Approximate Controllability of Semilinear Control System Using Tikhonov Regularization

Abstract

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 to an neighbourhood of the target state at time under the assumption that the nonlinear function 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.

1. Introduction

Controllability is one of the qualitative properties of a control system that occupies an important place in control theory. Controllable systems have many applications in different branches of science and engineering (see [1–12] for an extensive review on controllability literature).

Let and be Hilbert spaces called state and control spaces, respectively. Let and be the function spaces. The inner product and the corresponding norm on a Hilbert space are denoted by , , respectively.

Consider the semilinear control systemwhere is a densely defined closed linear operator which generates a semigroup , . is a bounded linear operator and is a nonlinear function where . If , then the resultant system is called the corresponding linear system which is denoted by .

For , the mild solution (see [13]) of (1) is given by

The control system (1) is said to be exactly controllable if, for every and , there exists such that the mild solution verifies the condition .

The control system (1) is said to be approximately controllable if, for every and for every 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.

Theorem 1 (see [3]). The semilinear control system (1) is approximately controllable under the following conditions:(i)The semigroup is compact .(ii)The nonlinear function is Lipschitz continuous with respect to ; that is, , , , where is Lipschitz constant.(iii), where is a positive constant.(iv)For every , there exists a such that , where is the range of the bounded linear operator and is a bounded linear operator defined as

Condition (iv) of Theorem 1 implies that the corresponding linear system 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 using Tikhonov regularization and the mild solutions corresponding to . 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.

2. Regularized Control

Definition 2 (well-posed problem). Let and be normed linear spaces and be a linear operator. The equationis said to be well-posed if the following holds: (i)For every , there exists a unique such that .(ii) is a bounded operator. Equivalently, for every and for every , there exists a with the following properties: If with and if are such that and , then .

Definition 3 (ill-posed problem). Equation (5) is said to be ill-posed if violates one of the conditions for well-posedness.

Theorem 4 (Tikhonov regularization, see [14]). Let and be Hilbert spaces and be a bounded linear operator. Then for each and , there exists a unique which minimizes the mapMoreover, for each , the mapis a bounded linear operator from to and , where is the unique adjoint of the bounded linear operator .

Theorem 5 (see [14]). For , the solution of the operator equationminimizes the function , and

Definition 6. For and , the element as in Theorems 4 and 5 is called the Tikhonov regularized solution of .

Lemma 7 (14). Let be Hilbert spaces and . Then for ,

For more details on ill-posed problems and regularization methods one can refer to [14–20].

Let be a linear operator defined as

Assumption 8. (i) System (1) is approximately controllable.
(ii) , where 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 to an neighbourhood of the final state at time

Considerwhere , for all , and is a control function such that . We start with an initial (guess) mild solution . To find such that , we need to solvewhere Since (13) is ill-posed in the sense of Hadamard [21], any small perturbations in 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 , , such that as . For this we shall use the Tikhonov regularization for obtaining the control function which is given below:Convergence of and . We have the sequence of regularized controls and the sequence of corresponding mild solutions for each , . The inner product and the corresponding norm on the function space are given below.

For ,

Theorem 9. Under Assumption 8 and for fixed , the sequences , are convergent with respect to in , , respectively.

Proof. As , are complete spaces, it is sufficient to prove and are Cauchy sequences in , , respectively.
We have ,where .
By Assumption 8 of (ii), ; hence for large value of , the sequence is Cauchy. Therefore converges.
Similarly, we haveThusWe haveFrom (19) and (20), we getSince , 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 ; that is, If 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.

3. Application for an Approximately Controllable System

In this section, we illustrate the theory for an approximately controllable semilinear system. Let be the regularized control. Let be the mild solution corresponding to .

Then from Theorem 4 we see thatwhich 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 equationwhere represents the temperature at position at time , 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 where Let , the identity operator on By using the notations , (24) takes the form of a control system defined on which is given below:The semigroup generated by the operator [22] isFor , the mild solution of (29) is given by Let be the operator defined by Then we haveSince 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:where , for all , and is a control function such that .We haveThus, using (36) in (37) we getError involved in the regularization procedure is given byFrom (40), it is clear that as .

Problem 12. Consider (24) and (25) with , and , , , ; that is, Here we have the system constants: , , , and is the Lipschitz constant.

In order to obtain better approximation to the regularized control, the regularization parameter can be chosen in such a way that . 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.

4. 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.

Competing Interests

The authors declare that they have no competing interests.

Acknowledgments

Ravinder Katta gratefully acknowledges the financial support of the University Grants Commission (UGC), New Delhi, India, for his research work.