )⟩
lim inf t →∞
log |x(t )|
⟨M (t )⟩
> 0 a.s.,
x˙ 2 (t ) = x2 (t ) a2 − b21 x1 (t ) − b22 x2 (t ) − b23 x3 (t )
x˙ 3 (t ) = x3 (t ) a3 − b31 x1 (t ) − b32 x2 (t ) − b33 x3 (t )
(39)
and, by the strong law of large numbers, therefore 2
x˙ 1 (t ) = x1 (t ) a1 − b11 x1 (t ) − b12 x2 (t ) − b13 x3 (t )
(40)
(46)
on t ≥ 0 with initial data x1 (0) > 0, x2 (0) > 0 and x3 (0) > 0. Assume that ai and bij , 1 ≤ i, j ≤ 3, are positive constants such
L. Huang / Systems & Control Letters 62 (2013) 163–169
condition, that is, there are nonnegative constants α , κ and γ such that
that ai >
bij aj
(47)
bjj
j̸=i
for 1 ≤ i ≤ 3, see, e.g., ai = bii = 3c for 1 ≤ i ≤ 3 and bij = c for i ̸= j with c > 0. According to the results in [19], we see 0 < inf xi (t ) ≤ sup xi (t ) < ∞ t ≥0
for 1 ≤ i ≤ 3. Clearly, system (45) is a specific case of (1) with
x=
x1 (a1 − b11 x1 − b12 x2 − b13 x3 ) x2 (a2 − b21 x1 − b22 x2 − b23 x3 ) . x3 (a3 − b31 x1 − b32 x2 − b33 x3 )
and f (x) =
xT f (x) ≤ κ|x|α+2 + γ |x|2 .
(53)
It is also assumed that f satisfies conditions (8) and (9). This problem was investigated in [24] with two Brownian motion sources w1 (t ) and w2 (t ) in the form of dx(t ) = f (x(t ))dt + qx(t )dw1 (t ) + σ |x(t )|β x(t )dw2 (t ).
t ≥0
x1 x2 x3
167
According to [24, Theorem 4.1], the solution of SDE (54) is almost surely stable if
2β > α,
Can this population system be destabilized by noise so that its solution goes to explosion with probability one, i.e., lim supt →τe |x(t )| = ∞ a.s.? It is noticed that the results in [1] are not prepared for such a purpose though [1, Theorem 12] states that, for an appropriate choice of g, the solution of SDE (7) obeys lim inft →τe |x(t )| > 0 a.s. Now let us turn to Theorem 4.1 with a design method of g derived from the examples in [4,1]. Let γ : R3 → R+ be a function as follows:
3 3 6 x2 b x ij j i=1 i j=1 γ (x) = . 3 2
(48)
xi
i =1
σ ̸= 0 and
σ2 > max − y2β + κ yα + γ . (55)
q2 2
2
y≥0
But this can be easily obtained from Theorem 3.1. It is observed that system (54) is indeed a special case of SDE (7) with g (x) = [qx σ |x|β x] and w(t ) = [w1 (t ) w2 (t )]T . Obviously, g is locally Lipschitz continuous with
|g (x)|2F = q2 |x|2 + σ 2 |x|2β+2 and
(56)
|xT g (x)|2 = q2 |x|4 + σ 2 |x|2β+4 . By inequalities (53) and (56), we have
1 2 |x| (2xT f (x) + |g (x)|2F ) − 2|xT g (x)|2 |x|4 ≤ 2κ|x|α + 2γ + q2 + σ 2 |x|2β − 2(q2 + σ 2 |x|2β ) = 2κ|x|α + 2γ − (q2 + σ 2 |x|2β ) =: −µ(x).
So we have
γ 2 (x)|x|2 + 6xT f (x) ≥ 6
ai x2i ≥ 6a|x|2 ,
x ∈ R3
(49)
i =1
with a = min1≤i≤3 ai > 0. Using the convention that x = [x1 x2 · · · xn ]T and xn+1 = x1 with n = 3 in this case, define g : R3 → R3×3 by g (x) = γ (x) diag{x2 , x3 , x1 }. Then g is locally Lipschitz continuous with
|g (x)|2F = γ 2 (x)|x|2 and |xT g (x)|2 = γ 2 (x)
3
x2i x2i+1
(57) 2β
Note that |x g (x)| /|x| = q + σ |x| = ν(x). It is observed that Assumption 3.2 holds if one of the following conditions is satisfied: T
3
(54)
2
4
2
2
(i) 2β > α > 0, σ 2 > 0 and q2 > maxy≥0 −σ 2 y2β + 2κ yα + 2γ ; (ii) 2β = α > 0, σ 2 > 2κ and q2 ≥ 2γ ; (iii) 2β = α = 0 and σ 2 + q2 > 2(k + γ ).
Therefore, by Theorem 3.1, the solution of (54) is almost surely stable in these cases. Alternatively, it is easy to observe that the sufficient condition (55) can be obtained by applying the existing result (4)–(5), i.e., Theorem 8 in [1] to system (54).
(50) 6. Conclusion
i=1 3
for all x ∈ R . Note that
|x|4 =
3 2
x2i
i=1
3
=
3
x4i + 2
i=1
3
x2i x2i+1 ≤ 3
i =1
x4i .
(51)
i =1
Substitution of (49)–(51) into (31) gives 1
|x|4 =
|x|2 (2xT f (x) + |g (x)|2F ) − 2|xT g (x)|2 1
|x|4
2|x|2 xT f (x) + γ 2 (x) |x|4 − 2
≥
1 1 2|x|2 xT f (x) + γ 2 (x)|x|4 4 |x| 3
=
1 T 6x f (x) + γ 2 (x)|x|2 ≥ 2a, 2 3|x|
3
x2i x2i+1
This paper has presented new results on stochastic stabilization and destabilization of nonlinear differential equations. It has been shown (see Remarks 3.1 and 4.1) and illustrated with examples that our proposed results reveal the more fundamental principle and are more applicable. It is noticed that the theorems in [24] can be easily obtained by applying our results or those in [1]. Recently, the techniques proposed in [24] have been extended to study stochastic stabilization of functional differential equations (see [25] and also [26]) x˙ (t ) = f (xt )
i=1
(58)
by introducing two Brownian motions in the form of dx(t ) = f (xt )dt + qx(t )dw1 (t ) + σ |x(t )|β x(t )dw2 (t ), (52)
which implies Assumption 4.1 holds. By Theorem 4.1, it follows the desired result lim supt →τe |x(t )| = ∞ a.s.
(59)
where f satisfies the local Lipschitz condition and, moreover, there are some nonnegative constants α , κ , κ¯ , γ and a probability measure µ on [−τ , 0] such that
|f (φ)| = |φ(0)| κ|φ(0)|α + κ¯
0
|φ(θ )|α dµ(θ ) + γ
(60)
−τ
Example 5.3. Let us study the problem of stochastic stabilization of ODE (1) with the so-called one-sided polynomial growth
for any (φ, t ) ∈ C ([−τ , 0]; Rn ) × R+ (see Assumptions 1.2 and 4.1 in [25]). Obviously, (58) is a special class of functional differential
168
L. Huang / Systems & Control Letters 62 (2013) 163–169
equations where its right-hand side vanishes whenever the current state is zero. Therefore, it shares some properties with differential equations (1), e.g., the current state φ(0) plays a dominant role in the system dynamics which can be verified, say, in the proof of θ0 = ∞a.s. if x(0) ̸= 0. It appears that the techniques proposed in [1] and our results could be also extended to such a class of functional differential equations.
(i) 2β > α > 0 and σ ̸= 0; (ii) 2β = α > 0 and σ 2 ≥ 2κ ; (iii) 2β ≥ α = 0 and σ ∈ R. So, by Proposition 3.1, there exists a unique continuous adapted process x, which is a solution of (63) such that τe = θ0 = ∞a.s. Appendix C. Proof of claim (23)
Acknowledgments The author gratefully acknowledges the reviewer’s comments. The author would like to thank Prof. G. Craciun for his lectures at ETH Zurich and thank Prof. H. Koeppel for the discussion, which helped with the Example 5.1 of mass-action kinetics.
We prove P(A1 ∩ E2 ) = 0 by contradiction. Suppose that P(A1 ∩ E2 ) > 0. There exist α0 > 0 and α1 > 0 such that P ω ∈ A1 : there are infinitely many j such that ρj < ∞ = P ω ∈ A1 : σj < ∞, j ∈ Z¯ ≥ α0 , (65)
Appendix A. Proof of Proposition 3.1
where {ρj }j≥1 are a sequence of stopping times defined by
By (13), we only need to prove τe = ∞ a.s. Since θ0 = ∞ a.s., by Itô’s formula, we obtain
ρ1 = inf{t ≥ 0 : µ(x(t )) ≥ 2α1 }, ρ2j = inf{t ≥ σ2j−1 : µ(x(t )) ≤ α1 }, ρ2j+1 = inf{t ≥ σ2j : µ(x(t )) ≥ 2α1 },
log |x(t )|2 = log |x0 |2 +
t
|x(s)|
4
0
≤ log |x0 |2 +
1
p(x(s))ds + M (t )
t
λ(x(s))ds + M (t )
(61)
0
on t ≥ 0, where p(x) = |x|2 (2xT f (x) + |g (x)|2F ) − 2|xT g (x)|2 M (t ) =
t
2xT (s)g (x(s))
0
|x(s)|
2
for all |x| ≤ k. Let τ (t ) = (ρj + t ) ∧ τk for t ≥ 0 and IA be the indicator of set A. For any j ∈ Z¯ , by Hölder’s inequality and Doob’s martingale inequality (see, e.g., [13]), we compute
E log |x(t ∧ τk )| ≤ E log |x0 | + E 2
t ∧τk
λ(x(s))ds
Appendix B. An application example of Proposition 3.1 Over the past few years, there have been many works dedicated to the study that environmental noise can be used to suppress explosions (in finite time) of differential equations (see, e.g., [27,22,24] and the references therein). For example, in [24], a scalar Brownian motion w(t ) is introduced into ODE (1) satisfying conditions (8), (9) and (53) in the form of dx(t ) = f (x(t ))dt + σ |x(t )| x(t )dw(t ),
(63)
which guarantees the existence and uniqueness of its global solutions. As an example of applications, we apply Proposition 3.1 to this case and figure out the sufficient conditions for existence and uniqueness of solutions to (63). Obviously, in system (63), g (x) = σ |x|β x is locally Lipschitz continuous with
|g (x)|2F = σ 2 |x|2β+2 and |xT g (x)|2 = σ 2 |x|2β+4 . Therefore, we have 1 2 |x| (2xT f (x) + |g (x)|2F ) − 2|xT g (x)|2 4 |x| 2β
=: λ(x).
j 2 τ j (t ) τ k (t ) k = E IA1 ∩{ρj <τk } sup g (x(s))dB(s) f (x(s))ds + 0≤t ≤T ρj σj j 2 τ k (t ) ≤ 2E IA1 ∩{ρj <τk } sup f (x(s))ds 0≤t ≤T ρj j 2 τ k (t ) + 2E IA1 ∩{ρj <τk } sup g (x(s))dB(s) 0≤t ≤T ρj j
τ k (T )
≤ 2E IA1 ∩{ρj <τk }
ρj
+ 8E IA1 ∩{ρj <τk }
|f (x(s))|2 ds
j
τ k (T ) ρj
|g (x(s))| ds 2
≤ 2Kk2 T (T + 4),
(67)
where T is some positive constant. Since µ(·) is continuous in Rn , it must be uniformly continuous in the closed ball S¯k = {x ∈ Rn : |x| ≤ k}. For any given b > 0, we can choose cb > 0 such that |µ(x) − µ(y)| < b whenever x, y ∈ S¯k and |x − y| < cb . Let us choose
ε=
α0 3
,
k ≥ kε
and b = α1 .
(68)
By (67) and Chebyshev’s inequality (see, e.g., [13]), we have
2β
≤ 2κ|x| + 2γ + σ |x| − 2σ |x| = 2κ|x|α + 2γ − σ 2 |x|2β 2
0≤t ≤T
≤ E log |x0 |2 + Hλ t =: H¯ t . (62) ¯ t is independent of k and log |x(τk )|2 = 2 log k. So Note that H ¯t. 2 log kP({τk ≤ t }) ≤ E log |x(t ∧ τk )|2 ≤ H Letting k → ∞ and then t → ∞, we obtain P({τe < ∞}) = 0, that is, τe = ∞a.s., which completes the proof.
β
j
E IA1 ∩{ρj <τk } sup |x(τk (t )) − x(ρj )|2
0
2
(66) j k
But this yields
α
and Z¯ is a set of natural numbers that includes infinitely many elements. Since x(t ) and hence µ(x(t )) are continuous on t ≥ 0, we see that ρj → ∞ a.s. as j → ∞. By local Lipschitz condition (9), for any given k > 0, there exists Kk > 0 such that
|f (x)| ∨ |g (x)| ≤ Kk
and
dw(s).
2
j = 1, 2, 3, . . . ;
P ω ∈ A1 : τk ≤ ρj + P ω ∈ A1 : ρj < τk and (64)
It is easy to see that λ ∈ CB (Rn ; R) and hence Assumption 3.1 holds if one of the following conditions is satisfied:
sup |µ(x(ρj + t )) − µ(x(ρj ))| ≥ α1
0≤t ≤T
≤ P ω ∈ A1 : τk ≤ ρj + P ω ∈ A1 : ρj < τk ≤ ρj + T
L. Huang / Systems & Control Letters 62 (2013) 163–169
+ P ω ∈ A1 : ρj + T < τk and
References
sup |µ(x(ρj + t )) − µ(x(ρj ))| ≥ α1 0 ≤t ≤T
≤ P ω ∈ A1 : τk ≤ ρj + T + P ω ∈ A1 : ρj + T < τk and sup x(ρj + t ) − x(ρj ) ≥ cα1 0 ≤t ≤T
≤
2Kk2 T (T + 4) cα21
+ (1 − 2ε) .
(69)
We furthermore choose T = T (ε, α1 , k) > 0 sufficiently small for 2Kk2 T (T + 4) cα21
≤ ε.
(70)
Inequalities (69) and (70) yield
P ω ∈ A1 : ρj < τk and sup |µ(x(ρj + t )) − µ(x(ρj ))| < α1
≥ ε.
(71)
0≤t ≤T
According to (65), j − 1 ∈ Z¯ whenever j ∈ Z¯ and j ≥ 2, which implies there are infinitely many even numbers in Z¯ . By (22) and (71), we have ∞
∞ > E IA1 0
≥
≥
≥
µ(x(t ))dt ρ2j
E IA1 ∩{ρ2j−1 <τk }
2j∈Z¯
ρ2j−1
µ(x(t ))dt
α1 E IA1 ∩{ρ2j−1 <τk } (ρ2j − ρ2j−1 )
2j∈Z¯
T α1 P ω ∈ A1 : ρ2j−1 < τk and
2j∈Z¯
sup |µ(x(ρ2j−1 + t )) − µ(x(ρ2j−1 ))| < α1
0 ≤t ≤T
≥
2j∈Z¯
T α1 ε =
1 3
T α0 α1 = ∞,
169
(72)
2j∈Z¯
which is a contradiction. So we must have P(A1 ∩ E2 ) = 0. The proof is complete.
[1] J.A.D. Appleby, X. Mao, A. Rodkina, Stabilization and destabilization of nonlinear differential equations by noise, IEEE Trans. Automat. Control 53 (2008) 683–691. [2] T. Caraballo, M. Garrido-Atienza, J. Real, Stochastic stabilisation of differential systems with general decay rate, Systems Control Lett. 48 (2003) 397–406. [3] R.Z. Has’minskii, Stochastic Stability of Differential Equations, Sithoff & Noordhoff, Alphen aan den Rijn, Netherlands, 1980. [4] X. Mao, Stochastic stabilisation and destabilisation, Systems Control Lett. 23 (1994) 279–290. [5] R. Bellman, J. Bentsman, S. Meerkov, Stability of fast periodic systems, IEEE Trans. Automat. Control 30 (1985) 289–291. [6] H.J. Kushner, On the stability of processes defined by stochastic differencedifferential equations, J. Differ. Equ. 4 (1968) 424–443. [7] S. Meerkov, Condition of vibrational stabilizability of a class of non-linear systems, IEEE Trans. Automat. Control 27 (1982) 485–487. [8] L. Arnold, H. Crauel, V. Wihstutz, Stabilization of linear systems by noise, SIAM J. Control Optim. 21 (1983) 451–461. [9] M. Scheutzow, Stabilization and destabilization by noise in the plane, Stoch. Anal. Appl. 11 (1993) 97–113. [10] X. Mao, Stability of stochastic differential equations with Markovian switching, Stochastic Process. Appl. 79 (1999) 45–67. [11] R.Z. Khasminskii, C. Zhu, G. Yin, Stability of regime-switching diffusions, Stochastic Process. Appl. 117 (2007) 1037–1051. [12] X. Mao, G.G. Yin, C. Yuan, Stabilisation and destabilisation of hybrid systems of stochastic differential equations, Automatica 43 (2007) 264–273. [13] X. Mao, Stochastic Differential Equations and Applications, second ed., Horwood Publishing, Chichester, UK, 2007. [14] R.Sh. Lipster, A.N. Shiryayev, Theory of Martingales, Kluwer Academic, Dordrecht, Netherlands, 1989. [15] L. Huang, X. Mao, On almost sure stability of hybrid stochastic systems with mode-dependent interval delays, IEEE Trans. Automat. Control 55 (2010) 1946–1952. [16] X. Mao, A note on the LaSalle-type theorems for stochastic differential delay equations, J. Math. Anal. Appl. 268 (2002) 125–142. [17] G. Craciun, F. Nazarov, C. Pantea, Persistence and permanence of mass-action and power-law dynamical systems, SIAM J. Appl. Math. (in press). [18] D.J. Wilkinson, Stochastic Modelling for Systems Biology, Chapman & Hall/CRC, London, UK, 2006. [19] S. Ahmad, A.C. Lazer, Average conditions for global asymptotic stability in a nonautonomous Lotka–Volterra system, Nonlinear Anal. 40 (2000) 37–49. [20] X. Mao, G. Marion, E. Renshaw, Environmental noise suppresses explosion in population dynamics, Stoch. Process. Appl. 97 (2002) 95–110. [21] X. Mao, S. Sabanis, E. Renshaw, Asymptotic behavior of the stochastic Lotka–Volterra model, J. Math. Anal. Appl. 287 (2003) 141–156. [22] Q. Luo, X. Mao, Stochastic population dynamics under regime switching II, J. Math. Anal. Appl. 355 (2009) 577–593. [23] X. Li, A. Gray, D. Jiang, X. Mao, Sufficient and necessary conditions of stochastic permanence and extinction for stochastic logistic populations under regime switching, J. Math. Anal. Appl. 376 (2011) 11–28. [24] F. Wu, S. Hu, Suppression and stabilisation of noise, Int. J. Control 82 (2009) 2150–2157. [25] F. Wu, X. Mao, S. Hu, Stochastic suppression and stabilization of functional differential equations, Systems & Control Letters 59 (2010) 745–753. [26] F. Wu, S. Hu, Stochastic suppression and stabilization of delay differential systems, Int. J. Robust Nonlinear Control 21 (2010) 488–500. [27] F. Deng, Q. Luo, X. Mao, S. Peng, Noise suppresses or expresses exponential growth, Systems & Control Letters 57 (2008) 262–270.