Approximate Boundary Controllability for Semilinear Delay Differential Equations
This paper considers the approximate controllability for a class of semilinear delay control systems described by a semigroup formulation with boundary control. Sufficient conditions for approximate controllability are established provided the approximate controllability of corresponding linear systems.
In this paper, we consider the boundary control system described by the following delay differential equation: where system state takes values in a Banach space ; control function takes values in another Banach space and for ; is a closed, densely defined linear operator; is a linear operator from to a Banach space ; is a linear bounded operator; is a nonlinear perturbation function, where is the Banach space of all continuous functions from to endowed with the supremum norm. For any and , is defined by for .
In most applications, the state space is a space of functions on some domain of the Euclidean space , is a partial differential operator on , and is a partial differential operator acting on the boundary of .
Several abstract settings have been developed to describe control systems with boundary control; see Barbu , Fattorini , Lasiecka , and Washburn . In this paper, we use the setting developed in  to discuss the approximate controllability of system (1.1).
The norms in spaces and are denoted by and , respectively. In other spaces, we use the norm notation with a space name in the subindex such as , , and .
Let be the linear operator defined by We impose the following assumptions throughout the paper. (H1) and the restriction of to is continuous relative to the graph norm of .(H2)The operator is the infinitesimal generator of an analytic semigroup for on .(H3)There exists a linear continuous operator and a positive constant such that (H4)For each and , one has . Also, there exists a positive function with such that (H5)There exists a positive number such that for all and .
The following system is called the corresponding linear system of (1.6)
Approximate controllability for semilinear control systems with distributed controls has been extensively studied in the literature under different conditions; see Fabre et al. , Fernandez and Zuazua , Li and Yong , Mahmudov , Naito , Seidman , Wang [11, 12], and many other papers. However, only a few papers dealt with approximate boundary controllability for semilinear control systems, in particular, semilinear delay control systems; the main difficulty is encountered in the construction of suitable integral equation to apply for different versions of fixed-point theorem. Balachandran and Anandhi  considered the controllability of boundary control integrodifferential system, Han and Park  studied the boundary controllability of nonlinear systems with nonlocal initial condition. MacCamy et al.  discussed the approximate controllability for the heat equations. The purpose of this paper is to study the approximate controllability for a class of semilinear delay systems with boundary control.
2. Mild Solutions
Theorem 2.1. If (H1)–(H5) are satisfied, then system (1.6) has a unique solution for each control .
and define . It is easy to know that satisfies
Let . Then, is a Banach space with supremum norm. For any , define an operator as follows:
We need to show that is well defined. First, we show that for any and . Indeed, we have from (H5) that , where . For any and , we have
and , where .
Note that and that Combining (2.5) and (2.6), we prove that for any and .
Next, we show that maps into , in other words, for any . Taking , with , then Since is an analytic semigroup, (2.6) implies that as Also, from (2.5), we have Notice that and as follow from estimates We have as and, hence, .
Now, we prove that is a contraction mapping for sufficiently large . In fact, for any , Therefore, Similarly, By mathematical induction, we have Hence, and is a contraction mapping for sufficiently large . The contraction mapping principle implies that has a unique fixed-point in , which is the unique solution of (1.6). The proof of the theorem is complete.
3. Approximate Controllability
The solution of (1.6) is denoted by to emphasize the initial time , initial state , and control function . is called the system state at time corresponding to initial pair and the control function . The set is called the reachable set of system (1.6) at time corresponding to initial pair . is the closure of in .
Definition 3.1. System (1.6) is said to be approximately controllable on if for any .
Definition 3.2. System (1.6) is said to be approximately null controllable on if for any and , there is a control function such that .
Similar to nonlinear system (1.6), we define the reachable set of system (1.7) at time corresponding to the initial pair as . The approximate controllability and approximate null controllability for system (1.7) can also be defined similarly.
To consider the approximate controllability of system (1.6), we need two new operators. For any with , , and are defined as: where is the solution of (1.6) with the initial pair and control function in the definition of .
The following result provides sufficient conditions for the approximate controllability of system (1.6).
Proof. We need to show that the reachable set of system (1.6) at time is dense in Banach space , in other words, for any . To this end, given any and . Since (1.7) is approximately controllable on , there exists a control function such that Note that , we can select a sequence such that and Let . Again, the approximate controllability of (1.7) on implies that a control exists such that Define Then . Repeating the procedure, we have three sequences , , and such that , , The solution of (1.6) under the control function is Therefore, for a sufficient large such that . Hence, (3.4) follows, and the proof is complete.
The next theorem is about the approximate null controllability of system (1.6).
Proof. For any and , we need to show that there exists a control function such that . Since system (1.7) is approximately null controllable on , there is a control function such that . Select a sequence as in the proof of Theorem 3.3. Let . There exists a control function such that
due to the assumption that (1.7) is approximately null controllable on .
Similar to the proof of Theorem 3.3, we obtain three sequences , , and such that , , Note that we have The proof of the theorem is complete.
In this section, we provide an example to illustrate the application of the results established in Section 3.
Example 4.1. Consider the following heat control system:
where is a bounded and open subset of the Euclidean space with a sufficiently smooth boundary .
To formulated this system as a boundary control system (1.1), we let , , , , , and . The operator is given by , . Then generates an analytic semigroup in . The operator is the trace operator which is well defined and belongs to for each . Clearly, assumptions (H1) and (H2) are satisfied. Define the linear operator by , where is the unique solution to the Dirichlet boundary-value problem It is proved in  that for every , (4.2) has a unique solution satisfying . This shows that (H3) is satisfied. It is proved in  that there exists a positive constant independent of and such that for all and . In other words, (H4) holds with . Therefore, system (4.1) can be formulated to the form (1.6). Since the corresponding linear system of (4.1) is approximately controllable on any interval with ; see . It follows from Theorem 3.3 that system (4.1) is approximately controllable on if the nonlinear perturbation function satisfies (H5).
X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhaäuser, Basel, Switzerland, 1994.