Practical stability of stochastic delay evolution equations

Practical stability of stochastic delay evolution equations Tom´as CaraballoaMohamed Ali HammamibLassad Mchirib aUniversidad de Sevilla, Dpto. Ecuaciones Diferenciales y An´alisis Num´erico, Facultad de Matem´aticas, Sevilla (Spain) bUniversity of Sfax, Faculty of Sciences of Sfax, Department of Mathematics, Tunisia E.mail: [email protected]u.tn Abstract In this paper we investigate the almost sure practical stability for a class of stochastic functional evolution equations. We establish some sufficient conditions based on the construction of appropriate Lyapunov functional. The abstract results are then applied to some illustrative examples. Keywords: Stochastic delay evolution equations, almost sure practical asymptotic stability, decay function. MSC: 60H15, 1 Introduction Stochastic differential equations are useful for modeling physical, technical, biological and economical dynamical systems in which significant uncertainty is present. In 1892, Lyapunov introduced the concept of stability for deterministic dynamical systems and firstly established the so-called Lyapunov second method of stability theory. The advantage of this method can judged the stability of systems without knowledge of the solution of the systems explicitly. It is therefore of great interest to be able to characterize the behavior of the solutions. Several interesting and important variants to Lyapunov’s original concepts of stability were proposed in [1]-[3] and [14]. However, when the origin is not necessarily an equilibrium point, it is still possible to analyze the asymptotic stability of solutions with respect to a small neighborhood of the origin, what yields 1 to the concept of practical stability. It is worth mentioning some previous works on practical stability in the deterministic framework, as for example, [1],[3]. Also we would like to mention here the references [4, 5, 19, 20], among others. As for the stochastic case, we initiate the study of this practical stability for non-delay stochastic partial differential equations in [14]. Needless to say that the asymptotic behavior of systems described by stochastic evolution equations is a very important topic as the vast literature on this field shows (see [7, 8, 9, 10, 11, 12, 13, 16, 19, 20, 22, 23, 25]. Moreover, the consideration of memory terms in the models are fully justified as it is sensible to think that not only the present state of the systems possesses influence in the future behavior of the problem, but also what has happened in the past. Additionally, in the nonlinear and/or nonautonomous situations, it may happen that the stability cannot always be exponential but can even be sub or super-exponential (see [12]). For this reason, in this paper we will study the almost sure practical stability with general decay rate and will illustrate the theory with some application examples. The content of the paper is as follows. In section 2, we introduce the basic notations and assumptions. In Section 3, we prove some sufficient conditions ensuring almost sure practical stability of solutions to stochastic delay evolution equations. First we prove a sufficient condition ensuring the convergence of solutions to zero with general decay rate, even in the case that zero is not an equilibrium point. Later, we weaken the assumptions on the operators involved in the model, obtaining a weaker result which will ensure the convergence to a ball with a small radius, instead of the convergence of solutions to zero. In Section 4, we finally study two examples to illustrate these results. 2 Basic notations and assumptions Let Vbe a reflexible Banach space and Ha real separable Hilbert space such that V⊂H≡H∗⊂V∗, where the injections are continuous and dense. In addition, we also assume both Vand V∗are uniformly convex. We denote by k.k,|.|and k.k∗the norms in V,Hand V∗respectively, by (., .) the inner product in H, and by < ., . > the duality between Vand V∗. Assume {Ω,F,P}is a complete probability space with a normal filtration {Ft}t≥0, i.e. F0 contains the null sets in Fand Ft=∩s>tFs, for all t≥0, and let us consider a real valued {Ft}-Wiener process {W(t)}t≥0. Given h≥0 , p≥2 and T > 0, we denote by Ip(−h, T, V ) the closed subspace of Lp(Ω ×(−h, T),F ⊗ B([−h, T]),P ⊗ dt;V) of all stochastic processes which are Ft-adapted 2 for almost every tin (0, T) (in what follows, a.e.t), where B([0, T]) denotes the Borel σ-algebra of subsets in [0, T] and we will set Ft=F0for t < 0. We write L2(Ω; C(−h, T;H)) instead of L2(Ω,F,P;C(−h, T;H)), where C(−h, T;H) denotes the space of all continuous functions from [−h, T] into H. Let CH=C(−h, 0; H) with norm |ψ|CH= sup−h≤s≤0|ψ(s)|,ψ∈CH;Lp V=Lp([−h, 0], V ) and Lp H=Lp([−h, 0], H). On the other hand, given a stochastic process X(t)∈Ip(−h, T, V )∩L2(Ω; C(−h, T;H)), we associate with another stochastic process Xt: Ω →Lp V∩CH, by means of the usual relation Xt(s)(ω) = X(t+s)(ω), 0 ≤t≤T,−h≤s≤0. Let us consider the following stochastic delay evolution equation in V∗: dX(t) = (A(t, X(t)) + F(t, Xt))dt +G(t, Xt)dWt, t ∈[0, T], X(t) = ψ(t), t ∈[−h, 0],(2.1) where T > 0 and the initial datum ψ∈Ip(−h, 0, V )∩L2(Ω; CH). As we are mainly interested in the stability analysis of solutions to equation (2.1), we shall assume that for each ψ∈Ip(−h, 0, V )∩L2(Ω; CH) there exists a process X(t)∈Ip(−h, T, V )∩L2(Ω; C(−h, T;H)), which is solution to equation (2.1) for every T > 0, in other words, X(t) satisfies the following integral equation in V∗: X(t) = ψ(0) + Rt 0(A(s, X(s)) + F(s, Xs))ds +Rt 0G(s, Xs)dWs,∀t∈[0, T], X(t) = ψ(t), t ∈[−h, 0].(2.2) This happens, for instance, if A(t, .) : V→V∗is a family of (nonlinear) operators defined a.e.t. satisfying there exists t∈R+such that A(t, 0) 6= 0 and fulfilling the following assumptions : 1.Measurability: for every x∈V, the map t∈(0, T)7→ A(t, x)∈V∗is Lebesgue measurable, a.e. t. 2.Hemicontinuity: The map θ∈R7→< A(t, x +θy), z >∈Ris continuous for every x, y, z ∈V, a.e. t. 3.Boundedness: There exists β > 0, c > 0 such that ||A(t, x)||∗≤c||x||p−1+βfor all x∈V, a.e.t. 3 4.Coercivity: There exist α > 0, λ,γ∈Rsuch that −2< A(t, x), x > +λ|x|2+γ≥α||x||pfor all x∈V, a.e.t. 5.Monotonicity: λ|x−y|2−2< A(t, x)−A(t, y), x −y > ≥0 for all x, y ∈V, a.e.t. Also F(t, .) : [0, T]×CH→V∗is a family of Lipschitz continuous operators defined a.e.t. and G(t, .) : [0, T]×CH→His another family of Lipschitz operators defined a.e.t. (see [7], [8] and also [24]) and satisfying there exists t∈R+such that G(t, 0) 6= 0. Unless otherwise is stated, we will assume that U(t, x) is a C1,2-positive appropriate Lyapunov functional such that for any x∈Vand t∈R+,U0 x(t, x)∈V, and satisfies some additional assumptions which enable us to apply the Itˆo formula for the process X(t), solution to equation (2.1) (see [21]). 3 Practical stability of stochastic delay evolution equation We first state the definition of almost surely convergence of solutions to a small ball centered at the origin. Definition 3.1. The ball Br:= {x∈H/|x| ≤ r},r > 0 is said to be almost surely globally practically uniformly exponentially stable with respect to the system (2.1) if: For any initial value ψ∈Ip(−h, 0, V )∩L2(Ω; CH) such that 0 <|X(t, ψ)|−r, for all t≥t0≥0, P-a.s., where X(t, ψ) denotes the solution of (2.1), it holds that lim t→∞ sup 1 tln(|X(t, ψ)| − r)<0,P−a.s. (3.1) System (2.1) is said to be almost surely globally practically uniformly exponentially stable if there exists r > 0 such that Bris almost surely globally practically uniformly exponentially stable. Next we state the definition of practical convergence to zero with general decay rate. Definition 3.2. Let λ(t) be a positive function defined for sufficiently large t > 0, say t≥T > 0, and satisfying that λ(t)↑+∞as t→+∞. A solution X(t) to equation (2.1) (defined in the future, i.e. for tlarge enough) is said to decay to zero almost surely with decay function λ(t) and order at least γ > 0, if there exists a function r(·) such that r(t)→0 as t→+∞, and its generalized Lyapunov exponent is less than or equal to −γwith probability one, i.e. lim t→+∞sup ln |X(t)| − r(t) ln(λ(t)) ≤ −γ, P−a.s. 4 If in addition 0 is solution to equation (2.1), the zero solution is said to be almost surely practically asymptotically stable with decay function λ(t) and order at least γ, if there exist r(t)→0 as t→+∞such that every solution to equation (2.1) decays to zero almost surely with decay function λ(t) and order at least γ. Now, we can state and prove one of our main results in this paper. Theorem 3.1. Let U(t, x)be an appropriate Lyapunov functional. Assume that ln λ(t)is uniformly continuous on t∈[T, +∞)and there exists a constant τ≥0such that lim t→∞ sup ln ln(t) ln λ(t)≤τ. Assume that there exist constants q∈N∗,m > 0,µ≥0,b≥0,ν,θ≥0, a non-increasing function ξ(t)>0and a continuous non-negative function ϕ(t)such that (a)|x|qλ(t)m≤U(t, x),∀(t, x)∈R+×V. (b) For a solution X(t)to (2.1), defined in the future, it holds Zt 0 U0 s(s, X(s))ds +Zt 0 < U0 x(s, X(s)), A(s, X(s)) + F(s, Xs)> ds +1 2Zt 0 (U00 xx(s, X(s))G(s, Xs), G(s, Xs))ds +Zt 0 ξ(s)(U0 x(s, X(s)), G(s, Xs))2ds ≤ζψ(t) + %(t) + Zt 0 ϕ(s)U(s, X(s))ds, where ζψ(t)is a non-negative function depending on the initial datum ψand %(t)6= 0, for all t≥0, is a function independent on ψsuch that limt→∞ %(t)=0. (c) lim t→∞ sup ln ζψ(t) ln λ(t)≤ν, lim t→∞ sup Rt 0ϕ(s)ds ln λ(t)≤θ. lim t→∞ sup ln c+%(t) λ(t)δ ln λ(t)≤b, lim t→∞ inf ln ξ(t) ln λ(t)≥ −µ. where δ= (µ+τ+ 2)∨(ν+)for arbitrary  > 0and c= exp((µ+)), if δ =µ+τ+ 2, c= 1, if δ =ν+. (3.2) 5 (d) q X k=1 |x|q−kh|%(t)| λ(t)mik−1 q≥1,for all t≥0. Then, lim t→+∞sup ln |X(t)| − |%(t)| λ(t)m1 q ln λ(t)≤ −hm−θ+b+(µ+τ)∨νi,P−a.s. In particular, if m>θ+b+(µ+τ)∨ν, the solution X(t)decays to zero almost surely with decay function λ(t)and order at least γ=hm−θ+b+(µ+τ)∨νi. Proof. Assume that λ(t)m|x|q− |%(t)|>0 for all t≥0 and x∈H. Then, we immediately have λ(t)m|x|q− |%(t)| ≤ λ(t)m|x|q≤U(t, x). Now, observe that λ(t)m|x|q− |%(t)|=λ(t)m|x|q−|%(t)| λ(t)m =λ(t)m|x|q−(|%(t)| λ(t)m)1 qq =λ(t)m|x| − (|%(t)| λ(t)m)1 q|x|q−1+|x|q−2(|%(t)| λ(t)m)1 q+... + ( |%(t)| λ(t)m)q−1 q =λ(t)m|x| − (|%(t)| λ(t)m)1 q q X k=1 |x|q−kh|%(t)| λ(t)mik−1 q ≥λ(t)m|x| − (|%(t)| λ(t)m)1 q. Therefore, λ(t)m|x| − (|%(t)| λ(t)m)1 q≤U(t, x) and ln hλ(t)m|x| − (|%(t)| λ(t)m)1 qi≤ln U(t, x). Thus ln h|x| − (|%(t)| λ(t)m)1 qi+mln(λ(t)) ≤ln U(t, x). Applying the Itˆo formula we obtain U(t, X(t)) = U(0, ψ(0)) + Zt 0 U0 s(s, X(s))ds +Zt 0 < U0 x(s, X(s)), A(s, X(s)) + F(s, Xs)> ds +1 2Zt 0 (U00 xx(s, X(s))G(s, Xs), G(s, Xs))ds +Zt 0 (U0 x(s, X(s)), G(s, Xs))dW(s).(3.3) 6 Now, from the uniform continuity of ln λ(t), we can ensure that for each  > 0 there exist two positive integers N=N() and k1() such that if k−1 2N≤t≤k 2N,k≥k1(), it follows ln λk 2N−ln λ(t)≤. On the other hand, due to the exponential martingale inequality Pnω; sup 0≤t≤whM(t)−u 2Zt 0 (U0 x(s, X(s)), G(s, Xs))2dsi> σo≤e−uν, for any positive constants u,σand w, where M(t) = Zt 0 (U0 x(s, X(s)), G(s, Xs))dW(s). In particular, for the preceding  > 0, if we set u= 2ξk 2N,σ=ξk 2N−1ln k−1 2N,w=k 2N, k= 2,3, ..., we can then apply the Borel-Cantelli lemma to obtain that, for almost all ω∈Ω, there exists an integer k0(, ω)>0 such that Zt 0 (U0 x(s, X(s)), G(s, Xs))dW(s)≤ξk 2N−1ln k−1 2N +ξk 2NZt 0 (U0 x(s, X(s)), G(s, Xs))2ds ≤ξk 2N−1ln k−1 2N +Zt 0 ξ(s)(U0 x(s, X(s)), G(s, Xs))2ds, for 0 ≤t≤k 2N,k≥k0(, ω). Replacing this into equation (3.3), we see that P-a.s. U(t, X(t)) ≤U(0, ψ(0)) + ξk 2N−1ln k−1 2N+Zt 0 U0 s(s, X(s))ds +Zt 0 < U0 x(s, X(s)), A(s, X(s)) + F(s, Xs)> ds +1 2Zt 0 (U00 xx(s, X(s))G(s, Xs), G(s, Xs))ds +Zt 0 ξ(s)(U0 x(s, X(s)), G(s, Xs))2ds, 7 for 0 ≤t≤k 2N,k≥k0(, ω). Using condition (b) we derive that P-a.s. U(t, X(t)) ≤U(0, ψ(0)) + ξk 2N−1ln k−1 2N+ζψ(t) + %(t), +Zt 0 ϕ(s)U(s, X(s))ds, for 0 ≤t≤k 2N,k≥k0(, ω). By virtue of the Gronwall lemma we derive that P-a.s. U(t, X(t)) ≤U(0, ψ(0)) + ξk 2N−1ln k−1 2N+ζψ(t) + %(t) ×exp Zt 0 ϕ(s)ds, for 0 ≤t≤k 2N,k≥k0(, ω). On the other hand, thanks to condition (c) and the uniform continuity of ln λ(t) there exists a positive integer k1() such that whenever t≥k1() ζψ(t)≤λ(t)ν+,Zt 0 ϕ(s)ds ≤(θ+) ln λ(t), ξk 2N−1≤e(µ+)λ(t)µ+, for k−1 2N≤t≤k 2N,k≥k1(). Also ln k−1 2N≤ln t≤ln k 2Nfor k−1 2N≤t≤k 2N. We therefore obtain that, for almost all ω∈Ω, ln U(t, X(t)) ≤ln U(0, ψ(0)) + λ(t)µ+τ+2e(µ+)+λ(t)ν++%(t) + (θ+) ln λ(t), for k−1 2N≤t≤k 2N,k≥k1(). Hence lim t→+∞sup ln U(t, X(t)) ln λ(t)≤δ+θ++b, P−a.s. Taking into account that  > 0 is arbitrary and using (a) and ln h|x| − (|%(t)| λ(t)m)1 qi+mln(λ(t)) ≤ln U(t, x), 8 we can deduce lim t→+∞sup ln |X(t)| − |%(t)| λ(t)m1 q ln λ(t)≤ −hm−θ+b+(µ+τ)∨νi,P−a.s. as required. 2 Remark 3.2. In the next theorem we will weaken the assumptions imposed on the operators in our model and, as a counterpart, we will obtain a weaker convergence too. In fact, instead of obtaining practical convergence to zero with general decay rate, we will deduce only practical convergence to a ball. Theorem 3.3. Let U(t, x)be an appropriate Lyapunov functional. Assume that ln λ(t)is uniformly continuous on t∈[T, +∞)and there exists a constant τ≥0such that lim t→∞ sup ln ln(t) ln λ(t)≤τ. Assume that there exist constants q∈N∗,m > 0,µ≥0,b > 0,ν,θ≥0and a small constant % > 0, a non-increasing function ξ(t)>0and a continuous non-negative function ϕ(t)such that (a)|x|qλ(t)m≤U(t, x),∀(t, x)∈R+×V. (b) For a solution X(t)to defined in the future it holds equation (2.1) Zt 0 U0 s(s, X(s))ds +Zt 0 < U0 x(s, X(s)), A(s, X(s)) + F(s, Xs)> ds +1 2Zt 0 (U00 xx(s, X(s))G(s, Xs), G(s, Xs))ds +Zt 0 ξ(s)(U0 x(s, X(s)), G(s, Xs))2ds ≤ζψ(t) + %+Zt 0 ϕ(s)U(s, X(s))ds, where ζψ(t)is a non-negative function depending on the initial datum ψ. (c) lim t→∞ sup ln ζψ(t) ln λ(t)≤ν, lim t→∞ sup Rt 0ϕ(s)ds ln λ(t)≤θ, lim t→∞ inf ln ξ(t) ln λ(t)≥ −µ. (d)λ(t)≥b; q X k=1 |x|q−kh% bmik−1 q≥1,for all t≥0. 9 Therefore, Zt 0 U0 s(s, X(s))ds +Zt 0 < U0 x(s, X(s)), A(s, X(s)) + F(s, Xs)> ds +1 2Zt 0 (U00 xx(s, X(s))G(s, Xs), G(s, Xs))ds +Zt 0 1 4e2s(U0 x(s, X(s)), G(s, Xs))2ds ≤Zt 0h4K2e2h+2+e−s 2+e1 2(3h−s)−4α+ 2K2e−3siU(s, X(s))ds +|ψ|2 CHh4K2he2(h+t)+he3 2(t+h)+ 2K2i+K2e−t ≤Zt 0 ϕ(s)U(s, X(s))ds +ζψ(t) + %(t), where ϕ(t) = 4K2e2h+2+e−t 2+e1 2(3h−t)−4α+ 2K2e−3t, %(t) = K2e−t, λ(t) = et and ζψ(t) = |ψ|2 CHh4K2he2(h+t)+he3 2(t+h)+ 2K2i, r(t) = Ke−3 2t. Therefore, constants in theorem 3.1 can be chosen as follows τ= 0,µ=ν=m=q= 2,θ= 2 + 4K2e2h−4α,b= 0. Hence we deduce that lim t→+∞sup ln |X(t)| − Ke−3 2t t≤ −h2−4+4K2e2h−4αi,P−a.s. If 2>4 + 4K2e2h−4α, the solution is almost surely exponentially stable with decay function λ(t) = etand order at least γ= 2 −4+4K2e2h−4α. Remark 4.3. 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