On Exponential Stability Criteria of Stochastic Partial Differential Equations

ON EXPONENTIAL STABILITY CRITERIA OF STOCHASTIC PARTIAL DIFFERENTIAL EQUATIONS TOM´ AS CARABALLO Dpto. Ecuaciones Diferenciales y An´alisis Num´erico Universidad de Sevilla. Apartado de Correos 1160 41080-SEVILLA, SPAIN e-mail: [email protected] KAI LIU Department of Mathematics University of Wales Swansea Singleton Park, Swansea SA2 8PP, UK e-mail: [email protected] ABSTRACT Some criteria for the mean square and almost sure exponential stability of nonlinear stochastic partial differential equations are shown in this paper. In particular, the main results obtained in Caraballo and Real (1994) are improved, since the new coercivity condition introduced in this work permits the state independent term γto be time dependent and nonnegative but of subexponential growth, while in Caraballo and Real (1994) this parameter is required to be constant and nonpositive. Several examples are studied to illustrate the theory. AMS Classification: 60H15, 34K40. Keywords: Stochastic partial differential equations; Almost sure stability; Mean square stability. 1 1. Introduction and Preliminaries The main aim of this paper is to establish some criteria for the mean square and almost sure exponential stability of a class of nonlinear stochastic partial differential equations of monotone type. In fact, a coercivity condition, extending the one considered by Chow (1982) and Caraballo and Real (1994), is introduced and will play the role of a stability criterion. To be precise, under the coercivity condition (Theorem 1.2 below) from Caraballo and Real (1994), almost sure exponential stability of solutions is obtained, while in Chow (1982) pathwise asymptotic stability is proved. However, as we will explain later, coercivity criteria from Caraballo and Real (1994) 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 nonautonomous stochastic differential equations in Hilbert spaces. For this purpose, let us first state some basic notations and notions (mainly from Caraballo and Real (1994) and Chow (1982)). Let Vbe a Banach space and H,Kreal, separable Hilbert spaces such that V ,→H≡H0,→V0, where the injections are continuous and dense. Let k·k,| · | and k·k∗denote the norms in V,Hand V0respectively, <·,·>the duality product between V0and V, (·,·) the inner product in H, and βa constant such that |x| ≤ βkxk,∀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. Consider the following nonlinear stochastic diffusion equation: Xt=X0+Zt 0 A(s, Xs)ds +Zt 0 B(s, Xs)dWs(1.1) where A(t, ·) : V→V0is a family of nonlinear operators defined a.e.t. satisfying A(t, 0) = 0 for all t∈R+; and where B(t, ·) : V→ L(K, H), the family of all bounded linear operators from Kinto H, satisfies 2 (b.1) B(t, 0) = 0; (b.2) There exists k > 0 such that kB(t, y)−B(t, x)k ≤ kky−xk,∀x, y ∈V, a.e.t; (b.3) t∈(0, T)→B(t, x)∈ L(K, H) is Lebesgue-measurable ∀x∈V,∀T > 0. Definition 1.1. Let (Ω,F,{Ft}, P) be the stochastic basis and WtaK-valued Wiener process with covariance operator Q. Suppose that X0is an H-valued random variable such that E|X0|2<∞. A stochastic process Xtis said to be a strong solution on Ω to the SDE (1.1) for t∈[0, T] if the following conditions are satisfied: (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), and satisfying EZT 0kXtkpdt < ∞. Here C(0, T;H) denotes the space of all continuous functions from [0, T] to H; (c) Equation (1.1) is satisfied for every t∈[0, T] with probability one. If Tis replaced by ∞,Xtis called a global strong solution of (1.1). As we are mainly interested in stability analysis, one always assumes that for each H-valued random variable X0with E|X0|2<∞, there exists a global strong solution to (1.1). In this situation, it is reasonable to assume the following (see Pardoux (1975)) (a.1) (Coercivity). There exist α > 0, p > 1 and λ,γ∈Rsuch that: 2< A(t, x), x > +kB(t, x)k2 2≤ −αkxkp+λ|x|2+γ, ∀x∈V, a.e.t. where k·k2denotes the Hilbert-Schmidt norm of nuclear operator, i.e., kB(t, x)k2 2=tr(B(t, x)QB(t, x)∗); 3 (a.2) (Boundedness). There exists c > 0 such that kA(t, x)k∗≤ckxkp−1,∀x∈V, a.e.t; (a.3) (Monotonicity). −2< A(t, x)−A(t, y), x −y > +λ|x−y|2 ≥ kB(t, x)−B(t, y)k2 2,∀x, y ∈V, a.e.t; (a.4) (Hemicontinuity). The map θ∈R7→< A(t, x +θy), z >∈Ris continuous ∀x, y, z ∈V, a.e.t; (a.5) (Measurability). t∈(0, T)7→ A(t, x)∈V0is Lebesgue-measurable ∀x∈V, a.e.t, ∀T > 0. The following stability criterion is proved in Caraballo and Real (1994): Theorem 1.2. Assume conditions (b.1)–(b.3) and (a.1) hold. We also suppose that Xtis a global strong solution to (1.1). Then, there exists r > 0such that E|Xt|2≤E|X0|2e−rt,∀t≥0,(1.2) if either one of the following hypotheses holds: (a) λ < 0,γ≤0, (∀p > 1); (b) λβ2−α < 0,γ≤0, (p= 2). Furthermore, under the same conditions the solution is almost surely stable. That is, there exist positive constants ξ,ηand a subset N0⊂Ωwith P(N0)=0such that, for each ω6∈ N0, there exists a positive random number T(ω)such that the following holds: |Xt(ω)|2≤η|X0|2e−ξt,∀t≥T(ω). However, when the time variable does appear in the operators A(t, ·) and B(t, ·) in an explicit way or the term γis finally positive so that neither hypothesis (a) nor (b) holds, this criterion cannot be applied (see the examples in Section 3). In the following section, we shall improve this theorem. 4 2. The Main Results In this section, we shall prove the mean square and almost sure exponential stability of the solutions to (1.1). Before introducing the coercivity condition which will guarantee such results, we are going to exhibit two simple examples of one-dimensional linear Itˆo equations in order to motivate the subexponential growth imposed on the state independent term appearing in such a condition: Examples 2.1. First, assume Xtsatisfies the following dXt=−pXtdt + (1 + t)−qdWt, t ≥0 with initial data X0= 0, where p,q > 0 are two positive constants and Wtis a onedimensional standard Brownian motion. Let <·,·>denote the standard inner product in Rand we set A(t, x) = −px, B(t, x) = (1 + t)−q. It easily follows that 2< A(t, x), x > +kB(t, x)k2=−2px2+ (1 + t)−2q,(2.1) and, consequently, Theorem 1.2 can not be applied to this example since (1 + t)−2q>0, for all t≥0, and so one cannot find a γ≤0 which satisfies (a.1). However, it is easy to obtain the explicit solution Xt=e−pt Zt 0 eps ·(1 + s)−qdWs≡e−ptMt, t ≥0. Noticing the law of the iterated logarithm lim sup t→∞ Mt √2< Mt>log log < Mt>= 1 a.s. and lim sup t→∞ log ³Rt 0e2ps(1 + s)−2qds´ t= 2p, we get Lyapunov exponent lim sup t→∞ 1 tlog |Xt|= 0 a.s. 5 which means that almost all the sample paths of the solution will not tend to zero exponentially. Next, suppose Ytsatisfies dYt=−pYtdt +e−qtdWt, t ≥0 with initial data Y0= 0, and p,qboth are positive constants. Assume A(t, x) = −px and B(t, x) = e−qt, then it is easy to deduce 2< A(t, x), x > +kB(t, x)k2=−2px2+e−2qt,(2.1) and again Theorem 1.2 cannot be applied. However, the explicit solution is now given by Yt=e−pt Zt 0 e(p−q)sdWs≡e−ptNt, t ≥0. Taking into account again the law of the iterated logarithm for the process Ntand lim sup t→∞ log ³Rt 0e2(p−q)sds´ t= 2(p−q), we can obtain Lyapunov exponent lim sup t→∞ 1 tlog |Yt|=−q a.s. That is, the solution is almost surely exponentially stable. Therefore, if the term γappearing in condition (a.1) is permitted to be nonnegative and time dependent, a polynomial decay of such a term is not sufficient, in general, to ensure exponential stability of the solutions. However, the solution could be exponentially stable provided the term tends to zero with an exponential decay. Bearing these examples in mind, we can now formulate our stability hypothesis. Once again, we consider the stochastic diffusion equation (1.1) where A(t, ·) : V→V0is supposed to be a measurable family of nonlinear operators defined a.e.t. and B(t, ·) : V→ L(K, H) a measurable family of operators. Note that, at the moment, we do not assume A(t, 0) = 0 and B(t, 0) = 0, t∈R+, as in Caraballo and Real (1994). 6 The following coercivity condition (CC) will play a key role in our stability result: 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 > +kB(t, v)k2 2≤ −αkvkp+λ|v|2+γ(t)e−µt, v ∈V, (2.2) where p > 1 and, for arbitrary δ > 0, γ(t) satisfies γ(t) = o(eδt), as t→ ∞, i.e., limt→∞ γ(t)/eδt = 0. Remark 2.1. Observe that, owing to the continuity and subexponential growth of the term γ(t)e−µt, there exists a positive constant ˜γsuch that γ(t)e−µt ≤˜γfor all t∈R+. As a consequence, (2.2) implies (a.1) (by replacing γby ˜γ), i.e., this assumption is compatible with the existence of the strong solutions to (1.1). Theorem 2.2. Assume conditions (CC), (b.2) and (b.3) hold. Then, if Xtis a global strong solution to the equation (1.1), there exist constants τ > 0,C > 0such that E|Xt|2≤C·e−τt,∀t≥0,(2.3) if either one of the following hypotheses holds (i) λ < 0,(∀p > 1); (ii) λβ2−α < 0,(p= 2). Proof. We only show case (ii). Case (i) can be proved similarly. Firstly, we can choose δ > 0 small enough such that µ−δ > 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. (2.4) Now, since Rt 0e(µ−δ)s< Xs, B(s, Xs)dWs>,t∈R+, is a continuous martingale, it follows that E³Zt 0 e(µ−δ)s< Xs, B(s, Xs)dWs>´= 0, t ∈R+. 7 Therefore, condition (2.2) 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, (2.5) where ν= (α−λβ2)/β2. If µ−ν≤0, it follows immediately e(µ−δ)tE|Xt|2≤E|X0|2+Zt 0 γ(s)e−δsds, which means that there exists a positive constant k=k(δ)>0 such that E|Xt|2≤³E|X0|2+k(δ)´e−(µ−δ)t. On the other hand, if µ−ν > 0, we can choose δ > 0 small enough such that µ−ν−δ > 0. Then, from (2.5) and Gronwall’s lemma one can obtain e(µ−δ)tE|Xt|2≤³E|X0|2+Zt 0 γ(s)e−δsds´et(µ−δ−ν), and, once again, there exists a positive constant k(δ)>0 such that E|Xt|2≤³E|X0|2+k(δ)´e−νt. Theorem 2.3. Assume the hypotheses in Theorem 2.2 hold. Then there exist positive constants M,εand a subset N0⊂Ωwith P(N0)=0such that, for each ω6∈ N0, there exists a positive random number T(ω)such that |Xt|2≤M·e−εt,∀t≥T(ω).(2.6) Proof. We only prove case (ii) as in the last proof. We shall split our proof into several steps, as follows. Step 1. We claim that there exists C > 0, independent of t∈R+, such that Zt s EkB(u, Xu)k2 2du ≤Ce−τs,0≤s≤t. (2.7) 8 Indeed, applying Itˆo’s formula to (1.1) as in Theorem 2.2, 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 (2.8) and e(µ−δ)tE|Xt|2≤E|X0|2+ (µ−δ+λ)Zt 0 e(µ−δ)sE|Xs|2ds +Zt 0 γ(s)e−δsds −αZt 0 e(µ−δ)sEkXsk2ds, (2.9) where ν= (α−λβ2)/β2. Now, if µ−ν≤0, it follows from (2.8) that Zt 0 e(µ−δ)sE|Xs|2ds ≤E|X0|2+Rt 0γ(s)e−δsds ν+δ−µ(2.10) which, together with (2.9), immediately implies Zt 0 e(µ−δ)sEkXsk2ds ≤1 α·E|X0|2+Zt 0 γ(s)e−δsds¸ +µ−δ+λ αZt 0 e(µ−δ)sE|Xs|2ds ≤1 α·µ−δ+λ ν+δ−µ+ 1¸·E|X0|2+Zt 0 γ(s)e−δsds¸ ≤1 α·µ−δ+λ ν+δ−µ+ 1¸£E|X0|2+k(δ)¤.(2.11) Consequently, for 0 ≤s≤t, Zt s EkXuk2du ≤Zt s e(µ−δ)(u−s)EkXuk2du ≤e−(µ−δ)sZt s e(µ−δ)uEkXuk2du ≤1 α·µ−δ+λ ν+δ−µ+ 1¸£E|X0|2+k(δ)¤e−(µ−δ)s(2.12) which, together with (b.2) and (2.2), immediately yields that Zt s EkB(u, Xu)k2 2du ≤2Zt s EkB(u, Xu)−B(u, 0)k2 2du + 2 Zt s EkB(u, 0)k2 2du ≤k1Zt s EkXuk2du +k2Zt s γ(u)e−µudu ≤C(δ)e−(µ−δ)s,(2.13) 9 where |h(t)|denotes the absolute value of h(t). Thus, if h(t) is of subexponential type as above, the hypotheses in Theorems 2.4 and 2.5 are fulfilled by taking σ= 1/2, provided that (2r0+k2)β2<2. Lastly, let us simply come back to an example investigated in Caraballo and Real (1994). Example 3.2. Let D= [0,1] and 2 < p < +∞,r > 0, and consider the following:            dXt(x) =h∂ ∂x³¯¯¯ ∂Xt(x) ∂x ¯¯¯ p−2∂Xt(x) ∂x ´−a(x)Xt(x)idt +g(Xt(x))dWt, t > 0, x ∈D X0(x) =x0(x), x ∈D, Xt(0) = Xt(1) = 0, a.s. (3.6) where a∈L∞(D) satisfies a(x)≥˜a > 0 a.s., x∈Dand g:R→Ris Lipschitz continuous with constant k > 0 such that k2<2˜aand g(0) = 0. Wtis a standard real Wiener process. Let H=L2(D), V=W1,p 0(D) be the Sobolev space with elements satisfying the above boundary conditions. At the moment, A(t, u) is nonlinear, B(t, u) = g(u),for all u∈V. It is easy to check that in this case (2.2) holds with γ(s) = 0, λ=−² < 0, p > 2, α= 2, where ² > 0 is such that k2<2˜a−². Using Theorem 2.2, we easily obtain the required exponential stability. 4. Remarks and conclusions We have proved some results which, in particular, extend the theory developed by Caraballo and Real (1994). In fact, our results can be applied to a number of examples where the criteria in that paper do not hold, since the coercivity condition assumed there requires a uniform bound on the operators. We no longer require the condition A(t, 0) = B(t, 0) = 0 from Caraballo and Real (1994); nevertheless, even in this case (when Xt≡0 is solution to (1.1)) our theory improves that which obtains exponential stability in the mean square and almost surely of the trivial solution to (1.1). However, the results proved in Section 2 are stronger still. Indeed, what we have shown is that, under the assumptions 16 in Theorem 2.2 (or Theorem 2.4) the strong solution to (1.1) exponentially converges in the mean square (and almost surely) to zero even if Xt≡0 is not a solution of the equation (1.1). Acknowledgements The authors would like to thank the referees for their very helpful comments which greatly improved this paper. Also, they wish express their thanks to Dr. James C. Robinson for his valuable comments on this paper. The research of T. Caraballo has been partially supported by D.G.I.C.Y.T (Spain) Proyecto No. PB95-1242. One of the authors, K. Liu would also like to acknowledge the support of EPSRC Grant GR/K 70397. References T. Caraballo and J. Real, On the pathwise exponential stability of non-linear stochastic partial differential equations, Stoch. Anal. Appl. 12(5) (1994) 517-525. P. L. Chow, Stability of nonlinear stochastic evolution equations, J. Math. Anal. Appl. 89 (1982) 400-419. U. G. Haussmann, Asymptotic stability of the linear Itˆo equation in infinite dimension, J. Math. Anal. Appl. 65 (1978) 219-235. X. R. Mao, Exponential stability of stochastic differential equations, Marcel Dekker, Inc. (1994). E. Pardoux, Equations aux d´eriv´ees partielles stochastiques nonlin´eaires monotones. Thesis, Universit´e Paris Sud, (1975). 17