Controllability Problems for the Heat Equation with Variable Coefficients on a Half-Axis Controlled by the Neumann Boundary Condition

In the paper, the problems of controllability and approximate controllability are studied for the control system $w_t=\frac{1}{\rho}\left(kw_x\right)_x+\gamma w$, $\left.\left(\sqrt{\frac{k}{\rho}}w_x\right)\right|_{x=0}=u$, $x>0$, $t\in(0,T)$, where $u$ is a control, $u\in L^\infty(0,T)$. It is proved that each initial state of the control system is approximately controllable to any target state in a given time $T>0$. To obtain this result, the transformation operator generated by the equation data $\rho$, $k$, $\gamma$ is applied. The results are illustrated by examples.


Introduction
Controllability problems for the heat equation with constant and variable coefficients were studied in a number of papers (see, e.g., [1-4, 9, 10, 13, 14, 17, 21-27]). However, there are only a few papers where these problems were investigated for the heat equation with constant coefficients on unbounded domains (see, e.g., [2,9,10,22,23]), and it seems these problems were investigated for the heat equation with variable coefficients on unbounded domains only in [11].
The paper deals with the controllability problems for the heat equation with variable coefficients on a half-axis controlled by the Neumann boundary condition. Consider the following control system: w(·, 0) = w 0 , x ∈ (0, +∞). (1.3) Here T > 0 is a constant; ρ, k, γ, and w 0 are given functions; u ∈ L ∞ (0, T ) is a control. We assume ρ, k ∈ C 1 [0, +∞) are positive on [0, +∞), (ρk) ∈ C 2 [0, +∞), (ρk) (0) = 0. Consider the even extensions of ρ, k, γ. Throughout the paper we will denote these extensions by the same symbols ρ, k, γ, respectively. Denote We assume σ(x) → +∞ as x → +∞. (1.5) Put Q 2 (ρ, k) = k/ρ Q 1 (ρ, k) + Q 1 (ρ, k) 2 , Q 1 (ρ, k) = k/ρ(kρ) /(4kρ). We also assume Q 2 (ρ, k) − γ ∈ L ∞ (0, +∞) C 1 [0, +∞) (1.6) and ρ k (Q 2 (ρ, k) − γ) σ ∈ L 1 (0, +∞). (1.7) We consider control system (1.1)-(1.3) in modified Sobolev spaces (see Section 2). In [11], controllability problems for the heat equation with variable coefficients on a half-axis controlled by the Dirichlet boundary condition are studied. The general methods applied in the present paper are similar to those from paper [11]. But for the case of the Neumann boundary condition, different spaces and operators are used that caused different technique of proofs of main results. Theorems 2.6 and 2.7 (see Section 2 below) are the main result of the paper. It is proved that each initial state of the control system is approximately controllable to any target state in a given time T > 0 (Theorem 2.7). In the case of constant coefficients (ρ = k = 1, γ = 0), the result of this theorem has been obtained earlier in [10]. In the case of variable coefficients, this result is similar to those of papers [5][6][7][8] for the wave equation with variable coefficients on a half-axis controlled either by the Dirichlet or by the Neumann boundary condition. However, the methods for obtaining the results are essentially different because of entirely different nature of the heat and wave equations. They are compared below. If an initial state of control system is controllable to the origin then the initial state is also the origin (see Theorem 2.6). In the case of constant coefficients (ρ = k = 1, γ = 0), the result of this theorem has been obtained earlier in [10]. This result is similar to that of the paper [22]. To study control system (1.1)-(1.3), we use the transformation operator T and the modified Sobolev spaces H s , s = −1, 1. This operator T : H −1 → H −1 together with the spaces H s , s = −1, 1, associated with the equation data (ρ, k, γ) are introduced and studied in [5][6][7][8]. The definitions of T, H s , and H s are given below in Section 2.
The operator T is a continuous one-to-one mapping between the spaces H s and H s . Moreover, it is one-to-one mapping between the set of the solutions to (1.1)-(1.3) with constant coefficients (ρ = k = 1, γ = 0) where u = u 110 ∈ L ∞ (0, T ) and the set of the solutions to this problem with variable coefficients ρ, k, γ where u = u ρkγ ∈ L ∞ (0, T ) (see below Theorems 3.3 and 3.6). Note that u 110 and u ρkγ are different generally speaking. The proofs of the main results of the paper are based on Theorems 3.3 and 3.6 proved in Section 3. The control system with variable coefficients ρ, k, γ replicates the controllability properties of the control system with constant coefficients (ρ = k = 1, γ = 0) and vice versa.
The last result also holds true for the wave equation on a half-axis [5][6][7][8]. But the proofs are essentially different for the cases of the wave and heat equations. Applying the operator T −1 to a solution to the equation with variable coefficients ρ, k, γ and a control u = u ρkγ ∈ L ∞ (0, T ), we obtain a solution to the equation with the constant coefficients ρ = k = 1, γ = 0 and a control u = u 110 ∈ L ∞ (0, T ) different from the control u ρkγ . To find and to estimate the control u 110 , we have to solve an integral equation of the form (1.8) In the case of the wave equation, it has been proved that f and P are bounded on [0, T ] [5][6][7][8]. Therefore, the integral operator in the right-hand side of (1.8) is of the Hilbert-Schmidt type. Hence, the Fredholm alternative together with the generalized Gronwall theorem can be applied to solve (1.8) in L 2 (0, T ) and estimate the solution u 110 in L ∞ (0, T ) when we deal with the wave equation [5][6][7][8].
In the case of the heat equation, it has been proved that f and (·)P are bounded [11]. That is why the integral operator in the right-hand side of (1.8) is not of the Hilbert-Schmidt type, and the Fredholm alternative is not applicable in the general case. The Banach fixedpoint theorem is also not applicable in general case. That is why the method of successive approximations has been used to construct a solution to (1.8) on [0, T ]. Then the Banach fixed-point theorem has been applied in L 2 -space on small intervals to prove the uniqueness of the solution [11]. This result is recalled in Lemma 3.5 below.
Since the control system with variable coefficients ρ, k, γ replicates the controllability properties of the control system with constant coefficients (ρ = k = 1, γ = 0), we obtain the controllability properties of the first control system from the controllability properties of the second one by applying the operator T, i.e., we obtain Theorems 2.6 and 2.7 by applying Theorems 3.3 and 3.6 in Section 2.
The obtained results are illustrated by examples in Section 4.

Spaces, operators and main results
Let us give definitions of the spaces used in the paper.
By H p , p = 0, 1, denote the Sobolev spaces with the norm and H −p = (H p ) * with the norm associated with the strong topology of the adjoint space. We have H 0 = L 2 (R) = H 0 * = H −0 . By f, ϕ , denote the value of a distribution f ∈ H −p on a test function ϕ ∈ H p , p = 0, 1. By H l , denote the subspace of all even distributions in H l , l = −1, 1. It is easy to see that H l is a closed subspace of H l , l = −1, 1.
Let ϕ ∈ L 2 loc (Ω). We define the derivative D ρk by the rule If, in addition, D ρk ϕ ∈ L 2 loc (Ω) and (D ρk ϕ) ∈ L 2 loc (Ω) (the derivative (·) is considered in D (Ω)), we can consider D 2 ρk ϕ. Then ϕ ∈ D (Ω) and with the norm For p = 0, 1, consider also the space and the dual space with the norm associated with the strong topology of the adjoint space. Evidently, In particular, we have Put Consider also the following modified Sobolev spaces introduced and studied in [6][7][8]. Denote , ϕ ∈ H p , p = 0, 1, and the dual space H −p = (H p ) * with the norm associated with the strong topology of the adjoint space. By f, ϕ , denote the value of a distribution f ∈ H −p on a test function ϕ ∈ H p , p = 0, 1. Evidently, H 0 = H 0 * = L 2 (R) and Put In [6], it has been proved that H m ⊂ H n is dense continuous embedding, However, the relation between the Schwartz space S and H p essentially depends on ρ and k. For example, if ρ = k then By H s , denote the subspace of all even distributions in H s , s = −1, 1. The even extension of a function from • H s belongs to H s , s = 0, 1 (see [8] We will use the transformation operator T = S T r : H −1 → H −1 to investigate controllability problems for system (1.1)-(1.3). The operators S and T r have been introduced and studied in [7,8]. 7,8]). The following assertions hold.

(i)
The operator T is an isomorphism of H m and H m , m = −1, 0, 1.
Here δ is the Dirac distribution.
A description and some properties of the operators S and T r are given in Section 3.

Main results
One can easily see that equation (1.1) can be rewritten in the form and condition (1.2) is equivalent to the condition Let w(·, t), w 0 ∈ • H 1 and let W (·, t), W 0 be their even extensions with respect to x, respectively, t ∈ [0, T ]. Let q be defined by (2.1). If w is a solution to control system (1.1)-(1.3), then using Theorem 2.1 and taking into account (2.2) and (2.3), we conclude that W is a solution to the system with respect to x. Let W (·, t), W 0 ∈ H 1 and let w(·, t), w 0 be their restrictions to (0, +∞) with respect to x, respectively, t ∈ [0, T ]. If W is a solution to control system (2.5), (2.6), then due to Corollary 3.4 (see Section 3 below), Hence, w is a solution to control system (1.1)-(1.3).
Let w T ∈ • H 1 and let W T ∈ H 1 be its even extension with respect to x. It is easy to see that w(·, Thus, control systems (1.1)-(1.3) and (2.5), (2.6) are equivalent. Taking into account this equivalence, we will further consider system (2.5), (2.6). Let , denote the set of all states W T ∈ H 1 for which there exists a control u ρkγ ∈ L ∞ (0, T ) such that there exists a unique solution W to (2.5), (2.6) with u = u ρkγ and W (·, T ) = W T . Definition 2.3. A state W 0 ∈ H 1 is said to be controllable to a state W T ∈ H 1 with respect to system (2.5), (2.6) in a given time T > 0 if W T ∈ R ρkγ T (W 0 ). Definition 2.4. A state W 0 ∈ H 1 is said to be approximately controllable to a state W T ∈ H 1 with respect to system (2.5), (2.6) in a given time T > 0 if where the closure is considered in the space H 1 . Thus, the main goal of the paper is to investigate whether the state W 0 is controllable (approximately controllable) to a target state W T with respect to system (2.5), (2.6) in a given time T .
(ii) A state Z 0 is controllable to a state Z T with respect to system (2.8), (2.9) in a time T iff a state W 0 is controllable to a state W T with respect to system (2.5), (2.6) in this time T .
(iii) A state Z 0 is approximately controllable to a state Z T with respect to system (2.8), (2.9) in a time T iff a state W 0 is approximately controllable to a state W T with respect to system (2.5), (2.6) in this time T .
Thus, control system (2.5), (2.6) with a general heat operator replicates the controllability properties of control system (2.8), (2.9) with the simplest heat operator and vice versa.
The main results of the paper are the following two theorems.
Theorem 2.6. If a state W 0 ∈ H 1 is controllable to 0 with respect to system (2.5), (2.6) in a time T > 0, then W 0 = 0. Theorem 2.7. Each state W 0 ∈ H 1 is approximately controllable to any target state W T ∈ H 1 with respect to system (2.5), (2.6) in a given time T > 0.
Taking into account the algorithm given in [10, Section 7], one can construct piecewise constant controls solving the approximate controllability problem for system (2.8), (2.9). Hence, using Theorem 3.3, one can obtain controls solving the approximate controllability problem for system (2.5), (2.6) (see Section 3 below).

The transformation operator T and it's application to a control system
In this section, we recall some properties of the operator T and apply it to control system (2.5), (2.6). We have T = S T r : The operator S : H −1 → H −1 has been introduced and studied in [7,8].
In particular, we have where, according to [20,Chap. 3], the kernel K ∈ C 2 (Ω) is a unique solution to the system Ω = {(y 1 , y 2 ) ∈ R 2 | y 2 > y 1 > 0}, and the kernel L ∈ C 2 (Ω) is determined by the following equation L(y 1 , y 2 ) + K(y 1 , y 2 ) + We also need the following estimates proved in [20,Chap. 3]: where M 0 > 0, M 1 > 0 are constants, and In the following theorems, the application of the transformation operator T to a control system is considered.

8)
where K is a solution to (3.3), r is defined by (3.1). Besides, (2.7) holds and where E 0 > 0 and E 1 > 0 are constants independent of T , Proof. The first part of this theorem is proved similarly to the first part of the corresponding theorem in [7,8] Thus, (2.7) is valid. It follows from Theorem 2.2 (i) that there exists a constant E 0 > 0 such that (3.9) holds.
Due to Theorems 3.3 and 3.6, the operator T not only is a continuous oneto-one mapping between the spaces H s and H s (see Theorem 2.2) but also is one-to-one mapping between the set of the solutions to (1.1)-(1.3) with constant coefficients (ρ = k = 1, γ = 0) where u = u 110 ∈ L ∞ (0, T ) and the set of the solutions to this problem with variable coefficients ρ, k, γ where u = u ρkγ ∈ L ∞ (0, T ), where u 110 and u ρkγ are different generally speaking.
In [10,Section 7], piecewise constant controls u 110 N,l , N, l ∈ N, solving the approximate controllability problem for system (2.8), (2.9), have been constructed. Moreover, the solution to this system with the controls u 110 N,l has been obtained: In addition, it has been proved that Therefore, according to Theorem 3.3, the controls solve the approximate controllability problem for system (2.5), (2.6) with u = u ρkγ N,l . In addition, u ρkγ N,l ∈ L ∞ (0, T ) due to Theorem 3.3. Moreover, W N,l (·, t) = TZ N,l (·, t), t ∈ [0, T ], and W T − W N,l (·, T ) 1 → 0 as N → ∞ and l → ∞.

Examples
We have Due to (1.4), we get Let us consider system (2.8), (2.9) with Z 0 (x) = e − |x| 2 and with the steering is the unique solution to this system and the state Z 0 is controllable to the state Z T with respect to system (2.8), (2.9) in the time T with the control Now consider system (2.5), (2.6) with the given q. According to Theorem 2.5 (ii), the state W 0 = TZ 0 is controllable to the state W T = TZ T with respect to system (2.5), (2.6) in the time T . Moreover, due to Theorem 3.3, a control u ρkγ solving controllability problem for system (2.5), (2.6) is defined by (3.8).
Let us find W 0 , W T and u ρkγ explicitly. Due to (3.1), r(λ) = q • σ −1 = 1 4 e −λ , λ > 0. The kernel of the transformation operator T r has been found in [5,18] for this r. We have where I n is the modified Bessel function, n = 0, ∞. It is well-known (see, e.g., [16, 9.6.28]) that y −n I n (y) = y −n I n+1 (y) and y n I n (y) = y n I n−1 (y), y > 0, n ∈ N. (4.2) Due to the first of these formulae with n = 1, we obtain , y 2 > y 1 > 0. Hence, Taking into account (3.8), we get Substituting y for 1 − e − x 2 and then integrating the first integral by parts, we get With regard to (4.2), we obtain According to the definition of the operator T, we have Due to the definition of the operator T r , we obtain

Controllability Problems for the Heat Equation with Variable Coefficients 19
Replacing by y in the integral, we get ( Hence, Thus, the initial state W 0 defined by (4.5) is controllable to the steering state W T defined by (4.6) with respect to system (2.5), (2.6) in the time T by the control (4.3).
Controllability problems for this system have been considered in Example 4 in [10]. Controls solving the approximate controllability problem for system (2.8), (2.9) have been found in the form