Stabilisation of linear PDEs by Stratonovich noise

Stabilisation of linear PDEs by Stratonovich noise Tom´as Caraballo a,∗and James C. Robinson b aDpto. Ecuaciones Diferenciales y An´alisis Num´erico, Universidad de Sevilla, Apdo. Correos 1160, 41080–Sevilla, Spain. bMathematics Institute, University of Warwick, Coventry CV4 7AL, U.K. Abstract Some results concerning the stability and stabilisation of stochastic linear partial differential equations in the sense of Stratonovich are proved. The main result ensures that a deterministic linear PDE can be stabilised by adding a suitable Stratonovich noise if and only if the linear partial differential operator has negative trace. Key words: exponential stability, stabilization, Ito’s noise, Stratonovich’s noise, stochastic PDE. 1991 MSC: 35R10, 35B40, 47H20, 58F39, 73K70 1 Introduction and statement of the problem The use of stochastic partial differential equations in Physics, Chemistry, Biology, Economics, Engineering, etc, is widespread. The addition of random elements (noise) is based on the assumption that such equations are a better model of reality than their deterministic counterparts. Depending on the situation being modelled one can find arguments justifying either of the canonical choices of noise (Itˆo or Stratonovich). We will not discuss this is detail here, but will emphasise that these different types of noise can produce very different long-time behaviour of solutions (see Caraballo & Langa [6]). A fundamental question is the following: assuming that the real world is actually non-deterministic, are deterministic models good approximations? If the answer is affirmative, then the use of such models would be justified, but if the answer is negative, then in some situations the addition of noise could ∗Corresponding author Email addresses: [email protected] (Tom´as Caraballo), [email protected] (James C. Robinson). Preprint submitted to Systems & Control Letters August 29th, 2003 produce dramatic changes in the behaviour. Here we consider the long-time behaviour of solutions, and investigate the stabilising (or destabilising) effect of the addition of noise. In the finite-dimensional context, there is a wide literature about such problems (see Arnold [3], Arnold et al. [4], Mao [13], Scheutzow [15], ...). Many results on the stabilisation and destabilisation produced by both Itˆo and Stratonovich noise have been obtained, and these have also been applied to construct feedback stabilisers (an important task in control problems). Although in each particular situation one or other choice of the noise may be more appropriate, it is stabilisation by Stratonovich noise that is more significant. This is the main motivating reason for this paper, and we now explain this in more detail. Consider the linear n-dimensional ordinary differential equation ˙x=Ax, (1) where Acould have unstable directions, and the following stochastic versions of this equation corresponding to the two different interpretations of the stochastic integral dx=Ax dt+σx ◦dW(t) (Stratonovich) (2) dy=Ay dt+σy dW(t),(Ito) (3) where W(t) is a standard Wiener process on the complete probability space (Ω,F,P).The initial value problems for (2) and (3), i.e., to find the solutions of these equations satisfying x(0) = x0, y(0) = y0, can be solved explicitly, since their solutions are given by x(t) = eσW (t)exp(tA)x0and y(t) = e−σ2 2t+σW (t)exp(tA)y0. Taking into account the properties of the Wiener process (see e.g. Arnold [1]), it is easy to see that for σlarge enough the zero solution is exponentially stable for the Itˆo equation (with probability one), while the same is not true for the Stratonovich equation. This seems to imply that Itˆo noise has a more profound stabilising effect than Stratonovich noise. However, this argument is somewhat misleading. Indeed, Eq. (1) can be obviously stabilised by using a simple deterministic feedback control, i.e., the new equation ˙x=Ax +λx (4) becomes exponentially stable provided λ < 0 and |λ|is large enough. Similarly, it could be stabilised with the periodic control λ(t)x, ˙x=Ax +λ(t)x, where λ(t) = λ0+ sin t, with λ0<0 and |λ0|large enough. The function sin t is, in some sense, a mean-zero function. The same is true with faster meanzero periodic fluctuations: stabilisation takes place because of the systematic 2 dissipative term λ0xin the equation, while the mean-zero property means that the other term has no influence on the asymptotic behaviour. We can write such an equation in the form ˙x=Ax +λ0x+x˙ Wε(t) where ˙ Wε(t) denotes the zero mean periodic term and may be considered as a physically realistic approximation of the ideal white noise ˙ W(t). It is therefore not surprising that the same result is true in the limit when the regular meanzero fluctuations tend to a mean-zero white noise, and the systematic term λ0xis still present. Now, it is well known that in such a limit (more precisely in all cases where there are rigorous results concerning the Wong-Zakai [17] approximation of a stochastic equation by a random equation with regularised noise) the correct stochastic interpretation for the equation is the Stratonovich one (see, e.g., Sussmann [16] for a more detailed analysis) dx=Ax dt+λ0xdt+σx ◦dW(t) where σ > 0 describes the intensity of the noise. While it should be intuitively clear that this equation is exponentially stable when λ0<0 is sufficiently small, a rigorous proof follows from the Itˆo formula, since as we previously mentioned an explicit form for the solution is x(t) = eλ0t+σW (t)exp(At)x(0). Notice that this result is independent of σ. However, the previous Stratonovich equation can be rewritten in its equivalent Itˆo form: dx=Ax dt+λ0xdt+σx dW(t) + σ2 2xdt. If we choose a white noise with intensity such that σ2 2=−λ0, we arrive at the Itˆo equation dx=Ax dt+σx dW(t).(5) As a consequence of this easy analysis, it is clear that this equation is exponentially stable with probability one for σlarge enough. In general, the “moral” of this example is that Itˆo equations with multiplicative noise correspond to the limit of deterministic equations with a mean-zero fluctuating control plus a stabilising systematic control. So, the fact that an Itˆo equation such as (5) is exponentially stable, even if the equation (1) is not, should not be much more surprising than the fact that Eq. (4) with sufficiently large λ < 0 is exponentially stable; it is only that the mathematics required for a proof is more elaborate. There is a non-trivial literature on stabilisation by Stratonovich noise, with both mathematical and engineering contributions (see [3],[4] and the references therein). Since such a noise is like a periodic zero-mean feedback control, its stabilising effect is unexpected and very intriguing. In the finite-dimensional 3 case, Arnold and his collaborators have proved that the linear differential system (1) can be stabilised by the addition of a collection of multiplicative noisy terms, dx=Ax dt+ d X i=1 Bix◦dWi(t),(6) where Wiare mutually independent Wiener process and Biare suitable skewsymmetric matrices, if and only if tr A < 0.(7) [Note that the form of the noise is more complex than just a single multiplicative term of the form σx ◦dWt.] The corresponding problem for linear partial differential equations has been open for a long time, perhaps because one avenue would be to follow a similar approach but in the infinite-dimensional case, e.g. by proving a version of the celebrated Oseledec Multiplicative Ergodic Theorem for infinite-dimensional spaces. In this paper we use a very simple argument to show that one can obtain a similar stabilisation result for a linear PDE du/dt=Au with a finite sum of Stratonovich terms as in (6). Essentially we will reduce the problem to a finite-dimensional one and then apply the result of Arnold et al. Although our main aim in this paper is to find out conditions under which we can stabilise an unstable linear PDE by using suitable linear Stratonovich noise 1, our analysis will be more complete. Indeed, we consider a linear evolution equation on a separable Hilbert space Hgiven by du dt=Au, (8) where Ais a linear (unbounded) operator, i.e., A:D(A)⊂H→H, and consider the stochastically perturbed evolution equation du=Au dt+ N X i=1 Biu◦dWi,(9) where Bi:D(Bi)⊂H→Hare linear operators and Wiare mutually independent Wiener process on the same probability space (Ω,F,P). In Section 2 we first prove that the stability of (8) and (9) are equivalent if the operators Biand Aare mutually commuting and satisfy other additional assumptions. It is remarkable that if the noise is considered in the Itˆo sense it may produce stabilisation or even destabilisation under the same assumptions (see Caraballo & Langa [6] for a detailed analysis). Section 3 gives some sufficient conditions that ensure the stability of the deterministic equation (8) 1The reader is referred to Caraballo et al. [7], [5], Caraballo & Langa [6], Kwiecinska [11], Leha et al. [12] for similar results but using noisy terms in the sense of Itˆo. 4 is transferred to the stochastic equation (9) when the operator Adoes not commute with some Bi.Finally, in Section 4, we will prove that (9) becomes exponentially stable with probability 1 for suitable operators Biif and only if the trace of Ais negative, an infinite-dimensional analogue of the results of Arnold et al. 2 Stability of stochastic PDEs: the commutative case In this section we will establish a result concerning the exponential stability of the null solution to a linear stochastic PDE. Roughly, we will show that the presence of linear (unbounded) operators in the diffusion part do not modify the exponential stability of the null solution provided that they commute with the linear operator in the drift. To start with, we will consider the problem in the Itˆo formulation, so that we can apply a result due to Da Prato & Zabczyk [8] which ensures the equivalence of the stochastic PDE to a nonautonomous deterministic one depending on a random parameter, i.e. a random PDE. Then, we will transform our Stratonovich model into its equivalent 2Itˆo one and will apply this result. Let us consider    du=Au dt+Pd k=1 BkudWk, u(0) = u0∈H. (10) where A:D(A)⊂H→H,Bk:D(Bk)⊂H→H,k= 1,2,···, d are generators of C0−semigroups SA(t) and Skrespectively, and W1,··· , Wdare independent real Wiener processes. We will need the following additional assumptions. The operators B1,···, Bdgenerate mutually commuting C0−groups Sk (11) D(B2 k)⊃D(A) for k= 1,···, d and d \ k=1 D((B∗ k)2) is dense in H, (12) where B∗ kdenotes the adjoint operator of Bk; C=A−1 2 d X k=1 B2 kgenerates a C0−semigroup SC(13) [Note that D(C) = D(A).] 2This equivalence has been proved by Kunita [10] for suitable partial differential operators. We implicitly assume that we are considering this case. It is undoubtedly an important task to develop a general theory of stochastic PDEs in the Stratonovich sense. 5 Now, given a fixed realisation of our Wiener processes Wk(t, ω), ω ∈Ω, in order to solve (10) we define Uω(t) = d Y k=1 Sk(Wk(t, ω)) and v(t) = U−1 ω(t)u(t), t ≥0,(14) and consider the equation    ˙v(t) = U−1 ω(t)CUω(t)v(t) v(0) = u0,(15) which is a deterministic equation depending on the parameter ω. The following result, along with the definition of a strong solution, can be found in Da Prato and Zabczyk [8] Proposition 2.1 Assume conditions (11)-(13). If uis a strong solution to (10), then the process v(t, ω)defined by (14) satisfies (15). Conversely, if v is a predictable process whose trajectories are continuously differentiable and satisfy (15) P-a.s., then the process u(t, ω) = Uω(t)v(t, ω)takes values in D(C) P-a.s and for almost all t, and is a strong solution to (10). Remark 2.2 One can also find in Da Prato & Zabczyk [8] a sufficient condition ensuring solvability of (15) which could be useful in applications (see [8, pp. 177-179] for more details). Now, we consider our original Stratonovich version of the problem,    du=Au dt+Pd k=1 Bku◦dWk, u(0) = u0∈H. (16) To ensure existence of solutions to this problem, we can consider its equivalent Itˆo version    du= (A+1 2Pd k=1 B2 k)udt+Pd k=1 BkudWk, u(0) = u0∈H. (17) If we now assume conditions (11), (12), and the following (18) instead of (13), C=A+1 2 d X k=1 B2 k,generates a C0−semigroup SC,(18) [we still have D(C) = D(A)] we can ensure (thanks to Proposition 2.1) that our problem (16) can be written in an equivalent way as    ˙v(t) = U−1 ω(t)AUω(t)v(t) v(0) = u0,(19) We now prove the following result which, in particular, generalises a similar one in Caraballo and Langa [6]. 6 Theorem 2.3 In addition to assumptions (11), (12), and (18), suppose that Acommutes with each Sk(t).Then the strongly continuous semigroup SA(t) generated by Ais exponentially stable, i.e. there exist M0, γ > 0such that |SA(t)| ≤ M0e−γt for all t > 0,if and only if there exist α, C > 0and Ω0⊂Ω with P(Ω0) = 0 such that for ω /∈Ω0there exists T(ω)>0such that the following holds for the solution of (16), |u(t)| ≤ C|u0|e−αt for t≥T(ω)almost surely. Proof. ⇒Let us denote by u(t) = u(t, ω; 0, u0) the solution of (16) for u0∈D(A), and by v(t) = v(t, ω; 0, u0) the corresponding solution to (19), i.e. v(t) = U−1 ω(t)u(t). But, owing to the commutativity of operator Aand operators Sk, the problem (19) can be written now as    ˙v(t) = Av(t) v(0) = u0∈D(A),(20) whose solution is given by v(t) = SA(t)u0, so we have an explicit expression for our solution u(t): u(t) = u(t, ω; 0, u0) = Uω(t)SA(t)u0.(21) Now, since we know that SA(t) is exponentially stable, there exist M0, γ > 0 such that |S(t)| ≤ M0e−γt for all t > 0. It is straightforward to bound |u(t)|as follows. First, we notice that (see e.g. Pazy [14]) there exists Mk>0, bk∈R such that |Sk(t)| ≤ Mkebk|t|,k= 1,···, d. Thus, |u(t)| ≤ |Uω(t)SA(t)u0| ≤ |Uω(t)||SA(t)||u0| ≤MM1···Md|u0|exp (−γt + d X k=1 bk|Wk(t)|) ≤C|u0|exp (t −γ+ d X k=1 bk|Wk(t)| t!),(22) and as lim t→+∞ |Wk(t, ω)| t= 0,for all k= 1,··· , d, (23) there exists Ω0⊂Ω with P(Ω0) = 0 such that if ω /∈Ω0,then lim t→+∞ γ− d X k=1 bk|Wk(t, ω)| t!=γ, and, there exists T(ω) such that for all t≥T(ω) γ− d X k=1 bk|Wk(t, ω)| t≥γ 2. 7 Thus, this part of the proof is complete if we set α=γ/2. If we now consider u0∈Hwe can argue by approximation since D(A) in dense in H. ⇐We omit the converse since the proof is similar taking into account now that v(t) = U−1 ω(t)u(t), the properties of strongly continuous semigroups and the Wiener processes. 3 Stability of stochastic PDEs: the noncommutative case Notice that under our commutativity assumptions in the previous section, the deterministic problem is exponentially stable if and only if the stochastically perturbed equation has the same property. However, an immediate question arises. What happens if no commutativity holds? In this case we will be able to prove some sufficient conditions for the exponential stability of the zero solution. Theorem 3.1 Assume that (11),(18) hold, and there exists γ > 0such that |SA(t)| ≤ e−γt for all t≥0. If in addition we assume that Uω(t)is unitary for all t≥0and almost all ω(i.e., U∗ ω(t) = U−1 ω(t)) then, there exists α > 0such that the following holds for the solution of (16), |u(t)| ≤ |u0|e−αt for t≥0almost surely. Proof. First, we notice that an operator Agenerates a strongly continuous semigroup SA(t) satisfying |SA(t)| ≤ eαt,α∈R, if and only if (Au, u)≤α|u|2, for all u∈D(A). We can now consider our problem    ˙v(t) = U−1 ω(t)AUω(t)v(t) v(0) = u0∈D(A), and observe that, thanks to our assumptions, d dt|v(t)|2= 2(U−1 ω(t)AUω(t)v(t), v(t)) ≤2(AUω(t)v(t), Uω(t)v(t)) ≤ −2γ|Uω(t)v(t)|2 =−2γ|v(t)|2. This implies that |v(t)|2≤ |v(0)|2e−2γt,for all t≥0, and as v(t) = U−1 ω(t)u(t), we immediately obtain |u(t)|=|v(t)| ≤ |u0|e−γt for u0∈D(A) and all t≥0. Finally, the result holds true for u0∈Hby an approximation argument. 8 Nevertheless, when the operator Uis not unitary, we can still prove a sufficient condition ensuring this exponential behaviour of solutions. Notice that we do not need the mutual commutativity of the operators Bk. Theorem 3.2 Assume that Agenerates a strongly continuous semigroup SA(t) such that |S(t)| ≤e−γt,for all t≥0and some γ > 0. Assume that Bk:H→H is a linear and bounded operator for k= 1,··· , d such that there exists bk≥0 satisfying bk|u|2≤(Bku, u),∀1≤k≤d, u ∈H. Then for every initial value u0∈D(A)such that u06= 0, the corresponding solution to (16), u(·) = u(·, ω; 0, u0), is defined in the future and satisfies |u(t)|>0for all t > 0almost surely, lim sup t→+∞ 1 tlog |u(t)| ≤ − γ− d X k=1 (|Bk|2−b2 k)!P−a.s. Proof. Take u0∈D(A) such that u06= 0, then it follows from the theory of linear stochastic partial differential equations that the solution u(t) to (16) (which in addition has a continuous version) is defined in the future and |u(t)|>0 for all t > 0, P−a.s. (see Da Prato & Zabczyk [8] p. 190-191). Then, in order to apply Itˆo’s formula, we consider the equivalent Itˆo formulation for our problem (17), i.e. u(t) = u0+Zt 0 Au(s) + 1 2 d X k=1 B2 ku(s)!ds+ d X k=1 Zt 0Bku(s) dWk(s). Thus, it follows (we omit the argument sin the integrals for short) log |u(t)|2(24) ≤log |u0|2+Zt 0 2u, Au +1 2Pd k=1 B2 ku+Pd k=1 |Bku|2 |u|2ds −1 2 d X k=1 Zt 0 4(u, Bku)2 |u|4ds+M(t), ≤log |u0|2+Zt 0 2(u, Au) + Pd k=1 1 2(u, B2 ku) + 1 2|Bku|2−b2 k|u|2 |u|2ds +M(t) ≤log |u0|2+ 2 Zt 0 −γ+ d X k=1 (|Bk|2−b2 k)!ds+M(t) where M(t), owing to our assumptions, is a continuous local martingale given by M(t) = 2 d X k=1 Zt 0 (u(s), Bku(s)) |u(s)|2dWk(s), 9