# Stabilization for the Stochastic Heat Equation with Boundary Control

Authors: Baolin Ma; Yiyue Sun; Guojie Zheng
Journal: Journal of Mathematics
 (2022)

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## Abstract

This paper investigates the stabilization of an unstable stochastic heat equation. By the backstepping technique, a boundary feedback control is designed to stabilize the unstable stochastic system, including the 2nd-moment and almost-sure exponential stabilization. We also prove that the system with a disturbance in the control channel possesses good robust stability under our control strategy and suitable conditions. Finally, we also provide numerical simulations which illustrate the effectiveness of the theoretical results.


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## Body

## 1. Introduction

From the application point of view, many practical systems are often subjected to stochastic disturbances. A natural and widely acceptable way of describing stochastic factors is using white noises, and such a class of induced systems is called stochastic systems (see [1, 2]). Since the early 1990s, the study of stochastic systems has become a hot topic in control theory. There are many studies on this topic, and we would mention [3–6] for the related work. In particular, Cheng et al. discussed the problem of hidden Markov model-based control for periodic systems subject to singular perturbations in [3]. However, there are very limited studies on the stabilization of stochastic partial differential equations. This paper aims to investigate the stabilization of the stochastic heat equation with boundary control.LetΩ,ℱ,ℱtt≥0,P be a complete filtered probability space, on which a one-dimensional standard Brownian motion Btt≥0 is defined. (The definition of the complete filtered probability space Ω,ℱ,ℱtt≥0,P can be found in page 201 of [1].) We denote by ω a sample point of Ω, by E⋅ the expectation with respect to the probability measure P, and by L2Ω,ℱ0,P;L20,1 the space consisting of all L20,1-valued ℱ0-measurable random variable X⋅ such that EX⋅2<+∞. Let a>0 and f⋅ be a real-valued ℱtt≥0-adapted process in Lℱ∞0,+∞;L∞0,1, which is the space consisting of all L∞0,1-valued ℱtt≥0-adapted bounded process.This paper investigates the following controlled stochastic heat equation driven by a multiplicative noise:(1)dux,t−uxxx,tdt−aux,tdt=ftux,tdBt,x,t∈0,1×0,+∞,u0,t=0,t∈0,+∞,u1,t=Ut,t∈0,+∞,ux,0=u0x,x∈0,1,where the initial datum u0x∈L2Ω,ℱ0,P;L20,1 and U⋅ is the control function. Here and in what follows, we omit the variable ω∈Ω in the defined functions if there is no risk of causing any confusion.For the deterministic case, namely,f⋅≡0, the stabilization of the heat equation has been widely studied (see [7, 8]). However, the study on the stability of stochastic controlled equations has turned out to be a difficult problem, even for linear finite-dimensional stochastic equations. When f⋅≠0, it is well known that stochastic heat equation (1) is unstable if the control U⋅≡0 and the number a is big enough. This point will be discussed in detail in the next section. In this article, we mainly study the stabilization for unstable stochastic heat equation (1). We mention [9] in this direction. The backstepping method is employed to deal with destabilizing terms in system equation (1). The backstepping method is a systematic boundary feedback control approach for partial differential equations (PDEs), which uses an integral transformation to convert an unstable PDE into a stable one. By this technique, the destabilizing terms are eliminated through an invertible integral transformation of the PDEs with boundary feedback. Nowadays, the backstepping method is successfully applied to many types of PDEs. A systematic discussion of this method can be found in the literature [7, 10, 11] and the references therein. The main contribution of the current work is the extension of the backstepping technique to the study of the stabilization of stochastic heat equations, even when there is a disturbance in the control channel.We organize the paper as follows: In Section2, some preliminary results are presented. Sections 3 and 4 are devoted to the main results and their proofs. Section 5 shows the numerical simulation results.


## 2. Preliminary Results

We first define that(2)μk=k2π2,andekx=2sinkπx,k=1,2,3,….Then,ekxk=1∞ constitutes an orthonormal basis of L20,1. Indeed, μkk=1∞ is the set of eigenvalues of the operator −△ with homogeneous Dirichlet boundary conditions, and ekk=1∞ is the set of the corresponding eigenfunctions. We denote by ⋅ and ⋅,⋅ the canonical norm and the inner product of L20,1, respectively, without specific explanation in what follows. It is now useful to provide several definitions of stability that we will use in this paper.Definition 1.
(i)
Stochastic equation (1) is said to be second-moment exponentially stable if there exist positive numbers C and ρ such that for any initial datum u0x∈L2Ω,ℱ0,P;L20,1, the solution to equation (1) satisfies(3)Eux,t2≤Ce−ρtu02,fort≥0.(ii)
Stochastic equation (1) is said to be second-moment asymptotically stable if for any initial datum u0x∈L2Ω,ℱ0,P;L20,1, the solution to equation (1) satisfies(4)limt⟶+∞Eux,t2=0.(iii)
Stochastic equation (1) is said to be almost surely exponentially stable if for any initial datum u0x∈L2Ω,ℱ0,P;L20,1, the solution to equation (1) satisfies(5)limsupt⟶+∞1tlnux,t<0,a.s..Remark 1.
(i)
Here and in what follows, the abbreviation a.s. stands for almost surely in the sense of probabilityP.(ii)
These definitions can be found in [2] (see page 119 and page 127 of [2]).Now, fixing a real numberc, we first discuss the following stochastic heat equation:(6)dwx,t−wxxx,tdt+cwx,tdt=ftwx,tdBt,x,t∈0,1×0,+∞,w0,t=w1,t=0,t∈0,+∞,wx,0=w0x,x∈0,1.It is well known that for anyw0∈L2Ω,ℱ0,P;L20,1, there exists a unique (weak) solution:(7)w∈Cℱ0,+∞;L2Ω;L20,1,for equation (6) (see [12, 13]). Moreover, we have the following result for the solution of equation (6).Lemma 1.
For any initial datumw0∈L2Ω,ℱ0,P;L20,1, the solution to equation (6) satisfies that(8)Ewx,t2≤e−2μ1+c−b2tEw02.Here and in what follows, b=f⋅Lℱ∞0,+∞;L∞Ω.The proof can be found in [14].Remark 2.
By virtue of Lemma1, we can get that equation (6) is second-moment exponentially stable if μ1+c>b2/2.Now, we will introduce the following well-known strong law of large numbers.Lemma 2.
LetXt be a real-valued continuous local martingale vanishing at t=0, and let the process Xt,Xtt be the quadratic variation of Xt. Then,(9)limsupt⟶+∞Xt,Xttt<+∞,a.s..⇒limt⟶+∞Xtt=0,a.s..This lemma can be found in [2]. Here, we omit the detailed proof. With the aid of Lemma 2, we can investigate the almost-sure exponentially stability for equation (6).Lemma 3.
For any initial datumw0∈L2Ω,ℱ0,P;L20,1, the solution to equation (6) satisfies(10)limsupt⟶+∞1tlnwx,t≤−μ1+c,a.s..Proof.
Writeλi=μi+ci=1,2,3,…, where μi is given in equation (2). By Itô’s formula,(11)dlnwx,t2=−2wxx,t2−2cwx,t2wx,t2−ft2wx,t4wx,t4dt+2ftdBt.
Integrating it over the interval0,t, we get(12)lnwx,t2=lnw02+∫0t2fsdBs+∫0t−2wxx,s2−2cwx,s2wx,s2−fs2ds.
This, together with the Poincaré inequality, yields(13)lnwx,t2≤lnw02−2λ1t+2∫0tfsdBs.
Sincef⋅∈Lℱ∞0,+∞;L∞Ω, Mt=∫0tfsdBs is a continuous martingale with the quadratic variation:(14)Mt,Mtt=∫0tfs2ds≤b2t,where b=f⋅Lℱ∞0,+∞;L∞Ω. It, along with equation (13) and Lemma 2, leads to equation (10). Hence, we complete the proof.Remark 3.
Ifμ1+c>0, then equation (6) is almost surely exponentially stable.


## 3. Stability for Stochastic Equation (1)

By Lemmas1 and 3, we have the following observations: if a≥μ1−b2/2 and Ut=0b=f⋅Lℱ∞0,+∞;L∞Ω, equation (1) is not second-moment exponentially stable or almost surely exponentially stable.In this section, we will design a suitable control functionU⋅ to stabilize stochastic heat equation (1) when a≥μ1+b2/2. For this purpose, we define a linear bounded operator as follows:(15)Λ:L20,1⟶L20,1,by setting(16)Λφx=φx−∫0xkx,yφydy,forφx∈L20,1,where the kernel function kx,y is the solution of the following PDE:(17)kxxx,y=kyyx,y+c+akx,y,kx,0=0,kxx,x+kyx,x+ddxkx,x=−c+a,where c is a fixed positive number. Indeed, equation (17) has a unique and twice continuously differentiable solution kx,y over the following domain (see [7, 8]):(18)Σ=x,y∈0,1×0,1;y≤x.Moreover, we have the following lemma.Lemma 4.
The linear operatorΛ defined in equation (16) is bounded from L20,1 to L20,1. Moreover, Λ has a linear bounded inverse Λ−1 from L20,1 to L20,1.The proof can be found in [7]. By the backstepping approach, we now introduce a new variable(19)wx,t=Λux,t,t≥0,and then apply transformation equation (16) to equation (1), where the boundary feedback law is given as follows:(20)Ut=∫01k1,yuy,tdx.It follows from the standard method in the stochastic analysis thatwx,t is a L20,1-valued ℱtt≥0-adapted process and Ut is a real-valued ℱtt≥0-adapted process. According to the well-posedness result for stochastic differential equations (see [12, 13]), system equation (1) admits a unique solution. By this boundary feedback law, we obtain the following result.Theorem 1.
Letc>0 in equation (17). Then, for any initial datum u0∈L2Ω,ℱ0,P;L20,1, the solution to equation (1) with boundary feedback control equation (20) satisfies(21)limsupt⟶+∞1tlnux,t<0,a.s..If we further assume thatμ1+c>b2/2, where b=f⋅Lℱ∞0,+∞;L∞Ω, then there exist positive numbers C and ρ such that for any initial datum u0∈L2Ω,ℱ0,P,L20,1,(22)Eux,t2≤Ce−ρtu02,fort≥0.Proof.
The proof will be organized in two steps as follows:Step 1. We prove equation (21) with the help of Lemma 4.Letwx,t=Λux,t, where the operator is given in equation (16). By direct computations, we can obtain that(23)dwx,t=dux,t−d∫0xkx,yuy,tdy=dux,t−∫0xkx,yuyyy,t+auy,tdydt−ft∫0xkx,yuy,tdydBt,∫0xkx,yuyyy,tdy=kx,xuxx,t−kyx,xux,t+∫0xkyyx,yuy,tdy,and(24)wxxx,t=uxxx,t−ddxkx,xux,t−kx,xuxx,t−kxx,xux,t−∫0xkxxx,yuy,tdy.Therefore,(25)dwx,t−wxxx,tdt+cwx,tdt−ftwx,tdBt,=du−a∫0xkx,yuy,tdydt−ft∫0xkx,yuy,tdydBt,−kx,xuxx,t−kyx,xux,tdt−∫0xkyyx,yuy,tdydt,−uxxx,tdt+ddxkx,xux,tdt+kx,xuxx,tdt+kxx,xux,tdt,+∫0xkxxx,yuy,tdydt+cux,tdt−c∫0xkx,yuy,tdydt,−ftux,tdBt+ft∫0xkx,yuy,tdydBt=0.These, along with equation (20), yield that(26)dwx,t−wxxx,tdt+cwx,tdt=ftwx,tdBt,x,t∈0,1×0,+∞,w0,t=w1,t=0,t∈0,+∞,wx,0=w0x,x∈0,1,wherew0x=Λu0x. By Lemma 3, we obtain that(27)limsupt⟶+∞1tlnwx,t<−μ1+c<0,a.s..It, together with Lemma4 and equation (19), indicates equation (21).Step 2. We prove equation (22).By Lemma1, the solution wx,t to equation (26) satisfies that(28)Ewx,t2≤e−2μ1+c−b2tEwx,02.
This, together with Lemma4, equation (19), and μ1+c>b2/2, yields equation (22). Hence, we complete the proof.


## 4. Robust Stability

In this section, we consider the stability of the perturbed controlled stochastic heat equation:(29)dux,t−uxxx,tdt−aux,tdt=ftux,tdBt,x,t∈0,1×0,+∞,u0,t=0,t∈0,+∞,u1,t=Ut+dt,t∈0,+∞,ux,0=u0x,x∈0,1,where the initial datum u0∈L2Ω,ℱ0,P;L20,1, a≥μ1+b2/2, U⋅ is the control function, and d⋅ is a real-valued ℱtt≥0-adapted process, which can be regarded as a disturbance in the control channel. Does the control in equation (20) work well for equation (29) with a disturbance d⋅ in the control channel? It should be pointed out that there are many studies on this topic for the deterministic parabolic system (see [15, 16]). We also mention [8, 17–22] for the related work. However, we can only find little work concerned with this problem for stochastic PDEs. Our objective is to stabilize the zero equilibrium of the unstable stochastic heat equation with a disturbance d⋅.


### 4.1. An Auxiliary Result

To study the problem of robust stability, we introduce the following nonhomogeneous Dirichlet boundary value problem:(30)dvx,t−vxxx,tdt+cvx,tdt=ftvx,tdBt,x,t∈0,1×0,+∞,v0,t=0,t∈0,+∞,v1,t=gt,t∈0,+∞,vx,0=v0x,x∈0,1,where the initial datum v0∈L2Ω,ℱ0,P;L20,1, and the real-valued ℱtt≥0-adapted process g⋅ satisfies(31)Hithere exists a positive numberM>0such thatEgt2<Mfort∈0,+∞;iilimt⟶+∞Egt2=0.Now, we introduce the following result for equation (30).Theorem 2.
Letg⋅ in equation (30) be a real-valued ℱtt≥0-adapted process satisfying H, and let μ1+c>b2/2, where b=f⋅Lℱ∞0,+∞;L∞Ω. Then, for any initial datum v0∈L2Ω,ℱ0,P;L20,1, the solution to equation (30) satisfies(32)limt⟶+∞Evx,t2=0.Proof.
Letvit=vx,t,eii=1,2,…. Then, we have that for any t∈0,+∞,(33)vx,t=∑i=1∞vitei,Evx,t2=∑i=1∞Evit2.
After some computations, we obtain(34)dvit=−λivit+−1i+1iπgtdt+ftvitdBt,t∈0,+∞,vi0=vi0,where vi0=v0,ei∈L2Ω,ℱ0,P and λi=μi+ci∈ℕ.
Letθii=1∞ be a sequence of positive numbers, which will be determined later. It follows from Itô’s formula that(35)deθitvit2=θieθitvit2dt+2eθitvitdvi+eθitftvit2dt,=θi−2λi+b2eθitvit2dt+2eθit−1i+1iπvigt+ftvit2dBt.
Integrating equation (35) over 0,t and demonstrating the mathematics expectation, we have(36)Evit2≤e−θitEvi02+θi−2λi+b2E∫0te−θit−svis2ds,+2E∫0te−θit−s−1i+1iπvigsds,i=1,2,3,…,where b=f⋅Lℱ∞0,+∞;L∞Ω.
Wheni=2,3,…, we take(37)θi=2λi−b2−μii=2,3,….
By the definition ofλii∈ℕ and b2<2λ1, we have the following observations:(38)i0<θ2<θ3<…<θn<…;ii∑i=2∞1θi<+∞.
Therefore, it follows from the Cauchy–Schwarz inequality that(39)2E∫0te−θit−s−1i+1iπvigsds,≤E∫0te−θit−sgs2ds+μiE∫0te−θit−svis2ds.
This, together with equations (36) and (37), indicates that(40)Evit2≤e−θitEvi02+∫0te−θit−sEgs2ds.
It, along withi of equation (38), yields(41)E∑i=2∞vit2≤e−θ2tE∑i=2∞vi02+∑i=2∞∫0te−θit−sEgs2ds.
Now, we claim that(42)E∑i=2∞vit2⟶0,ast⟶+∞.
It follows fromi of equation (38) that(43)limt⟶+∞e−θ2tE∑i=2∞vi02=0.
Now, we are going to prove that(44)limt⟶+∞∑i=2∞∫0te−θit−sEgs2ds=0.
Byii of H, we have that for any ε>0, there exists a positive number N>0, which only depends on ε, such that(45)Egs2<ε,ass>N.
Therefore,(46)θi∫Nte−θit−sEgs2dsε,astN,fori=2,3,….
Byi of H, we obtain(47)θi∫0Ne−θit−sEgs2ds,≤Me−θit−N+e−θit,fori=2,3,….
Thus, there exists a positive numberR>N, which only depends on ε, such that when t>R,(48)θi∫0Ne−θit−sEgs2ds<ε,fori=2,3,….
Together with equations (46) and (48), it shows that when t>R,(49)∫0te−θit−sEgs2ds<εθi,fori=2,3,….
This, along withii of equation (38), yields equation (44). Thus, equation (42) can be obtained by equations (41), (43), and (44).
Sinceb2<2λ1=2π2+2c, we can find a positive number ϖ such that 2λ1−b2>ϖ. When i=1, we take(50)θ1=2λ1−b2−ϖ>0.
Therefore, it follows from the Cauchy–Schwarz inequality that(51)2E∫0te−θ1t−sπv1gsds,≤μ1ϖE∫0te−θ1t−sgs2ds+ϖE∫0te−θ1t−sv1s2ds.
This, together with equations (36) and (50), indicates that(52)Ev1t2≤e−θ1tEv102+μ1ϖ∫0te−θ1t−sEgs2ds.
By the same argument in the proof of equation (44), we can also obtain that(53)μ1ϖ∫0te−θ1t−sEgs2ds⟶0,ast⟶+∞.
It, along with equation (52), yields(54)Ev1t2⟶0,ast⟶+∞.
By equations (33), (42), and (54), we can obtain equation (32). This completes the proof. □


### 4.2. Stability for Stochastic Equation (29)

A result on the stability for stochastic equation (29) is given as follows.Theorem 3.
Letd⋅ in equation (29) be a real-valued ℱtt≥0-adapted process satisfying H. Then, for any initial datum u0∈L2Ω,ℱ0,P;L20,1, the solution to equation (29) with boundary feedback control equation (20) satisfies that(55)limt⟶+∞Eux,t2=0.Proof.
Letwx,t=Λux,t, where the operator Λ is given in equation (16). By the same argument in Theorem 1, we have that(56)dwx,t=wxxx,t−cwx,tdt+ftwx,tdBt,x,t∈0,1×0,+∞,w0,t=0,t∈0,+∞,w1,t=dt,t∈0,+∞,wx,0=w0x,x∈0,1.
Then, it follows from Theorem2 that(57)limt⟶+∞Ewx,t2=0.
By Lemma4,(58)Eux,t2≤Λ−12Ewx,t2.
This, together with equation (57), shows that equation (55) holds. This completes the proof.


## 4.1. An Auxiliary Result

To study the problem of robust stability, we introduce the following nonhomogeneous Dirichlet boundary value problem:(30)dvx,t−vxxx,tdt+cvx,tdt=ftvx,tdBt,x,t∈0,1×0,+∞,v0,t=0,t∈0,+∞,v1,t=gt,t∈0,+∞,vx,0=v0x,x∈0,1,where the initial datum v0∈L2Ω,ℱ0,P;L20,1, and the real-valued ℱtt≥0-adapted process g⋅ satisfies(31)Hithere exists a positive numberM>0such thatEgt2<Mfort∈0,+∞;iilimt⟶+∞Egt2=0.Now, we introduce the following result for equation (30).Theorem 2.
Letg⋅ in equation (30) be a real-valued ℱtt≥0-adapted process satisfying H, and let μ1+c>b2/2, where b=f⋅Lℱ∞0,+∞;L∞Ω. Then, for any initial datum v0∈L2Ω,ℱ0,P;L20,1, the solution to equation (30) satisfies(32)limt⟶+∞Evx,t2=0.Proof.
Letvit=vx,t,eii=1,2,…. Then, we have that for any t∈0,+∞,(33)vx,t=∑i=1∞vitei,Evx,t2=∑i=1∞Evit2.
After some computations, we obtain(34)dvit=−λivit+−1i+1iπgtdt+ftvitdBt,t∈0,+∞,vi0=vi0,where vi0=v0,ei∈L2Ω,ℱ0,P and λi=μi+ci∈ℕ.
Letθii=1∞ be a sequence of positive numbers, which will be determined later. It follows from Itô’s formula that(35)deθitvit2=θieθitvit2dt+2eθitvitdvi+eθitftvit2dt,=θi−2λi+b2eθitvit2dt+2eθit−1i+1iπvigt+ftvit2dBt.
Integrating equation (35) over 0,t and demonstrating the mathematics expectation, we have(36)Evit2≤e−θitEvi02+θi−2λi+b2E∫0te−θit−svis2ds,+2E∫0te−θit−s−1i+1iπvigsds,i=1,2,3,…,where b=f⋅Lℱ∞0,+∞;L∞Ω.
Wheni=2,3,…, we take(37)θi=2λi−b2−μii=2,3,….
By the definition ofλii∈ℕ and b2<2λ1, we have the following observations:(38)i0<θ2<θ3<…<θn<…;ii∑i=2∞1θi<+∞.
Therefore, it follows from the Cauchy–Schwarz inequality that(39)2E∫0te−θit−s−1i+1iπvigsds,≤E∫0te−θit−sgs2ds+μiE∫0te−θit−svis2ds.
This, together with equations (36) and (37), indicates that(40)Evit2≤e−θitEvi02+∫0te−θit−sEgs2ds.
It, along withi of equation (38), yields(41)E∑i=2∞vit2≤e−θ2tE∑i=2∞vi02+∑i=2∞∫0te−θit−sEgs2ds.
Now, we claim that(42)E∑i=2∞vit2⟶0,ast⟶+∞.
It follows fromi of equation (38) that(43)limt⟶+∞e−θ2tE∑i=2∞vi02=0.
Now, we are going to prove that(44)limt⟶+∞∑i=2∞∫0te−θit−sEgs2ds=0.
Byii of H, we have that for any ε>0, there exists a positive number N>0, which only depends on ε, such that(45)Egs2<ε,ass>N.
Therefore,(46)θi∫Nte−θit−sEgs2dsε,astN,fori=2,3,….
Byi of H, we obtain(47)θi∫0Ne−θit−sEgs2ds,≤Me−θit−N+e−θit,fori=2,3,….
Thus, there exists a positive numberR>N, which only depends on ε, such that when t>R,(48)θi∫0Ne−θit−sEgs2ds<ε,fori=2,3,….
Together with equations (46) and (48), it shows that when t>R,(49)∫0te−θit−sEgs2ds<εθi,fori=2,3,….
This, along withii of equation (38), yields equation (44). Thus, equation (42) can be obtained by equations (41), (43), and (44).
Sinceb2<2λ1=2π2+2c, we can find a positive number ϖ such that 2λ1−b2>ϖ. When i=1, we take(50)θ1=2λ1−b2−ϖ>0.
Therefore, it follows from the Cauchy–Schwarz inequality that(51)2E∫0te−θ1t−sπv1gsds,≤μ1ϖE∫0te−θ1t−sgs2ds+ϖE∫0te−θ1t−sv1s2ds.
This, together with equations (36) and (50), indicates that(52)Ev1t2≤e−θ1tEv102+μ1ϖ∫0te−θ1t−sEgs2ds.
By the same argument in the proof of equation (44), we can also obtain that(53)μ1ϖ∫0te−θ1t−sEgs2ds⟶0,ast⟶+∞.
It, along with equation (52), yields(54)Ev1t2⟶0,ast⟶+∞.
By equations (33), (42), and (54), we can obtain equation (32). This completes the proof. □


## 4.2. Stability for Stochastic Equation (29)

A result on the stability for stochastic equation (29) is given as follows.Theorem 3.
Letd⋅ in equation (29) be a real-valued ℱtt≥0-adapted process satisfying H. Then, for any initial datum u0∈L2Ω,ℱ0,P;L20,1, the solution to equation (29) with boundary feedback control equation (20) satisfies that(55)limt⟶+∞Eux,t2=0.Proof.
Letwx,t=Λux,t, where the operator Λ is given in equation (16). By the same argument in Theorem 1, we have that(56)dwx,t=wxxx,t−cwx,tdt+ftwx,tdBt,x,t∈0,1×0,+∞,w0,t=0,t∈0,+∞,w1,t=dt,t∈0,+∞,wx,0=w0x,x∈0,1.
Then, it follows from Theorem2 that(57)limt⟶+∞Ewx,t2=0.
By Lemma4,(58)Eux,t2≤Λ−12Ewx,t2.
This, together with equation (57), shows that equation (55) holds. This completes the proof.


## 5. Numerical Simulations

In this section, we will carry out a numerical simulation to illustrate the theoretical results. In equation (1), we take a=12 and ft=sint. Let U⋅=0, and let the initial datum be(59)y0x=10x1−x2.Now, we discretize the stochastic heat equation using the finite difference method. The time and the space steps are chosen ask=0.00005 and h=0.01, respectively. We performed the numerical simulation 30 times under the Matlab environment and presented one of them in this section.Figures1 and 2 show that equation (1) is unstable without control. In equation (29), we still take a=12 and b=1. Now, we let c=1 in the kernel function and let the disturbance dt=1/t+5 in equation (29). It is obvious that dt satisfies H given in Section 4. According to [11], we have(60)kx,s=−11sI111x2−s211x2−s2,where I1 is the modified Bessel function of order one. Using this kernel function, we can construct boundary feedback law equation (20).Figure 1
A sample path of the solutionux,t to equation (1) without control.Figure 2
TheL20,1 norm of the sample path of the solution ux,t to equation (1) without control.Figures3 and 4 confirm that the experiment agrees with the theoretical results in this paper. We performed it 30 times, and these results all show that our method yields satisfactory performance.Figure 3
A sample path of the solutionux,t to equation (29) with control (20).Figure 4
TheL20,1 norm of the sample path of the solution ux,t to equation (29) with control (20).



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*Source: 5623756-2022-05-27.xml*