Approximate Controllability of Semilinear Control System Using Tikhonov Regularization

For an approximately controllable semilinear system, the problem of computing control for a given target state is converted into an equivalent problem of solving operator equation which is ill-posed. We exhibit a sequence of regularized controls which steers the semilinear control system from an arbitrary initial state 𝑥 0 to an 𝜖 neighbourhood of the target state 𝑥 𝜏 at time 𝜏 > 0 under the assumption that the nonlinear function 𝑓 is Lipschitz continuous. The convergence of the sequences of regularized controls and the corresponding mild solutions are shown under some assumptions on the system operators. It is also proved that the target state corresponding to the regularized control is close to the actual state to be attained.

Let and be Hilbert spaces called state and control spaces, respectively. Let = 2 ( , ) and = 2 ( , ) be the function spaces. The inner product and the corresponding norm on a Hilbert space are denoted by ⟨⋅, ⋅⟩, ‖⋅‖, respectively.
For ∈ , the mild solution (see [13]) of (1) is given by (2) The control system (1) is said to be exactly controllable if, for every 0 and ∈ , there exists ∈ such that the mild solution ∈ verifies the condition ( ) = .
The control system (1) is said to be approximately controllable if, for every > 0 and for every 0 and ∈ , there exists ∈ such that the corresponding mild solution ∈ satisfies In [3], Naito proved the approximate controllability of semilinear system (1) under some assumptions which are given below.
(iv) For every ∈ , there exists a ∈ ( ) such that L 1 = L 1 , where ( ) is the range of the bounded linear operator and L 1 : → is a bounded linear operator defined as Condition (iv) of Theorem 1 implies that the corresponding linear system (1) * is approximately controllable; for more details one can see the proof in [3].
In this paper, we study the problem of computing control for an approximately controllable semilinear system for a given target state by converting it into an equivalent linear operator equation which is ill-posed. We find sequence of regularized controls { , : ∈ N, > 0} using Tikhonov regularization and the mild solutions { , : ∈ N, > 0} corresponding to { , : ∈ N, > 0}. Under some assumptions we prove the convergence of { , } and { , }.
The outline of the paper is as follows. In Section 2, regularized control, its corresponding mild solutions, their convergence, and limitations due to the presence of nonlinearity are discussed. Section 3 is devoted to illustrating our theory through an example. Conclusions are made in Section 4.

Regularized Control
Definition 2 (well-posed problem). Let U and V be normed linear spaces and L : U → V be a linear operator. The equation is said to be well-posed if the following holds: (i) For every V ∈ V, there exists a unique ∈ U such that L = V.
(ii) L −1 is a bounded operator. Equivalently, for every V ∈ V and for every > 0, there exists a > 0 with the following properties: IfṼ ∈ V with ‖Ṽ − V‖ ≤ and if ,̃∈ U are such that L = V and L̃=Ṽ, then ‖ −̃‖ ≤ . Theorem 4 (Tikhonov regularization, see [14]). Let U and V be Hilbert spaces and L : U → V be a bounded linear operator. Then for each V ∈ V and > 0, there exists a unique (V) ∈ U which minimizes the map Moreover, for each > 0, the map is a bounded linear operator from V to U and V fl (L * L + ) −1 L * V, where L * is the unique adjoint of the bounded linear operator L.
Theorem 5 (see [14]). For > 0, the solution of the operator equation Definition 6. For V ∈ V and > 0, the element ∈ U as in Theorems 4 and 5 is called the Tikhonov regularized solution of L = V.
Let : → be a linear operator defined as > 0 is a regularization parameter (to be chosen appropriately) and , are given by In our analysis, we assume that the control system (1) satisfies Assumption 8. We obtain a sequence of controls and corresponding mild solutions for semilinear system (1) iteratively and also prove that this sequence of controls steers the semilinear control system from an initial state 0 to an neighbourhood of the final state at time > 0. Consider where ( ) = 1 , for all = 0, 1, 2, . . ., and ( ) is a control function such that ( ) = 1 . We start with an initial (guess) International Journal of Differential Equations 3 mild solution 0 ( ). To find ( ) such that ( ) = 1 , we need to solve Since (13) is ill-posed in the sense of Hadamard [21], any small perturbations in V can lead to large deviations in the solution. Hence, in practice it is not advisable to solve (13) directly to obtain ; one has to look for stable approximations , , > 0, such that ‖ , − V ‖ → 0 as → 0. For this we shall use the Tikhonov regularization for obtaining the control function , which is given below: . . .
Similarly, we have International Journal of Differential Equations We have From (19) and (20), we get ) . . .
Since < 1, for large value of , the sequence { , } is also Cauchy; hence it converges. This completes the proof.

Remark 10.
In practice, to obtain better approximation to the sequence of controls, (regularization parameter) can be chosen such that < 1; that is, If ( − 0 ) ≪ 1 then is very small. Then we get better approximation.
In many practical semilinear control systems, the nonlinear part is a perturbation, in the sense that the Lipschitz constant is sufficiently small so that the system is approximately controllable. In particular, the regularization parameter > , where is very small. Then can also be chosen sufficiently small. Hence we get a regularized control close to the exact solution.

Application for an Approximately Controllable System
In this section, we illustrate the theory for an approximately controllable semilinear system. Let fl lim →∞ , be the regularized control. Let be the mild solution corresponding to .
Then from Theorem 4 we see that which shows that the target state corresponding to the regularized control ( ( )) is close to the actual state ( ( )) to be attained.
Example 11. Consider the semilinear heat equation given by the partial differential equation (0, ) = 0 = (ℓ, ) , where ( , ) represents the temperature at position at time , 0 ( ) is the initial temperature profile, and ( , ) is the heat input (control) along the rod and : × → is a nonlinear function which is Lipschitz continuous. We have Define the operator by International Journal of Differential Equations 5 The 0 semigroup generated by the operator [22] is For̃∈ U, the mild solution of (29) is given bỹ Let 0 : U → V be the operator defined by Then we have Since the semigroup (31) is compact, is a compact operator; consequently the control system (24) is approximately controllable. The control system (24) satisfies Assumption 8. Hence, the regularized control of system (24) for a given target state (desired temperature profile) is obtained as follows:̃, wherẽ, ( ) = , for all = 0, 1, 2, . . ., and̃, ( ) is a control function such that̃, ( ) = .
In order to obtain better approximation to the regularized control, the regularization parameter can be chosen in such a way that > 8 2 /(2 − 4 ) 2 , ̸ = 1/2. Then the semilinear control system (41) satisfies Assumption 8. Hence, the convergence of the sequences of regularized controls { , } and the corresponding mild solutions { , } follows from Theorem 9.

Conclusions
In the mathematical control theory literature, Tikhonov regularization is not given much attention to the problems related to approximately controllable system. We use the Tikhonov regularization method and exhibited a sequence of regularized controls and their corresponding mild solutions. The convergence of the sequences under some assumptions has also been established. The results are illustrated with an example. However, the case where ̸ = should be considered for future work as the theory will change substantially.