Approximate Controllability of a Semilinear Heat Equation

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

see (Figure 1), where

Remark 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).

Proposition 3. Let (X, Σ, μ) be a measure space with μ(X) < ∞ and 1 ≤ q < r < ∞. Then, L_r(μ) ⊂ L_q(μ) and

Proof. The proof of this proposition follows from Theorem I.V.6 from [12] by putting p = r/q > 1 and considering the relation

Theorem 4 (Rothe’s theorem [13, page 129]). Let E(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.

Let Φ : B → E be a continuous mapping with Φ(B) relatively compact in E and Φ(∂B) ⊂ B.

Then, there is a point x^* ∈ B such that Φ(x^*) = x^*.

Lemma 5 (see Lemma 3.14 from [15], page 62.)Let {α_j} _j≥1 and {β_i,j : i = 1,2, …, m} _j≥1 be two sequences of real numbers such that α₁ > α₂ > α₃⋯. Then,

If and only if

Proposition 6. Under condition (12), the function g^e : [0, τ] × Z × U → Z defined by g^e(t, z, u)(x) = g(t, z(x), u(x)), for all x ∈ Ω, satisfies for all u, z ∈ Z = L²(Ω):

Proof

Now, since 1/2 ≤ β < 1⇔1 ≤ 2β < 2, applying Proposition 3, we obtain that

Definition 7. For system (21), we define the following concept: the controllability map (for τ > 0) G_a : L²(0, τ; U) → Z is given by

whose adjoint operator $$ is given by

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)

  $$.

• (b)

  $$.

• (c)

  $$, z ≠ 0  in Z.

• (d)

  $$.

• (e)

  $$, for  all t ∈ [0, τ], ⇒z = 0.

• (f)

  For all z ∈ Z , we have $$, where

  $$ ()

So, lim _α→0G_au_α = z and the error E_αz of this approximation is given by

Remark 9. Lemma 8 implies that the family of linear operators Γ_α : Z → L²(0, τ; U), defined for 0 < α ≤ 1 by

is an approximate inverse for the right of the operator G_a in the sense that

Proposition 10 (see [7].)If $$, 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].

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

and the error of this approximation E_α is given by

Definition 13. For the system (22), we define the following concept: the nonlinear controllability map (for τ > 0) G_g : L²(0, τ; U) → Z is given by

where H : L²(0, τ; U) → Z is the nonlinear operator given by

Lemma 14. Equation (22) is approximately controllable on [0, τ] if and only if $$.

Definition 15. The following equation shall be called the controllability equation associated with the nonlinear equation (22):

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

and the error of this approximation E_αz is given by

Proof. For each z ∈ Z fixed, we shall consider the following family of nonlinear operators K_α : L²(0, τ; U) → L²(0, τ; U) given by

First, we shall prove that for all α ∈ (0,1] the operator K_α has a fixed point u_α. In fact, since g^e 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 K_α is compact.

Moreover,

In fact, putting C = 2cμ(Ω) ^{β/(2β−1/2)} and $$, we get from Proposition 6 that and from the definition of the operator H(u), Proposition 3, and (19) we have, for u ∈ L²(0, τ; U), the following estimate: Now, since 1/2 ≤ β < 1⇔1 ≤ 2β < 2, applying Proposition 3, we obtain that Therefore, Consequently, Then, from condition (45), we obtain that, for a fixed 0 < ρ < 1, there exists R_α > 0 big enough such that Hence, if we denote by B(0, R_α) the ball of center zero and radio R_α > 0, we get that K_α(∂B(0, R_α)) ⊂ B(0, R_α). Since K_α is compact and maps the sphere ∂B(0, R_α) into the interior of the ball B(0, R_α), we can apply Rothe′s fixed point Theorem 4 to ensure the existence of a fixed point u_α ∈ L₂(0, τ; U) such that Claim. The family of fixed pint {u_α} _0<α≤1 is bounded.

In fact, for the purpose of contradiction, let us assume the contrary. Then, there exists a subsequence $$ such that

On the other hand, we have that $$. So, Hence, which is evidently a contradiction. Then, the claim is true and there exists γ > 0 such that Therefore, without loss of generality, we can assume that the sequence H(u_α) converges to y ∈ Z. So, if then Hence, To conclude the proof of this theorem, it is enough to prove that From Lemma 8 (d), we get that On the other hand, from Proposition 10, we get that Therefore, since H(u_α) converges to y, we get that Consequently, So, by putting z = z₁ − e^aτT(τ)z₀ and using (38), we obtain the nice result:

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:

where a ≥ 0 and b > 0 are constants, Ω is a domain in ℝ^N, ω is an open nonempty subset of Ω, 1_ω 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:

in the space Z_1/2 = D((−Δ) ^1/2) × L²(Ω), where Ω is a bounded domain in ℝⁿ, ω is an open nonempty subset of Ω, 1_ω 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:

where Ω is a bounded domain in ℝⁿ, ω is an open nonempty subset of Ω, 1_ω denotes the characteristic function of the set ω, the distributed control u ∈ L²(0, τ; L²(Ω)), and $$.

Theorem 20. If vectors B^*ϕ_j,k are linearly independent in Z, then the system (69) is approximately controllable on [0, τ].