Practical exponential stability in mean square of stochastic partial differential equations

Practical exponential stability in mean square of stochastic partial differential equations Tom´as Caraballoa∗Mohamed Ali HammamibLassaad 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 The main aim of this paper is to establish some criteria for the mean square and almost sure practical exponential stability of a nonlinear monotone stochastic partial differential equations. Keywords. Stochastic partial differential equations, almost sure uniform stability, mean square practical stability. Mathematics Subject Classification: Primary 93E03; Secondary 60H10. 1 Introduction We are mainly interested in the stability of a class of nonlinear stochastic partial differential equations of monotone type. The question of the asymptotic stability of the second moment of Xt(which is the solution of equation (2.1) below) has received considerable attention in the literature. Willems [7], [18] have established sufficient conditions which guarantee asymptotic stability when the spaces are finite dimensional. Wonham [20], and Willems [19] have considered a related problem, the stabilization problem, again in finite dimension. Recently Ichikawa [14] have extended these results to infinite dimensions. In fact, a coercivity condition, extending the one considered by Chow [11] and Caraballo and Real [8], is introduced and will play the role of ∗The research of Tom´as Caraballo has been partially supported by the Spanish Ministerio de Econom´ıa y Competitividad project MTM2011-22411 and the Consejer´ıa de Innovaci´on, Ciencia y Empresa (Junta de Andaluc´ıa) under grant 2010/FQM314 and Proyecto de Excelencia P12-FQM-1492. 1 a stability criterion. To be precise, under the coercivity condition from Caraballo and Real [8], almost sure exponential stability of solutions is obtained, while in Chow [11] pathwise asymptotic stability is proved. However, as we will explain later, coercivity criteria from Caraballo and Real [8] are too restrictive to be applied to a number of interesting and, in our opinion, important examples, especially in the non-autonomous case. In this work, we shall improve their results to cover the general non-autonomous stochastic differential equations in Hilbert spaces. The organization 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 exponential stability in mean square of solutions of a class of nonlinear stochastic partial differential equation, and study an example to illustrate these results. 2 Preliminaries Let Vbe a Banach space and H,Kreal, separable Hilbert spaces such that V ,→H≡H0,→V0, where the injections are continuous and dense. We denote by k.k,|.|and k.k∗the norms in V,Hand V0respectively, by (., .) the inner product in H, and by < ., . > the duality product between Vand V0, and βis a constant such that |x| ≤ β||x||,∀x∈V. Let Wtbe a Wiener process defined on some complete probability space {Ω,F,P}and taking its values in the separable Hilbert space K, with increment covariance operator Q, and let (Ft)t≥0be the usual family of subt-σ-algebras of Fsuch that, for each t≥0, Ftis generated by {Ws,0≤s≤t}. Consider the following nonlinear stochastic diffusion equation: Xt=X0+Zt 0 A(s, Xs)ds +Zt 0 B(s, Xs)dWs,(2.1) where A(t, .) : V→V0is a family of nonlinear operators defined a.e.t. satisfying there exists t∈R+such that A(t, 0) 6= 0, and where B(t, .) : V→ L(K, H), the family of all bounded linear operators from Kinto H, satisfies (b.1) There exists t∈R+such that B(t, 0) 6= 0, (b.2) There exist continuous non-negative functions k(t), ψ(t) and positive constants θand ξ such that θ:= Z+∞ 0 k2(t)dt, ξ := Z+∞ 0 ψ2(t)dt, and ||B(t, x)||2≤k(t)||x|| +ψ(t),for all x∈V, a.e.t., 2 where ||.||2denotes the Hilbert-Shmidt norm of nuclear operators, i.e., ||B(t, x)||2 2=tr(B(t, x)QB(t, x)∗). (b.3) The map t∈(0, T)7→ B(t, x)∈ L(K, H) is Lebesgue-measurable ∀x∈V,∀T > 0. Definition 2.1. Let {Ω,F,(Ft)t≥0,P}be the stochastic filter associated to the K-valued Wiener process Wtwith covariance operator Q. Suppose that X0∈L2(Ω,F0,P;H), i.e, X0is an Hvalued F0-measurable random variable such that E|X0|2<∞. A stochastic process Xtis said to be a strong solution on Ω to the SDE (2.1) for t∈[0, T] if the following conditions are satisfied (see [12]): (a)Xtis a V-valued Ft-measurable random variable; (b)Xt∈Ip(0, T;V)∩L2(Ω; C(0, T;H)), p > 1, T > 0, where Ip(0, T;V) denotes the space of all V-valued processes (Xt)t∈[0,T ](we will write Xtfor short) measurable (from [0, T]×Ω into V), satisfying that Xtis Ft-measurable (hence Xtis Ft-adapted) for almost all t∈[0, T], and EZT 0 ||Xt||pdt < ∞. Here C(0, T;H) denotes the space of all continuous functions from [0, T] into H. (c)EZT 0 ||A(t, Xt)||2 ∗dt < ∞. (d) Eq. (2.1) is satisfied for every t∈[0, T] with probability one. If Tis replaced by ∞,Xtis called a global strong solution of (2.1). As we are mainly interested in the stability analysis, we always assume that for each X0∈ L2(Ω,F0,P;H), there exists a global strong solution to (2.1). This happens, for instance, if the following assumptions hold true (see, for example, Pardoux [17]). (a.1) Coercivity: There exist α > 0, p > 1 and λ,γ∈R∗such that 2< A(t, x), x > +||B(t, x)||2 2≤ −α||x||p+λ|x|2+γfor all x∈V, a.e.t. (a.2) Boundedness: There exists β > 0, c > 0 such that ||A(t, x)||∗≤c||x||p−1+βfor all x∈V, a.e.t. (a.3) Monotonicity: ||B(t, x)−B(t, y)||2≤λ|x−y|2−2< A(t, x)−A(t, y), x −y > for all x, y ∈V, a.e.t. (a.4) Hemicontinuity: The map θ∈R7→< A(t, x +θy), z >∈Ris continuous for every x, y, z ∈V, a.e. t. 3 (a.5) Measurability: for every x∈V, the map t∈(0, T)7→ A(t, x)∈V0is Lebesgue measurable, a.e. t., ∀T > 0. Now we establish a version of the Itˆo formula (see Pardoux [17]) which will be needed later in this paper. Let C(1,2)([0,∞)×H, R+) denote the space of all R+-valued functions Ψ defined on [0,∞)×Hwith the following properties: (1) Ψ(t, x) is differentiable in t∈[0,∞) and twice Frechet differentiable in xwith Ψt(t, .), Ψx(t, .) and Ψxx(t, .) locally bounded on H, (2) Ψ(t, .), Ψt(t, .) and Ψx(t, .) are continuous on H, (3) for all trace class operators R, tr (Ψxx(t, .)R) is continuous from Hinto R, (4) if v∈Vthen Ψx(t, v)∈V, and u→ hΨx(t, u), v∗iis continuous for each v∗∈V0, (5) kΨx(t, v)k ≤ C0(t)(1 + kvk), C0(t)>0, for all v∈V. Theorem 2.1. (Itˆo’s formula). Let Ψ∈C(1,2)([0,∞)×H, R+). If the stochastic process X(t) is a weak solution to (2.1), then it holds that Ψ(t, X(t)) = Ψ(0, X(0)) + Zt 0 LΨ(s, X(s))ds, +Zt 0 (Ψx(s, X(s)), B(s, X(s))dW(s)), where LΨ(s, X(s)) = Ψt(s, X(s)), +hA(s, X(s)),Ψx(s, X(s))i, +1 2tr(Ψxx(s, X(s))B(s, X(s))QB(s, X(s))∗). Remark 2.2. Notice that any strong solution in the sense of Definition 2.1 is a weak solution in the weak or variational sense in Theorem 2.1 (see e.g. [8, 9, 17]). We state now the definitions of the almost surely convergence of solutions to a small closed ball Br⊂Hcentered at zero with radius r(see [1]-[6], [10]), and we will consider initial values in the space X0∈L2(Ω,F0,P;H). Definition 2.2. The ball Bris said to be almost surely globally practically uniformly exponentially stable if: For any initial value X0∈L2(Ω,F0,P;H), such that its corresponding strong solution X(t) := X(t, X0) to (2.1) satisfies 0 <|X(t)| − r, for all t≥0, it holds that lim sup t→∞ 1 tln(|X(t, X0)| − r)<0,a.s. (2.2) 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. 4 Definition 2.3. The ball Bris said to be almost surely globally practically uniformly exponentially stable in mean square if: For any initial value X0∈L2(Ω,F0,P;H), such that its corresponding strong solution X(t) := X(t, X0) to (2.1) satisfies 0 <E|X(t, X0)|2−r, for all t≥0, it holds that lim sup t→∞ 1 tln E(|X(t, X0)|2)−r<0,a.s. (2.3) System (2.1) is said to be almost surely globally practically uniformly exponentially stable in mean square if there exists r > 0 such that Bris almost surely globally practically uniformly exponentially stable in the mean square. Definition 2.4. The system (2.1) is said to be almost surely globally practically uniformly exponentially convergent to zero in mean square if there exists a function r(·) such that: For any initial value X0∈L2(Ω,F0,P;H), such that its corresponding strong solution X(t) := X(t, X0) to (2.1) satisfies 0 <E|X(t, X0)|2−r(t), for all t≥0, it holds that lim sup t→∞ 1 tln E(|X(t, X0)|2)−r(t)<0,a.s. (2.4) with limt→+∞r(t) = 0. Definition 2.5. The ball Bris said to be uniformly stable in probability if the strong solution X(t) := X(t, X0) to (2.1) satisfies: For each ∈]0,1[ and k > r, there exists δ=δ(, k)> r such that P|X(t, X0)|< k, ∀t≥0≥1−for all |X0|< δ. (2.5) Remark 2.3. Noting that if r→0 we have the classical definition of the stability in probability. We write in the definition (2.5) that δ=δ(, k)> r because if we take δ=δ(, k)< r and letting r→0 we get |X0|<0 which contradicts with the classical definition of the stability in probability when 0 is an equilibrium point. 3 Practical exponential stability in mean square Now we shall impose the following coercivity condition (CC): There exist constants α > 0, µ > 0, λ∈R, and a nonnegative continuous function γ(t), t∈R+, such that 2< A(t, v), v > +||B(t, v)||2 2≤ −α||v||p+λ|v|2+γ(t)e−µt, v ∈V, (3.1) where p > 1 and, for arbitrary δ > 0, γ(t) satisfies γ(t) = o(eδt), as t→ ∞, i.e., lim t→∞ γ(t) eδt = 0 and Z+∞ 0 γ(t)e−δtdt ≤Kwith K > 0. 5 Remark 3.1. Observe that, owing to the continuity and subexponential growth of the term γ(t)e−µt, there exists a positive constant eγsuch that γ(t)e−µt ≤eγfor all t∈R+. As a consequence, (3.1) implies (a.1) (by replacing γby eγ), i.e., this assumption is compatible with the existence of the strong solutions to (2.1). Theorem 3.2. Assuming conditions (CC) and (b.3), there exists a constant τ > 0such that if Xtis a global strong solution to Eq. (2.1) corresponding to an initial value X0∈L2(Ω,F0,P;H), satisfying that E|Xt|2> r(t) := Ke−τt, for all t≥0, then E|Xt|2≤E|X0|2e−τt +r(t),∀t≥0,(3.2) if either one of the following hypotheses holds (i) λ < 0,(∀p > 1); (ii) λβ2−α < 0,(p= 2). Then, system (2.1) is almost surely globally practically uniformly exponentially convergent to zero in mean square. Proof. Firstly, let us denote ν=(α−λβ2) β2for case (ii) and ν=−λ β2for case (i), which are positive by assumption (ii) and (i) respectively, and the rest of the proof is the same for both cases. Then, if µ−ν≤0, we can choose δ > 0 small enough such that µ−δ > 0 and define τ:= µ−δ. If, on the other hand, µ−ν > 0, then we can choose δ > 0 small enough such that µ−ν−δ > 0 and, in this case, we define τ:= ν. Now, let us suppose that E|Xt|2> r(t), for all t≥0. Then, Itˆo’s formula implies e(µ−δ)t|Xt|2− |X0|2= (µ−δ)Zt 0 e(µ−δ)s|Xs|2ds + 2 Zt 0 e(µ−δ)s< A(s, Xs), Xs> ds, + 2 Zt 0 e(µ−δ)s< Xs, B(s, Xs)dWs>+Zt 0 e(µ−δ)str(B(s, Xs)QB(s, Xs)∗)ds. (3.3) Now, since Zt 0 e(µ−δ)s< Xs, B(s, Xs)dWs>,t∈R+, is a continuous martingale, it follows that EZt 0 e(µ−δ)s< Xs, B(s, Xs)dWs>= 0, t ∈R+. Therefore, condition (3.1) and the continuous injection V ,→Hyield e(µ−δ)tE|Xt|2≤E|X0|2+ (µ−δ−ν)Zt 0 e(µ−δ)sE|Xs|2ds +Zt 0 γ(s)e−δsds. (3.4) If µ−ν≤0, it follows immediately e(µ−δ)tE|Xt|2≤E|X0|2+Zt 0 γ(s)e−δsds ≤E|X0|2+K, 6 thus E|Xt|2≤E|X0|2e−(µ−δ)t+Ke−(µ−δ)t≤E|X0|2e−τt +r(t). On the other hand, if µ−ν > 0, as we have chosen δ > 0 small enough such that µ−ν−δ > 0, then, from (3.4) and Gronwall’s lemma one can obtain e(µ−δ)tE|Xt|2≤E|X0|2+Zt 0 γ(s)e−δsdse(µ−δ−ν)t≤E|X0|2+Ke(µ−δ−ν)t, finally E|Xt|2≤E|X0|2e−νt +Ke−νt ≤E|X0|2e−τt +r(t), as required. 2 Remark 3.3. Notice that we can have a second version of Theorem 3.2 under the same hypotheses as it is straightforward to prove that E|Xt|2≤E|X0|2e−τt +K, ∀t≥0. Then, system (2.1) is almost surely globally practically uniformly exponentially in mean square. Theorem 3.4. In addition to hypotheses in Theorem 3.2, assume that b2) also holds and Z+∞ 0 γ(s)e−µsds ≤η < +∞and sup u∈[s,t) k2(u)≤ϕ < +∞for 0≤s≤t,µ > 0,η > 0and ϕ is a positive constant independent of tand s. Then, there exist positive constants M,and a subset N0⊂Ωwith P(N0)=0such that, for each ω∈Ω\N0, there exists a positive random number T(ω)such that |Xt|2≤Me−t +η, ∀t≥T(ω).(3.5) Then, the ball B√η⊂His uniformly stable in probability. Proof. We only prove case (ii). Case (i) can be proved similarly. We shall split our proof into several steps, as follows. Step 1: We will find three constants C=C(δ, X0)>0, ζ > 0 and τ > 0, independent of t∈R+, such that Zt s E||B(u, Xu)||2 2du ≤Ce−τs +ζ, 0≤s≤t. (3.6) Applying Itˆo’s formula to (2.1) as in theorem 3.2, we get that for any δ > 0 with µ−δ > 0 e(µ−δ)tE|Xt|2≤E|X0|2+ (µ−δ−ν)Zt 0 e(µ−δ)sE|Xs|2ds +Zt 0 γ(s)e−δsds, (3.7) and e(µ−δ)tE|Xt|2≤E|X0|2+ (µ−δ+λ)Zt 0 e(µ−δ)sE|Xs|2ds 7 +Zt 0 γ(s)e−δsds −αZt 0 e(µ−δ)sE||Xs||2ds, (3.8) where ν=(α−λβ2) β2. Now, if µ−ν≤0, it follows from (3.7) that Zt 0 e(µ−δ)sE|Xs|2ds ≤E|X0|2+Rt 0γ(s)e−δsds ν+δ−µ,(3.9) which, together with (3.8), immediately implies Zt 0 e(µ−δ)sE||Xs||2ds ≤1 αhE|X0|2+Zt 0 γ(s)e−δsdsi, +µ−δ+λ αZt 0 e(µ−δ)sE|Xs|2ds, ≤1 αhµ−δ+λ ν+δ−µ+ 1ihE|X0|2+Zt 0 γ(s)e−δsdsi, ≤1 αhµ−δ+λ ν+δ−µ+ 1iE|X0|2+K.(3.10) Consequently, for 0 ≤s≤t, Zt s E||Xu||2du ≤Zt s e(µ−δ)(u−s)E||Xu||2du, ≤e−(µ−δ)sZt 0 e(µ−δ)uE||Xu||2du, thus, Zt s E||Xu||2du ≤1 αhµ−δ+λ ν+δ−µ+ 1iE|X0|2+Ke−(µ−δ)s,(3.11) which, together with (b.2) immediately yields that Zt s E||B(u, Xu)||2 2du ≤2Zt s k2(u)E||Xu||2du + 2 Zt s ψ(u)2du ≤2 sup u∈[s,t) k2(u)Zt s E||Xu||2du + 2 Z+∞ 0 ψ(u)2du ≤2ϕZt s E||Xu||2du + 2ξ therefore, Zt s E||B(u, Xu)||2 2du ≤Ce−(µ−δ)s+ζ, (3.12) 8 where k1is a positive constant, C=C(δ, X0) = 2ϕ αhµ−δ+λ ν+δ−µ+ 1iE|X0|2+Kand ζ= 2ξ. On the other hand, if µ−ν > 0, it is always possible to choose a suitable δ > 0 such that ν−δ > 0. Then, by applying Itˆo’s lemma to the strong solution Xt, it is easy to deduce e(ν−δ)tE|Xt|2≤E|X0|2+ (ν−δ+λ)Zt 0 e(ν−δ)sE|Xs|2ds, +Zt 0 γ(s)e−(µ−ν+δ)sds −αZt 0 e(ν−δ)sE||Xs||2ds, ≤E|X0|2+ (ν−δ+λ)Zt 0 e(ν−δ)sE|Xs|2ds, +Zt 0 γ(s)e−δsds −αZt 0 e(ν−δ)sE||Xs||2ds. (3.13) Noticing that, in this case, the parameter τin theorem 3.2 turns out to be ν, (3.13) yields αZt 0 e(ν−δ)sE||Xs||2ds ≤E|X0|2+ (ν−δ+λ)Zt 0 e−δsds +K, and we can argue in a similar manner as we did previously. Hence our claim is proved. Step 2: We claim that there exists a positive constant M > 0 such that Esup 0≤t<∞ |Xt|2≤M. Indeed, Itˆo’s formula implies |Xt|2− |X0|2= 2 Zt 0 < A(s, Xs), Xs> ds +Zt 0 trB(s, Xs)QB(s, Xs)∗ds, + 2 Zt 0 < Xs, B(s, Xs)dWs> . (3.14) On the other hand, from Burkholder-Davis-Gundy’s inequality, we get for any T∈R+ 2Ehsup t∈[0,T ]Zt 0 < Xs, B(s, Xs)dWs>i, ≤K1EhZT 0 |Xs|2||B(s, Xs)||2 2ds1 2i, ≤K1Ensup 0≤s≤T |Xs|hZT 0 ||B(s, Xs)||2 2dsi1 2o, 9