Approximate Controllability of a Semilinear Heat Equation

We apply Rothe’s type fixed point theorem to prove the interior approximate controllability of the following semilinear heat equation: z t (t, x) = Δz(t, x) + 1 ω u(t, x) + f(t, z(t, x), u(t, x)) in (0, τ] × Ω, z = 0, on (0, τ) × ∂Ω, z(0, x) = z 0 (x), x ∈ Ω, where Ω is a bounded domain in RN (N ≥ 1), z 0 ∈ L 2 (Ω), ω is an open nonempty subset of Ω, 1 ω denotes the characteristic function of the set ω, the distributed control u belongs to L2(0, τ; L(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 ofΩ, the system is approximately controllable on [0, τ]. Moreover, we could exhibit a sequence of controls steering the nonlinear system from an initial state z 0 to an ε neighborhood of the final state z 1 at time τ > 0.

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.

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 space  =  2 (Ω) and 0 <  1 <  2 < ⋅ ⋅ ⋅ <   → ∞ the eigenvalues of −Δ with the Dirichlet homogeneous conditions, each one with finite multiplicity   equal to the dimension of the corresponding eigenspace.Then, we have the following well-known properties.
(ii) For all  ∈ (), we have where ⟨⋅, ⋅⟩ is the inner product in  and So, {  } is a family of complete orthogonal projections in  and  = ∑ ∞ =1   ,  ∈ .

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  0 ∈  and  ∈  2 (0, ; ), the initial value problem where the control function  belongs to  2 (0, ; ), admits only one mild solution given by Definition 7.For system (21), we define the following concept: the controllability map (for  > 0)  a :  2 (0, ; ) →  is given by whose adjoint operator  * a :  →  2 (0, ; ) is given by The following lemma holds in general for a linear bounded operator  :  →  between Hilbert spaces  and .
International Journal of Partial Differential Equations So, lim  → 0  a   =  and the error    of this approximation is given by Remark 9. Lemma 8 implies that the family of linear operators Γ  :  →  2 (0, ; ), defined for 0 <  ≤ 1 by is an approximate inverse for the right of the operator  a in the sense that lim Proposition 10 (see [7]).If Rang( a ) = , then 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].
and the error of this approximation   is given by

Controllability of the Semilinear System
In this section, we shall prove the main result of this paper, the interior approximate controllability of the semilinear  heat equation given by (1), which is equivalent to prove the approximate controllability of the system (22).To this end, for all  0 ∈  and  ∈  2 (0, ; ), the initial value problem where  :  2 (0, ; ) →  is the nonlinear operator given by The following lemma is trivial.
Now, we are ready to present and prove the main result of this paper, which is the interior approximate controllability of the semilinear  heat equation ( 1) Theorem 16.The system (22) is approximately controllable on [0, ].Moreover, a sequence of controls steering the system (22) from initial state  0 to an -neighborhood of the final state  1 at time  > 0 is given by and the error of this approximation    is given by Proof.For each  ∈  fixed, we shall consider the following family of nonlinear operators   :  2 (0, ; ) →  2 (0, ; ) given by First, we shall prove that for all  ∈ (0, 1] the operator   has a fixed point   .In fact, since   is smooth and satisfies (12) and the semigroup {()} ≥0 given by ( 19) is compact (see [19,20]), then using the result from [1], we obtain that the operator  is compact, which implies that the operator   is compact.Moreover, lim In fact, putting  = 2(Ω) /(2−1/2) and  = 2√(Ω), we get from Proposition 6 that and from the definition of the operator (), Proposition 3, and ( 19) we have, for  ∈  2 (0, ; ), the following estimate: Now, since 1/2 ≤  < 1 ⇔ 1 ≤ 2 < 2, applying Proposition 3, we obtain that Therefore, lim Consequently, lim Then, from condition (45), we obtain that, for a fixed 0 <  < 1, there exists   > 0 big enough such that Hence, if we denote by (0,   ) the ball of center zero and radio   > 0, we get that   ((0,   )) ⊂ (0,   ).Since   is compact and maps the sphere (0,   ) into the interior of the ball (0,   ), we can apply Rothe's fixed point Theorem 4 to ensure the existence of a fixed point   ∈  2 (0, ; ) such that Claim.The family of fixed pint {  } 0<≤1 is bounded.
In fact, for the purpose of contradiction, let us assume the contrary.Then, there exists a subsequence On the other hand, we have that which is evidently a contradiction.Then, the claim is true and there exists  > 0 such that           2 ≤ , (0 <  ≤ 1) .
Therefore, without loss of generality, we can assume that the sequence (  ) converges to  ∈ .So, if So, by putting  =  1 −   () 0 and using (38), we obtain the nice result:

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.