On Stabilization of Partial Differential Equations by Noise

T. Caraballo, K. Liu and X. Mao Nagoya Math. J. Vol. ? (?), 1–16 ON STABILIZATION OF PARTIAL DIFFERENTIAL EQUATIONS BY NOISE TOM´ AS CARABALLO, KAI LIU and XUERONG MAO §1. Introduction In this work, we shall present an investigation to stabilization of stochastic partial differential equations by noise. In order to motivate our theory, let us firstly analyze the following example from Haussmann [6]. Consider a one-dimensional rod of length πwhose ends are maintained at 0◦and whose sides are insulated. Assume that there is an exothermic reaction taking place inside the rod with heat being produced proportionally to the temperature. The temperature in the rod may be modeled to satisfy              ∂X ∂t =∂2X ∂y2+rX, t > 0,0< y < π, X(t, 0) = X(t, π) = 0, X(0, y) = x0(y), where rdepends on the rate of reaction. If we assume r=r0, a constant, then we can solve X(t, y) = ∞ X n=1 ane−(n2−r0)tsin ny, where x0(y) = P∞ n=1 ansin ny. Hence we obtain exponential stability if n2> r0for all n∈N, or equivalently, r0<1. Observe that, in general, for r0≥1 the trivial solution is not stable. Suppose now that ris random, and assume it is modeled as r0+r1˙w, so that the equation becomes dX =∂2 ∂y2+r0X dt +r1X dw, (∗) Received ??. 1 2T. CARABALLO, K. LIU AND X. MAO where wis a one-dimensional Wiener process. Haussmann proved in [6] that when r0<1 (i.e. the unperturbed system is very stable), the perturbed system (∗) remains pathwise exponentially stable if r2 1<2(1 −r0), i.e., if the perturbation is sufficiently small so that this relation is satisfied. On the other hand, since a new noise term is involved in the perturbed system (∗), a natural problem now arises: as r0>1 (a case where Haussmann’s results fail to be applied), is it possible to deduce any exponential stability results for the perturbed system? As we are going to see, by virtue of some general stabilization results to be derived in Section 3 below, we can actually prove that for arbitrary r0∈R1, if r2 1>2(r0−1), the perturbed system becomes exponentially stable. In other words, the multiplicative noise now plays the role to stabilize unstable deterministic systems. It is also worth pointing out that if r0<1, by employing our results we deduce meanwhile that for arbitrary r1∈R1, the system (∗) is still almost sure exponentially stable. This result greatly improves the original one in Haussmann [6]. Another more important fact we want to mention is that noise can also be used to stabilize some stochastic partial differential equations. Indeed, in the previous situation, i.e., if r0>1 and r1∈R1satisfies r2 1≤2(r0−1), we do not know whether the trivial solution is stable or not. But, if the system is perturbed by another multiplicative noise of the same type, say r2X˙w1, where w1is another one-dimensional Wiener process independent of w, as a consequence of our main result (Theorem 4.1), it can be deduced that the system dX =∂2 ∂y2+r0X dt +r1X dw +r2Xdw1,(∗∗) becomes once more pathwise exponentially stable if r2is chosen large enough. There existed some work on stabilization of stochastic differential equations by noise in finite dimension. The first one is due to Has’minskii [5], who stabilized a system by using two white noise sources. Later, Arnold et al. [1] showed, in particular, that the system ˙x(t) = Ax(t) can be stabilized by zero mean stationary parameter noise if and only if trace A < 0. On the other hand, in the nonlinear case, Scheutzow [11] provided some examples on stabilization and destabilization in the plane, and Mao [9] obtained some results on stabilization and destabilization by Brownian motion for certain class of nonlinear stochastic differential equations. However, as far as we know, there existed no papers devoting themselves to the investigation of STABILIZATION OF PDES BY NOISE 3 the analogous problems in infinite dimension, that is, stabilization of (nonlinear) partial differential equations. In this paper, we shall fill this gap and meanwhile improve some stability criteria in Caraballo and Liu [3], Chow [4] and Haussmann [6]. The organization of this paper is as follows. In Section 2, we shall prove some results in order to stabilize systems modeled by deterministic partial differential equations. Section 3 is devoted to applying the results obtained in Section 2 to analyze, in particular, the exponential stability of a stochastic nonlinear monotone equation which improves a stability result from Caraballo and Liu [3] and Chow [4]. Finally, we shall establish a stabilization result for stochastic partial differential equations in Section 4 which permits us to stabilize the stochastic systems studied in Section 3. §2. Stabilization of deterministic partial differential equations First of all, we introduce the framework where our analysis is going to be carried out. Let Hbe a real, separable Hilbert space and Va real, reflexive and separable Banach space such that V ֒→H≡H′֒→V′, where the injections are continuous and dense. In particular, 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 h·,·i the duality product between V , V ′, and by (·,·) the scalar product in H. Let βbe the constant of the injection V ֒→H, i.e. |x| ≤ βkxk,∀x∈V. Consider the following problem dx(t) = F(t, x(t)) dt x(0) = x0∈H, (1) where F(t, ·) : V→V′,t∈R+, is a family of (nonlinear) operators satisfying F(t, 0) = 0 and the following hypothesis: There exist a continuous function ν(·) and a real number ν0∈R1such that 2hx, F(t, x)i ≤ ν(t)|x|2,∀x∈V,(2) 4T. CARABALLO, K. LIU AND X. MAO where lim sup t→∞ 1 tZt 0 ν(s)ds ≤ν0.(3) Assume that for each x0∈Hthere exists a unique strong solution to problem (1), x(t;x0)∈L2(0, T ;V)∩C(0, T ;H). Observe that, when F(t, ·) satisfies a coercivity condition of the type 2hx, F(t, x)i ≤ −εkxkp+α|x|2, ε > 0, α ∈R1, p > 1,(4) and a monotonicity hypothesis, there exists a unique strong solution to (1) in Lp(0, T ;V)∩C(0, T ;H) (see, for instance, Lions [7]). Note that this coercivity assumption obviously implies (2). Now we can propose the following question: If problem (1) is not stable, can it be stabilized by using a stochastic perturbation of the type g(t, x(t))dw(t)? Here w(t) is (for simplicity) a standard real Wiener process defined on a certain complete probability space (Ω,F, P) with filtration (Ft)t≥0, and g(t, ·) : H→Hsatisfies g(t, 0) = 0 and the following condition |g(t, x)−g(t, y)|2≤λ(t)|x−y|2, t ∈R+, x, y ∈H,(5) where λ(·) is a nonnegative continuous function such that lim sup t→∞ 1 tZt 0 λ(s)ds ≤λ0∈R+.(6) The answer to this question is affirmative for a suitably chosen g. Indeed, consider the following perturbed problem du(t) = F(t, u(t))dt +g(t, u(t))dw(t), t > 0 u(0) = u0∈H. (7) As we are mainly interested in stability properties of the trivial solution to problem (7), we suppose that for each u0∈Hthere exists a unique strong solution to (7) in Ip(0, T ;V)∩L2(Ω; C(0, T ;H)) for all T > 0, and certain p > 1,where Ip(0, T ;V) denotes the space of all V-valued processes u(t), measurable and satisfying ERT 0ku(t)kpdt < ∞. (See, for instance, Pardoux [10] or Caraballo [2] for conditions under which there exists such a unique solution for initial values u0∈L2(Ω,F0, P;H)) STABILIZATION OF PDES BY NOISE 5 Assume V(t, x) : R+×H→R+is a C1,2-positive functional such that for any x∈V,t∈R+,V′ x(t, x)∈V. We define operators Land Qas follows: for x∈V,t∈R+ LV (t, x) = V′ t(t, x)+ < V ′ x(t, x), F(t, x)>+1 2(V′′ xx(t, x)g(t, x), g(t, x)) and QV (t, x) = (V′ x(t, x), g(t, x))2. Theorem 2.1. Assume the solution of (7) satisfies that |u(t)| 6= 0 for all t≥0a.s. provided |u0| 6= 0 a.s.. Let V(x)∈C2(H;R+)and ψ1(t)∈R1, ψ2(t)≥0be two continuous functions. Assume that there exist constants p > 0,γ≥0and θ∈R1such that (a).|x|p≤V(x),∀x∈V; (b). LV (t, x)≤ψ1(t)V(x),∀x∈V, ∀t∈R+; (c). QV (t, x)≥ψ2(t)V2(x),∀x∈V, ∀t∈R+; (d).lim sup t→∞ Rt 0ψ1(s)ds t≤θ, lim inf t→∞ Rt 0ψ2(s)ds t≥2γ. Then the strong solution of equation (7) satisfies lim sup t→∞ log |u(t, u0)| t≤ − γ−θ pa.s. where u0∈His an F0-measurable random vector such that |u0| 6= 0 a.s. In particular, if γ > θ, the solution is almost sure exponentially stable. Proof. Fix u0∈Hsuch that |u0| 6= 0 a.s.. It is easy to deduce by Itˆo’s formula that log V(u(t, u0)) ≤log V(u(0)) + M(t) +Zt 0LV (s, u(s)) V(u(s)) −1 2 QV (s, u(s)) V2(u(s)) ds,(8) where M(t) = Rt 0 1 V(u(s)) (V′ x(u(s)), g(s, u(s)))dw(s). By applying the exponential martingale inequality (e.g., see Lemma 1.1 in [8]), we can deduce Pnω: sup 0≤t≤whM(t)−Zt 0 u 2 1 V2(u(s))QV (s, u(s))dsi> vo≤e−uv 6T. CARABALLO, K. LIU AND X. MAO for any positive constants u, v and w. Assigning ǫ > 0 arbitrarily and taking u=α, v = 2α−1log k, w =kǫ, k = 1,2,3,..., where 0 < α < 1, we then apply the well-known Borel-Cantelli lemma to obtain that there exists an integer k0(ǫ, ω)>0 for almost all ω∈Ω such that M(t)≤2α−1log k+α 2Zt 0 QV (s, u(s)) V2(u(s)) ds for all 0 ≤t≤kǫ,k≥k0(ǫ, ω). Substituting this into (8) and using conditions (b)(c), we see that there exists a positive random integer k1(ǫ) such that almost surely log V(u(t)) ≤log V(u(0)) + 2α−1log k+Zt 0 ψ1(s)ds −1 2(1 −α)Zt 0 ψ2(s)ds for all (k−1)ǫ≤t≤kǫ,k≥k0(ǫ, ω)∨k1(ǫ) which for the preceding ǫ > 0, together with (d), immediately implies that log |u(t)| t≤log V(u(t)) pt ≤1 ptlog V(u(0)) + 2α−1log k+ (θ+ǫ)t−1 2(1 −α)(2γ+ǫ)t. Therefore, lim sup t→∞ log |u(t)| t≤1 ph(θ+ǫ)−(1 −α)(γ+ǫ 2)ia.s. Letting α→0, ǫ→0, we can immediately obtain lim sup t→∞ log |u(t, u0)| t≤ −γ−θ pa.s. 2 As a direct consequence, we immediately obtain the following: Theorem 2.2. Assume the solution of (7) satisfies that |u(t, u0)| 6= 0 for all t≥0almost surely provided |u0| 6= 0 almost surely. In addition to hypotheses (2), (3), (5) and (6), assume the following one holds: (g(t, x), x)2≥ρ(t)|x|4,∀x∈H,(9) STABILIZATION OF PDES BY NOISE 7 where ρ(·)is a nonnegative continuous function such that lim inf t→∞ 1 tZt 0 ρ(s)ds ≥ρ0, ρ0∈R+.(10) Then, if u(t, u0)denotes the solution to (7), it follows lim sup t→∞ 1 tlog |u(t, u0)|2≤ −(2ρ0−ν0−λ0), P −a.s.(11) for any u0∈H. In particular, if 2ρ0> ν0+λ0the equation (7) is almost surely exponentially stable. Proof. The proof follows immediately from Theorem 2.1 by setting V(t, x) = |x|2,x∈H.2 Remarks. 1. Observe that this result can be interpreted as follows. Given the deterministic system (1) satisfying (2)–(3), if a stochastic multiplicative noise, say g(t, u) ˙w(t),appears in the system, the perturbed one (7) becomes exponentially stable with probability one (w.p.1 in the sequel) for suitable g. In particular, one can always stabilize the system (1) by considering linear perturbations of the type g(t, u) = ρ0u , ρ0∈R1. 2. An analogous result holds when g(t, ·) : V→H, assuming the following condition instead of (2)–(6): 2hx, F(t, x)i+|g(t, x)|2≤δ(t)|x|2, t ∈R+, x ∈V,(12) where δ(·) is a nonnegative continuous function such that lim sup t→∞ 1 tZt 0 δ(s)ds ≤δ0∈R.(13) In this case, (11) is replaced by lim sup t→∞ 1 tlog |u(t, u0)|2≤ −(2ρ0−δ0), P −a.s.(14) §3. Applications 8T. CARABALLO, K. LIU AND X. MAO 3.1. A linear example First of all, let us consider the simple linear case described in the beginning of Section 1. We put this problem into our formulation by setting V=H1 0([0, π]), H=L2([0, π]), F(t, u) = ∂2u ∂y2+r0u, u ∈V, g(t, u) = r1u, u ∈H. Now, it is easy to check that Theorem 2.2 can be applied to this problem with ρ0=λ0=r2 1and ν0= 2(r0−1).Thus, one can obtain for the solutions to (∗) lim sup t→+∞ 1 tlog |X(t)| ≤ −(r2 1−2(r0−1)) with probability 1. In other words, we get pathwise exponential stability if r2 1>2(r0−1). Observe that, in particular, when r0<1, the trivial solution to Eq. (∗) is P-almost surely exponentially stable for all r1∈R, which greatly improves the results obtained in Haussmann [6] since the restriction imposed on r1is not necessary here. 3.2. A nonlinear example Now we are going to apply Theorem 2.2 to analyze the pathwise stability of a nonlinear stochastic partial differential equation (which contains the first example as a special case). Consider the following problem previously studied, among others, by Pardoux [10], Caraballo and Liu [3] (and also by Chow [4] in the autonomous case): du(t) = A(t, u(t))dt +B(t, u(t))dw(t) u(0) = u0∈H, (15) where A(t, ·) : V→V′is a family of nonlinear operators defined a.e.t. satisfying A(t, 0) = 0 for all t∈R+;B(t, ·) : V→H, satisfies (b.1) B(t, 0) = 0; (b.2) There exists k > 0 such that |B(t, y)−B(t, x)| ≤ kky−xk,∀x, y ∈V, a.e.t; In [3] it is proved the following result: Theorem 3.1. In addition to (b.1)–(b.2), assume the following coercivity condition: STABILIZATION OF PDES BY NOISE 9 There exist α > 0, p > 1and λ∈R1such that for almost all t∈R+ and all x∈V, 2hx, A(t, x)i+|B(t, x)|2≤ −αkxkp+λ|x|2.(16) Then, there exists r > 0such that E|u(t;u0)|2≤E|u0|2e−rt,∀t≥0, if either one of the following hypotheses holds: (i) λ < 0(∀p > 1); (ii) λβ2−α < 0(p= 2). Furthermore, under the same assumptions the solution is almost surely exponentially stable. That is, there exist positive constants ξ,ηand a subset Ω0⊂Ωwith P(Ω0) = 0 such that, for each ω6∈ Ω0, there exists a positive random number T(ω)such that |u(t, ω;u0)|2≤η|u0|2e−ξt,∀t≥T(ω).2 Observe that, in the practical applications, conditions (i) and (ii) mean that the term containing Bmust be small enough with respect to A. For example, let Obe an open, bounded subset in RNwith regular boundary and let 2 ≤p < +∞. Consider the Sobolev spaces V=W1,p 0(O) , H= L2(O) with their usual inner products, and the monotone operator A:V→ V′defined as hv, Aui=− N X i=1 ZO ∂u(x) ∂xi p−2∂u(x) ∂xi ∂v(x) ∂xi dx+ZO au(x)v(x)dx ∀u, v ∈V , where a∈R1. We also consider B(t, u) = bu , u ∈H, where b∈R1. Finally, let w(t) be a standard real Wiener process. Then, 2hx, A(t, x)i+|B(t, x)|2=−2kxkp+ 2a|x|2+b2|x|2, x ∈V,(17) so (16) holds with equality for α= 2 and λ= 2a+b2. Now, condition (i) requires 2a+b2<0, so a < 0 and b2<−2a. On the other hand, (ii) will hold whenever (2a+b2)β2−2<0, that is, b2<2β−2−2a. Therefore, Theorem 2.2 guarantees exponential stability of paths w.p.1 only for these 16 T. CARABALLO, K. LIU AND X. MAO Xuerong Mao Department of Statistics and Modelling Science, University of Strathclyde, Glasglow G1 1XH, Scotland, United Kingdom [email protected]