# Approximate Controllability of a Semilinear Heat Equation

Authors: Hugo Leiva; N. Merentes; J. Sanchez
Journal: International Journal of Partial Differential Equations
 (2013)

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## Abstract

We apply Rothe’s type fixed point theorem to prove the interior approximate controllability of the following semilinear heat equation:zt(t,x)=Δz(t,x)+1ωu(t,x)+f(t,z(t,x),u(t,x)) in (0,τ]×Ω,z=0, on (0,τ)×∂Ω,z(0,x)=z0(x),x∈Ω, where Ω is a bounded domain in ℝN(N≥1), z0∈L2(Ω), ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u belongs to L2(0,τ;L2(Ω)), and the nonlinear function f:[0,τ]×ℝ×ℝ→ℝ is smooth enough, and there are a,b,c∈ℝ, R>0 and 1/2≤β<1 such that |f(t,z,u)-az|≤c|u|β+b, for all u,z∈ ℝ,|u|,|z|≥R. Under this condition, we prove the following statement: for all open nonempty subset ω of Ω, the system is approximately controllable on [0,τ]. Moreover, we could exhibit a sequence of controls steering the nonlinear system from an initial state z0 to an ϵ neighborhood of the final state z1 at time τ>0.


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## Body

## 1. Introduction

In this paper, we prove the interior approximate controllability of the following semilinear heat equation:(1)zt(t,x)=Δz(t,x)+1ωu(t,x)+f(t,z,u(t,x)),00000000000000000000000000000in(0,τ]×Ω,z=0,on(0,τ)×∂Ω,z(0,x)=z0(x),x∈Ω,
where Ω is a bounded domain in ℝN(N≥1), z0∈L2(Ω), ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u belong to L2(0,τ;L2(Ω)), and the nonlinear function f:[0,τ]×ℝ×ℝ→ℝ is smooth enough, and there are a,b,c∈ℝ, R>0 and 1/2≤β<1 such that
(2)|f(t,z,u)-az|≤c|u|β+b,∀u,z∈ℝ,|u|,|z|≥R,
which implies that
(3)|f(t,z,u)|≤c|u|β+e|z|+b,∀u,z∈ℝ,|u|,|z|≥R.Definition 1 (approximate controllability).
The system (1) is said to be approximately controllable on [0,τ] if for every z0, z1∈Z=U=L2(Ω), ɛ>0, there exists u∈L2(0,τ;U) such that the solution z(t) of (1) corresponding to u verifies
(4)z(0)=z0,∥z(τ)-z1∥z<ɛ;
see (Figure 1), where
(5)∥z(τ)-z1∥Z=(∫Ω‍|z(τ,x)-z1(x)|2dx)1/2.Figure 1Remark 2.
It is clear that exact controllability of the system (1) implies approximate controllability, null controllability, and controllability to trajectories of the system. But it is well known (see [1]) that due to the diffusion effect or the compactness of the semigroup generated by -Δ, the heat equation can never be exactly controllable. We observe also that in the linear case, controllability to trajectories and null controllability are equivalent. Nevertheless, the approximate controllability and the null controllability are in general independent. Therefore, in this paper, we shall concentrate only on the study of the approximate controllability of the system (1).Now, before proceeding with the introduction of this paper, we should mention the work of other authors and show that ours is different in relation with new perturbation and the technique used. The approximate controllability of the heat equation under nonlinear perturbationsf(z) independents of t and u variable
(6)zt(t,x)=Δz(t,x)+1ωu(t,x)+f(z),in(0,τ]×Ω,z=0,on(0,τ)×∂Ω,z(0,x)=z0(x),x∈Ω
has been studied by several authors, particularly in [2–4], depending on conditions that impose to the nonlinear term f(z). For instance, in [3, 4], the approximate controllability of the system (6) is proved if f(z) is sublinear at infinity, that is,
(7)|f(z)|≤d|z|+e.
Also, in the above reference, they mentioned that when f is superlinear at the infinity, the approximate controllability of the system (6) fails.Recently, the interior controllability of the semilinear heat equation (1) has been proved in [5–7] under the following condition: there are constants a,c∈ℝ, with c≠-1, such that
(8)sup(t,z,u)∈Qτ|f(t,z,u)-az-cu|<∞,
where Qτ=[0,τ]×ℝ×ℝ.We note that the interior approximate controllability of the linear heat equation(9)zt(t,x)=Δz(t,x)+1ωu(t,x),in(0,τ]×Ω,z=0,on(0,τ)×∂Ω,z(0,x)=z0(x),x∈Ω,
has been studied by several authors, particularly by [8–10]; and in a general fashion in [11].Now, we shall describe the strategy of this work.First, we prove that the linear system(10)zt(t,x)=Δz(t,x)+1ωu(t,x)+az(t,x),in(0,τ]×Ω,z=0,on(0,τ)×∂Ω,z(0,x)=z0(x),x∈Ω
is approximately controllable.After that, we write the system (1) as follows:
(11)zt(t,x)=Δz(t,x)+1ωu(t,x)+az+g(t,z,u),0000000000000000000i000000000in(0,τ]×Ω,z=0,on(0,τ)×∂Ω,z(0,x)=z0(x),x∈Ω,
where g(t,z,u)=f(t,z,u)-az is a smooth enough functions satisfying the following conditions: there are a,b,c∈ℝ, R>0 and 1/2≤β<1 such that
(12)|g(t,z,u)|≤c|u|β+b,∀u,z∈ℝ,|u|,|z|≥R.
Finally, the approximate controllability of the system (11) follows from the controllability of (10), the compactness of the semigroup generated by the Laplacian operator -Δ, the condition (12) satisfied by the nonlinear term g, and the following results.Proposition 3.
Let(X,Σ,μ) be a measure space with μ(X)<∞ and 1≤q<r<∞. Then, Lr(μ)⊂Lq(μ) and
(13)∥f∥q≤μ(X)(r-q)/rq∥f∥r,f∈Lr(μ).Proof.
The proof of this proposition follows from Theorem I.V.6 from [12] by putting p=r/q>1 and considering  the relation
(14)∫X‍(|f|q)pdμ=∫X‍|f|rdμ,∀f∈Lr(μ).Theorem 4 (Rothe’s theorem [13, page 129]).
LetE(τ) be a Hausdorff topological vector space. Let B⊂E be a closed convex subset such that the zero of E is contained in the interior of B.
LetΦ:B→E be a continuous mapping with Φ(B) relatively compact in E and Φ(∂B)⊂B.
Then, there is a pointx*∈B such that Φ(x*)=x*.The technique we use here to prove the approximate controllability of the linear part of (10) is based on the classical unique continuation for elliptic equations (see [14]) and the following lemma.Lemma 5 (see Lemma 3.14 from [15, page 62]).
Let{αj}j≥1 and {βi,j:i=1,2,…,m}j≥1 be two sequences of real numbers such that α1>α2>α3⋯. Then,
(15)∑j=1∞‍eαjtβi,j=0,∀t∈[0,τ],i=1,2,…,m.
If and only if
(16)βi,j=0,i=1,2,…,m;j=1,2,…,∞.


## 2. Abstract Formulation of the Problem

In this section, we choose a Hilbert space where system (1) can be written as an abstract differential equation; to this end, we consider the following results appearing in [15, page 46], [16, page 335], and [17, page 147].Let us consider the Hilbert spaceZ=L2(Ω) and 0<λ1<λ2<⋯<λj→∞ the    eigenvalues of -Δ with the Dirichlet homogeneous conditions, each one with finite multiplicity γj equal to the dimension of the corresponding eigenspace. Then, we have the following well-known properties.(i)
There exists a complete orthonormal set{ϕj,k} of eigenvectors of A=-Δ.(ii)
For allz∈D(A), we have
(17)Az=∑j=1∞‍λj∑k=1γj‍〈ξ,ϕj,k〉ϕj,k=∑j=1∞‍λjEjz,where〈·,·〉 is the inner product in Z and
(18)Ejz=∑k=1γj‍〈z,ϕj,k〉ϕj,k.So,{Ej} is a family of complete orthogonal projections in Z and z=∑j=1∞‍Ejz, z∈Z.(iii)
-A generates an analytic semigroup {T(t)} given by
(19)T(t)z=∑j=1∞‍e-λjtEjz,∥T(t)∥≤e-λ1t,t≥0.Consequently, systems (1), (10), and (11) can be written, respectively, as an abstract differential equations in Z:
(20)z′=-Az+Bωu+fe(t,z,u),z∈Z,t≥0,(21)z′=-Az+Bωu+az,z∈Z,t≥0,(22)z′=-Az+Bωu+az+ge(t,z,u),z∈Z,t≥0,
where u∈L2([0,τ];U), U=Z, Bω:U→Z, Bωu=1ωu is a bounded linear operator, fe:[0,τ]×Z×U→Z is defined by fe(t,z,u)(x)=f(t,z(x),u(x)), for all x∈Ω, and ge(t,z,u)=fe(t,z,u)-az. On the other hand, from condition (12), we get the following estimate.Proposition 6.
Under condition (12), the function ge:[0,τ]×Z×U→Z defined by ge(t,z,u)(x)=g(t,z(x),u(x)), for all x∈Ω, satisfies for all u,z∈Z=L2(Ω):
(23)∥ge(t,z,u)∥Z≤2cμ(Ω)β/(2β-1/2)∥u∥Zβ+2bμ(Ω),000000000000000000000000∥z∥,∥u∥≥Rμ(Ω).Proof
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(25)∥ge(t,z,u)∥Z≤2cμ(Ω)(1-β)/β∥u∥Zβ+2bμ(Ω).


## 3. Interior Controllability of the Linear Equation

In this section, we shall prove the interior approximate controllability of the linear system (21). To this end, we note that, for all z0∈Z and u∈L2(0,τ;U), the initial value problem
(26)z′=-Az+Bωu(t)+az(t),z∈Z,z(0)=z0,where the control functionu belongs to L2(0,τ;U), admits only one mild solution given by
(27)z(t)=eatT(t)z0+∫0t‍ea(t-s)T(t-s)Bωu(s)ds,t∈[0,τ].Definition 7.
For system (21), we define the following concept: the controllability map (for τ>0) Ga:L2(0,τ;U)→Z is given by
(28)Gau=∫0τ‍ea(τ-s)T(τ-s)Bωu(s)ds,
whose adjoint operator Ga*:Z→L2(0,τ;Z) is given by
(29)(Ga*z)(s)=Bω*ea(τ-s)T*(τ-s)z,∀s∈[0,τ],∀z∈Z.The following lemma holds in general for a linear bounded operatorG:W→Z between Hilbert spaces W and Z.Lemma 8 (see [11, 15, 18]).
Equation (21) is approximately controllable on [0,τ] if and only if one of the following statements holds.(a)
Rang(Ga)¯=Z.(b)
Ker(Ga*)={0}.(c)
〈GaGa*z,z〉>0, z≠0  inZ.(d)
limα→0+α(αI+GaGa*)-1z=0.(e)
Bω*eatT*(t)z=0, for  allt∈[0,τ],⇒z=0.(f)
For allz∈Z , we haveGuα=z-α(αI+GaGa*)-1z, where(30)uα=Ga*(αI+GaGa*)-1z,α∈(0,1].
So,limα→0Gauα=z and the error Eαz of this approximation is given by
(31)Eαz=α(αI+GaGa*)-1z,α∈(0,1].Remark 9.
Lemma8 implies that the family of linear operators Γα:Z→L2(0,τ;U), defined for 0<α≤1 by
(32)Γαz=Bω*ea(·)T*(·)(αI+GaGa*)-1z=Ga*(αI+GaGa*)-1z,
is an approximate inverse for the right of the operatorGa in the sense that
(33)limα→0GaΓα=I.Proposition 10 (see [7]).
IfRang(Ga)¯=Z, then
(34)supα>0∥α(αI+GaGa*)-1∥≤1.Remark 11.
The proof of the following theorem follows from foregoing characterization of dense range linear operators and the classical unique continuation for elliptic equations (see [14]), and it is similar to the one given in Theorem 4.1 in [6].Theorem 12.
System (21) is approximately controllable on [0,τ]. Moreover, a sequence of controls steering the system (21) from initial state z0 to an ϵ neighborhood of the final state z1 at time τ>0 is given by
(35)uα(t)=Bω*ea(τ-t)T*(τ-t)(αI+GaGa*)-1×(z1-eaτT(τ)z0),
and the error of this approximation Eα is given by
(36)Eα=α(αI+GaGa*)-1(z1-eaτT(τ)z0).


## 4. Controllability of the Semilinear System

In this section, we shall prove the main result of this paper, the interior approximate controllability of the semilinearnD heat equation given by (1), which is equivalent to prove the approximate controllability of the system (22). To this end, for all z0∈Z and u∈L2(0,τ;U), the initial value problem
(37)z′=-Az+Bωu+az+ge(t,z,u),z∈Z,t≥0z(0)=z0
admits only one mild solution given by
(38)zu(t)=eatT(t)z0+∫0t‍ea(t-s)T(t-s)Bωu(s)ds+∫0t‍ea(t-s)T(t-s)ge(s,zu(s),u(s))ds,000000000000000000000000000t∈[0,τ].Definition 13.
For the system (22), we define the following concept: the nonlinear controllability map (for τ>0) Gg:L2(0,τ;U)→Z is given by
(39)Ggu=∫0τ‍ea(τ-s)T(τ-s)Bωu(s)ds+∫0τ‍ea(τ-s)T(τ-s)ge(s,zu(s),u(s))ds=Ga(u)+H(u),
where H:L2(0,τ;U)→Z is the nonlinear operator given by
(40)H(u)=∫0τ‍ea(τ-s)T(τ-s)ge(s,zu(s),u(s))ds,00000000000000000000000000u∈L2(0,τ;U).The following lemma is trivial.Lemma 14.
Equation (22) is approximately controllable on [0,τ] if and only if Rang(Gg)¯=Z.Definition 15.
The following equation shall be called the controllability equation associated with the nonlinear equation (22):
(41)uα=Γα(z-H(uα))=Ga*(αI+GaGa*)-1(z-H(uα)),00000000000000000000000000000000000(0<α≤1).Now, we are ready to present and prove the main result of this paper, which is the interior approximate controllability of the semilinearnD heat equation (1)Theorem 16.
The system (22) is approximately controllable on [0,τ]. Moreover, a sequence of controls steering the system (22) from initial state z0 to an ϵ-neighborhood of the final state z1 at time τ>0 is given by
(42)uα(t)=Bω*ea(τ-t)T*(τ-t)(αI+GaGa*)-1×(z1-eaτT(τ)z0-H(uα)),
and the error of this approximation Eαz is given by
(43)Eαz=α(αI+GaGa*)-1(z1-eaτT(τ)z0-H(uα)).Proof.
For eachz∈Z fixed, we shall consider the following family of nonlinear operators Kα:L2(0,τ;U)→L2(0,τ;U) given by
(44)Kα(u)=Γα(z-H(u))=Ga*(αI+GaGa*)-1(z-H(u)),000000000000000000000000000000000000(0<α≤1).
First, we shall prove that for all α∈(0,1] the operator Kα has a fixed point uα. In fact, since ge is smooth and satisfies (12) and the semigroup {T(t)}t≥0 given by (19) is compact (see [19, 20]), then using the result from [1], we obtain that the operator H is compact, which implies that the operator Kα is compact.
Moreover,(45)lim¯∥u∥L2→∞∥Kα(u)∥L2∥u∥L2=0.
In fact, putting C=2cμ(Ω)β/(2β-1/2) and B=2bμ(Ω), we get from Proposition 6 that
(46)∥ge(t,z,u)∥Z≤C∥u∥Zβ+B,∥z∥,∥u∥≥Rμ(Ω),
and from the definition of the operator H(u), Proposition 3, and (19) we have, for u∈L2(0,τ;U), the following estimate:
(47)∥H(u)∥≤∫0τ‍e(a+λ1)(τ-s)∥ge(s,zu(s),u(s))∥ds≤(∫0τ‍e2(a+λ1)(τ-s)ds)1/2×(∫0τ‍∥ge(s,zu(s),u(s))∥2ds)1/2=N(∫0τ‍∥ge(s,zu(s),u(s))∥2ds)1/2≤N(∫0τ‍(C∥u(s)∥β+B)2ds)1/2≤N(∫0τ‍(4C2∥u(s)∥2β+4B2)ds)1/2≤2NC(∫0τ‍∥u(s)∥2β)1/2+2NBτ≤2NC{(∫0τ‍∥u(s)∥2β)1/2β}β+2NBτ=2CN∥u∥L2ββ+2NBτ.
Now, since 1/2≤β<1⇔1≤2β<2, applying Proposition 3, we obtain that
(48)∥H(u)∥≤2CNτ(1-β)/β∥u∥L2β+2NBτ.
Therefore,
(49)lim¯∥u∥L2→∞∥H(u)∥L2∥u∥L2=0.
Consequently,
(50)lim¯∥u∥L2→∞∥Kα(u)∥∥u∥L2=0.
Then, from condition (45), we obtain that, for a fixed 0<ρ<1, there exists Rα>0 big enough such that
(51)∥Kα(u)∥L2≤ρ∥u∥L2,∥u∥L2=Rα.
Hence, if we denote by B(0,Rα) the ball of center zero and radio Rα>0, we get that Kα(∂B(0,Rα))⊂B(0,Rα). Since Kα is compact and maps the sphere ∂B(0,Rα) into the interior of the ball B(0,Rα), we can apply Rothe's fixed point Theorem 4 to ensure the existence of a fixed point uα∈L2(0,τ;U) such that
(52)uα=Kαuα=Γα(z-H(uα))=G*(αI+GG*)-1(z-H(uα)),00000000000000000(0<α≤1).Claim. The family of fixed pint {uα}0<α≤1 is bounded.
In fact, for the purpose of contradiction, let us assume the contrary. Then, there exists a subsequence{uαn}n≥1⊂{uα}0<α≤1 such that
(53)limn→∞∥uαn∥L2=∞.
On the other hand, we have that uαn=Kαuαn. So,
(54)∥uαn∥L2=∥Kαuαn∥L2⟺∥Kα(uαn)∥L2∥uαn∥L2=1.
Hence,
(55)lim¯n→∞∥Kα(uαn)∥L2∥uαn∥L2=1,
which is evidently a contradiction. Then, the claim is true and there exists γ>0 such that
(56)∥uα∥L2≤γ,(0<α≤1).
Therefore, without loss of generality, we can assume that the sequence H(uα) converges to y∈Z. So, if
(57)uα=Γα(z-H(u))=Ga*(αI+GaGa*)-1(z-H(uα))
then
(58)Gauα=GaΓα(z-H(u))=GaGa*(αI+GaGa*)-1(z-H(uα))=(αI+GaGa*-αI)(αI+GaGa*)-1(z-H(uα))=z-H(uα)-α(αI+GaGa*)-1(z-H(uα)).
Hence,
(59)Gauα+H(uα)=z-α(αI+GaGa*)-1(z-H(uα)).
To conclude the proof of this theorem, it is enough to prove that
(60)limα→0{-α(αI+GaGa*)-1(z-H(uα))}=0.
From Lemma 8 (d), we get that
(61)limα→0{-α(αI+GaGa*)-1(z-H(uα))}=-limα→0{-α(αI+GaGa*)-1H(uα)}=-limα→0-α(αI+GaGa*)-1y-limα→0-α(αI+GaGa*)-1(H(uα)-y)=limα→0-α(αI+GaGa*)-1(H(uα)-y).
On the other hand, from Proposition 10, we get that
(62)∥α(αI+GaGa*)-1(H(uα)-y)∥≤∥(H(uα)-y)∥.
Therefore, since H(uα) converges to y, we get that
(63)limα→0{-α(αI+GaGa*)-1(H(uα)-y)}=0.
Consequently,
(64)limα→0{-α(αI+GaGa*)-1(z-H(uα))}=0.
So, by putting z=z1-eaτT(τ)z0 and using (38), we obtain the nice result:
(65)z1=limα→0+{eaτT(τ)z0+∫0τ‍ea(τ-s)T(τ-s)Bωuα(s)ds00000000+∫0τ‍ea(τ-s)T(τ-s)ge(s,zuα(s),uα(s))ds}.


## 5. Final Remark

Our technique is simple and can be applied to those systems involving compact semigroups like some control system governed by diffusion processes. For example, the Benjamin-Bona-Mahony equation, the strongly damped wave equations, beam equations, and so forth.Example 17.
The original Benjamin-Bona-Mahony equation is a nonlinear one in [21] the authors proved the approximate controllability of the linear part of this equation, which is the fundamental base for the study of the controllability of the nonlinear BBM equation. So, our next work is concerned on the controllability of nonlinear BBM equation:
(66)zt-aΔzt-bΔz=1ωu(t,x)+f(t,z,u(t)),0000000000000000000000t∈(0,τ),x∈Ω,z(t,x)=0,t≥0,x∈∂Ω,
where a≥0 and b>0 are constants, Ω is a domain in ℝN, ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u∈L2(0,τ;L2(Ω)), and f(t,z,u(t)) is a nonlinear perturbation.Example 18.
We believe that this technique can be applied to prove the interior controllability of the strongly damped wave equation with Dirichlet boundary conditions:(67)wtt+η(-Δ)1/2wt+γ(-Δ)w=1ωu(t,x)+f(t,z,u(t)),00000000000000000000000000000000000in(0,τ)×Ω,w=0,in(0,τ)×∂Ω,w(0,x)=w0(x),wt(0,x)=w1(x),inΩ,
in the space Z1/2=D((-Δ)1/2)×L2(Ω), where Ω is a bounded domain in ℝn, ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u∈L2(0,τ;L2(Ω)), and η, γ are positive numbers.Example 19.
Another example where this technique may be applied is partial differential equations modeling the structural damped vibrations of a string or a beam:(68)ytt-2βΔyt+Δ2y=1ωu(t,x)+f(t,z,u(t)),00000000000000000000000000on(0,τ)×Ω,y=Δy=0,on(0,τ)×∂Ω,y(0,x)=y0(x),yt(0,x)=y1(x),inΩ,
where Ω is a bounded domain in ℝn, ω is an open nonempty subset of Ω, 1ω denotes the characteristic function of the set ω, the distributed control u∈L2(0,τ;L2(Ω)), and y0∈H2(Ω)∩H01,y1∈L2(Ω).Moreover, our result can be formulated in a more general setting. Indeed, we can consider the following semilinear evolution equation in a general Hilbert spaceZ:
(69)z´=-Az+Bu(t)+fe(t,z,u),z∈Z,t∈(0,τ],z(0)=z0,
where A:D(A)⊂Z→Z is an unbounded linear operator in Z with the following spectral decomposition:
(70)Az=∑j=1∞‍λj∑k=1γj‍〈z,ϕj,k〉ϕj,k,
with the eigenvalues 0<λ1<λ2<⋯<⋯λn→∞ of A having finite multiplicity γj equal to the dimension of the corresponding eigenspaces, and {ϕj,k} is a complete orthonormal set of eigenfunctions of A. The operator -A generates a strongly continuous compact semigroup {TA(t)}t≥0 given by
(71)TA(t)z=∑j=1∞‍e-λjt∑k=1γj‍〈z,ϕj,k〉ϕj,k.
The control u∈L2(0,τ;U), with U=Z, B:Z→Z is a linear and bounded operator (linear and continuous), and the function fe:[0,τ]×Z×U→Z is smooth enough, and
(72)∥fe(t,z,u)-az∥Z≤c∥u∥Zβ+b,000000∀u,z∈Z,∥u∥,∥z∥≥R.
In this case, the characteristic function set is a particular operator B, and the following theorem is a generalization of Theorem 16.Theorem 20.
If vectorsB*ϕj,k are linearly independent in Z, then the system (69) is approximately controllable on [0,τ].



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*Source: 424309-2013-11-03.xml*