Sensitivity to small delays of mean square stability for stochastic neutral evolution equations

In this work, we are concerned about the mean square exponential stability property for a class of stochastic neutral functional differential equations with small delay parameters. Both distributed and point delays under the neutral term are considered. Sufficient conditions are given to capture the exponential stability in mean square of the stochastic system under consideration. As an illustration, we present some practical systems to show their exponential stability which is not sensitive to small delays in the mean square sense.

In addition to the condition α+β < σ 2 /2, it was shown by Bierkens [3] that the null solution of (2) is exponentially stable in the almost sure sense if the delay parameter r > 0 is sufficiently small. Also, Appleby and Mao [1] obtained some similar results for some nonlinear stochastic systems. When we turn to consider an infinite dimensional system, the situation becomes complicated. It was observed by Datko et al. [4] that small delays may destroy stability for a partial differential equation. On the other hand, it is noticed that if some natural conditions are imposed, a similar result to those as above in finite dimensional spaces could be true. To state this, let us denote by L (X, Y ) the space of all bounded, linear operators from X into Y where X and Y are arbitrary Banach spaces with their respective norms · X and · Y . If X = Y , we simply write L (X) for L (X, X). Consider the following deterministic differential equation in a Hilbert space H, where A generates a C 0 -semigroup e tA , t ≥ 0, on H and A 0 ∈ L (H). It was shown in Theorem 7.5, [2] that if e tA , t ≥ 0, is a norm continuous C 0 -semigroup, i.e., the mapping t → e tA is continuous from (0, ∞) to L (H), and the C 0 -semigroup generated by A + A 0 is exponentially stable, then there exists r 0 > 0 such that the null solution of (3) is exponentially stable for all r ∈ [0, r 0 ). In other words, the exponential stability property is not sensitive to small delays in this situation. In particular, consider a linear partial differential equation in H = L 2 (0, π), where r ≥ 0 and α ∈ R. Let A = ∂ 2 /∂ξ 2 , which generates a norm continuous C 0 -semigroup on H and A 0 = αI, where I is the identity operator on H. If r = 0, it is well known that when α < 1, the trivial solution of (4) is exponentially stable. If r = 0 and α < 1, we have as above that the trivial solution of (4) is exponentially stable when r ∈ (0, r 0 ) for some r 0 > 0.
We turn our attention to stochastic systems and consider their sensitivity problem of exponential stability to small delays. As a motivation example, let r ≥ 0 and consider a stochastic version of the delay partial differential equation (4), where α, σ ∈ R and w is a standard real Brownian motion. If r > 0, it was shown in [5] that under the condition α − 1 < σ 2 /2, the pathwise exponential stability of the trivial solution to (5) is not sensitive to small delays r > 0. Now let us consider a time delay version of (5) of neutral type in the following form where γ ∈ R. In comparison with (5), the novelty of system (6) is that a delay term appears under the differentiation and the second order derivative, i.e., Laplacian, operator of (6) on one hand, and a time delay appears in the diffusion term on the other. Here we want to know whether, in addition to some natural conditions on σ, γ, the trivial solution of equation (6) can still secure its exponential stability, at least for sufficiently small delay parameter r > 0. In [6], the answer was shown to be affirmative for the pathwise exponential stability for a similar system to (6).
In this work, we shall consider the sensitivity problem to small delays of exponential stability in the mean square sense for such stochastic neutral functional differential equations as (6). The organization of this work is as follows. In Section 2, we first develop a C 0 -semigroup theory of functional differential equations, which enables us to lift up stochastic time delay systems into non time delay ones in the subsequent section. To justify the stochastic stability for our systems, it is important to know when the associated "lift-up" solution semigroups are exponentially stable. To this end, we distinguish in Section 3 between two kinds of the most popular delays, i.e., distributed and point delays, and treat the corresponding systems separately. Last, we apply the results established in this work to various examples to illustrate our theory.
2. Deterministic neutral systems. In the sequel, we shall focus on a class of norm continuous semigroups e tA , t ≥ 0, formulated by a variational approach. Precisely, let V be a separable Hilbert space and a : V × V → R a bilinear form satisfying the so-called Gårding's inequalities for some constants β > 0, α > 0. In association with the form a(·, ·), let A be a linear operator defined by where V * is the dual space of V and ·, · V,V * is the dual pairing between V and V * . Then A ∈ L (V, V * ). Moreover, it can be shown (see, e.g., [9]) that A generates a bounded, analytic semigroup e tA , t ≥ 0, on V * such that e tA : V * → V for each t > 0 and for some constant M > 0, We also introduce the standard interpolation Hilbert space H = (V, V * ) 1/2,2 , which is described by We identity the dual H * of H with H, then it is easy to see that where the imbedding → is dense and continuous with Hence, x, Ay H = x, Ay V,V * for all x ∈ V and y ∈ V with Ay ∈ H. Moreover, for any T ≥ 0 it is well known that . We introduce two linear mappings D and F on C([−r, T ], V ), respectively, by and Both the mappings D and F have a bounded, linear extension to Let H = H × L 2 ([−r, 0], V ) and consider the following deterministic functional differential equation of neutral type in V * , or its integral form It is known (see [8]) that there is a unique solution x of (14) in [0, T ] such that and the equation (14) is The family t → S(t) is a strongly continuous semigroup on H. Moreover, we have (see [8]) the following result which completely describes the generator A of semigroup S(t) or e tA , t ≥ 0.
Theorem 2.1. The generator A of the strongly continuous semigroup S(t), t ≥ 0, is given by 3. Stochastic neutral differential equations. In this work, we are concerned about a class of infinite dimensional stochastic systems, especially stochastic functional differential equations of neutral type. To this end, assume that {Ω, F , P} is a complete probability space equipped with some filtration {F t } t≥0 and K is a separable Hilbert space with the inner product ·, where Q is a positive, self-adjoint and trace class operator on K. We introduce the subspace K Q = Ran(Q 1/2 ), the range of Q 1/2 , of K and let L 2 = L 2 (K Q , H) denote the space of all Hilbert-Schmidt operators from K Q into H.
To proceed further, we split this section into two parts. The first part concentrates on the distributed delay case and the other concentrates on the point delay one.
3.1. Distributed delays under neutral terms. Let A be a linear operator given as in (8). Consider the following retarded differential equation of neutral type in space V * , and W Q (·) is a Q-Wiener process satisfying (16).
In association with (17), it is immediate by Theorem 2.1 that and we further define a bounded, linear operator B : H → L 2 (K Q , H) by By exploiting the standard lift-up method (see, e.g., Liu [8]), we can rewrite (17) as an equivalent stochastic evolution equation without delay, where is the lift-up process of y(t), t ≥ −r. It can be easily shown that the stochastic exponential stability of the null solution of (18) is equivalent to the corresponding exponential stability of (17).
The following proposition which is taken from Theorem 2.2.2 in Liu [7] is important in establishing the main results in this work.
where Σ ∈ L (H, L 2 ) and W Q (t), t ≥ 0, is a Q-Wiener process in K. If there exists a constant γ > 0 such that and ∞ 0 e tΛ * ∆(I)e tΛ dt < 1, t ≥ 0, then there exist positive constants M ≥ 1, µ > 0 such that where ∆(I) ∈ L (H) is the unique operator defined by the form Here T r{·} means the trace of operators.
Now we are in a position to state one of the main results in this work.
for all x ∈ V and some constant α > 0. Assume that the delay parameter r ∈ [0, 1] and for some 0 < λ < α, then we have the relation Proof. We intend to find an equivalent inner product (·, ·) H to ·, To this end, define (·, ·) H by where γ(·) is given by From (22), it is immediate that γ(−r) > 0. Hence, it is easy to see that (·, ·) H satisfy all the conditions of a valid inner product due to the fact that γ(θ) ≥ γ(−r) > 0 for θ ∈ [−r, 0]. Since x, Ax V,V * ≤ −α x 2 V for any x ∈ V , it follows for any φ ∈ D(A) that By using integration by parts, one can easily obtain for φ ∈ D(A) that Substituting (25) into (24), we thus get for φ ∈ D(A) that On the other hand, it is easy to see by the well-known Hölder's inequality and assumption r ∈ [0, 1] that By substituting (27) into (26), it follows immediately that If φ 0 V = 0, it follows from (28) that If φ 0 V = 0, we have from (28) and the assumption r ∈ [0, 1] that Hence, we get the relation Further, it follows from Proposition 2.1.4 in Liu [5] that the C 0 -semigroup e tA , t ≥ 0 is exponentially stable.  H)) . Assume that the following relation holds: then the null solution of (17) is exponentially stable in the mean square sense.
Proof. From Theorem 3.1, we know that That is, (20) is satisfied for the system (18). Let I denote the identity operator in H and define a linear operator ∆(I) ∈ L (H) by the relation Then, by virtue of (30), we have On the other hand, for any φ ∈ H, we easily get by Hölder's inequality that Therefore, it follows that Substituting (33) into (31) and using the condition we obtain the desired relation ∞ 0 e tA * ∆(I)e tA dt < 1.
Therefore, the conditions (20) and (21) in Proposition 1 are satisfied and the null solution of (17) is exponential stability in mean square. The proof is thus complete.
Corollary 1. Let K = R, W Q (t) = w(t), a standard real Brownian motion, B ∈ L (H) and assume that the delay parameter r ∈ [0, 1]. Further, suppose that for some λ ∈ (0, α), e tA ≤ e −αt , then the null solution of (17) is exponentially stable in the mean squarer sense.

3.2.
Point delays under neutral terms. In this subsection, we shall consider the following stochastic functional differential equation of neutral type in space V * , To this end, let and meanwhile we define We can rewrite (39) as an equivalent stochastic evolution equation without delay, where Y (t) = (y(t) − G 2 y(t − r), y(t + ·)), t ≥ 0 is the lift-up process of y(t).
Proof. As before, we intend to find an equivalent inner product (·, ·) H to ·, · H on where γ(θ) is given by It can be verified that (·, ·) H satisfies the definition of a valid inner product due to the fact that γ(θ) ≥ γ(−r) > 0 for θ ∈ [−r, 0]. For any φ ∈ D(A), it follows that where A 2 := A 2 L (V,V * ) . Using integration by parts, one can derive for φ ∈ D(A) that Substituting (45) into (44), we immediately obtain On the other hand, it is immediate that (47) and (46) we have for any φ ∈ D(A) that If φ 0 V = 0, it follows from (41) that Thus, it is immediate that Substituting (53) into (51) and using the condition λ > B 2 · ν 2 /2, we get immediately that ∞ 0 e tA * ∆(I)e tA dt < 1.
Therefore, the two conditions in Proposition 1 are satisfied for the stochastic system (39) and the mean square exponential stability is thus obtained. The proof is complete now.
As an immediate consequence of Theorem 3.4, we have the following result.
On this occasion, it is easy to see that α in Corollary 2 is equal to 1. Hence, by virtue of (54), if there exists λ > 0 such that then the null solution of (55) is exponentially stable in the mean square sense. It may be verified that λ := σ 2 2 + ε with ε > 0 sufficient small satisfies the first condition of (58). The second condition in (58), by letting ε → 0, reduces to 0 < r < 1 2σ 2 ln .
In summary, in the case γ = 0 and σ 2 < 2, we have that whenever |γ| < 1 + 2(1 − σ 2 2 ) 2 − 1 2 − σ 2 and r is so small, satisfying (59), the null solution of (55) is exponentially stable in the mean square sense. In other words, the mean square exponential stability of (55) is not sensitive to small delays in this situation.