1. Introduction

Up to now, the overwhelming majority of stability analysis and control theory concerning the distributed parameter systems are all limited to the case where distributed parameter is real valued [1, 2]. In this work, complex-valued systems that appear in such fields as quantum mechanics [3] and neural network [4] have been, for the first time, extended to the case of distributed complex-valued parameter systems where delay-dependent sufficient conditions for the input-to-state stability in complex Hilbert spaces are established in terms of linear operator inequality.

In this work, two new crucial lemmas used in complex Hilbert spaces will be developed and thereby our main results are given with detailed illustrations.

2. Preliminaries

Quantum control system, one of the major study intensities of control system, is a typical complex-valued distributed parameter system as also complex-valued neural network. Owing to the significance of this type of distributed parameter system, in view of the typical nonlinearity of Lur'e control system, consider the following Lur'e hyperbolic distributed complex-valued parameter control systems: with the Neumann boundary condition and the initial condition in complex Hilbert spaces where is the complex-valued state, is the imaginary unit, , and and where is an abstract nonlinear function satisfying the following sector condition: with operators Before proceeding, we shall introduce some notations and definitions as follows.

The set of such controls that are measurable and locally essentially bounded in complex Hilbert spaces with the supremum norm is denoted by .

For each and , we denote by the solution trajectory of systems (1) with initial state and control input .

Definition 1. A function is said to be a class -function if it is continuous, zero at zero and strictly increasing. A function is said to be a class -function if for each fixed , the function is a class -function and for each fixed , the function is decreasing and as .
In what follows, we will have a position to define the concept of input-to-state stability (ISS) in complex Hilbert spaces.

Definition 2. System (1) is called input-to-state stable (ISS) in complex Hilbert spaces if there exist a class -function and a class -function such that for any initial state and any bounded control input , it holds that where .

As a key tool for developing the input-to-state stability in this work, some lemmas will be presented and proved as follows.

Lemma 3 (See [5]). The following inequality holds:

Lemma 4 (see [5]). The following inequality holds:

Lemma 5 (see comparison principle [6]). If the function is continuous and satisfies a Lipschitz condition, then the implication is true for continuous functions and .

In the sequel, we shall give our main results using Lemmas 3, 4, and 5.

3. Main Results

Theorem 6. Given a scalar , if there exist scalars and positive definite real-valued matrices and such that the following LMIs hold: where Then system (1) is input-to-state stable with decay rate .

Proof. Using the loop transformation technique [7], it comes to conclude that the absolute input-to-state stability of system (1) in the sector is equivalent to that of the following system: in the sector , where abstract nonlinear function satisfies Choose the following Lyapunov-Krasovskii functional in complex Hilbert spaces:
It follows from (18) that where
Taking the operators
the proof is given in the following steps.
Step  1. To prove that operator is self-adjoint positive definite operator:
Using Lemmas 3 and 4 and inequality (10), we have that from which it follows that In view of LMI (13), positive definiteness of operator is verified.
Step  2. In view of Lemmas 3 and 4 and inequalities (11)-(12), direct computation can obtain that from which it is easy to obtain, in view of LMI (14), that which implies that the inequality holds for any satisfying (17) and hence along the solution trajectories of system (16), by virtue of Lemma 5, we have that It follows from (27) that And hence from Definition 2, the proof is completed.

Remark 7. To illustrate the utility of stability criteria established in this paper, applying Theorem 6 to the Lur'e distributed complex-valued parameter control systems (1) with coefficients , and yields that system (1) is input-to-state stable with decay rate and maximum delay .

Acknowledgments

This work was partially supported by the National Natural Science Foundation of China (60976071, 61174058, 60974052, 61134001, and 60974071), the National Key Basic Research Program, China (2012CB215202), and the 111 Project (B12018).