Input-to-State Stability of Lur ’ e Hyperbolic Distributed Complex-Valued Parameter Control Systems : LOI Approach

1 Institute of Dynamics and Control, School of Astronautics and Aeronautics, Tsinghua University, Beijing 100084, China 2Department of Automation, School of Information Science and Technology, Dalian Maritime University, Dalian 116026, China 3 School of Industrial Engineering, Tokai University, Kumamoto 862-8652, Japan 4Department of Engineering, Faculty of Engineering and Science, University of Agder, 4898 Grimstad, Norway


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.

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  () (0, ) =  () (, ) = 0 ( = 0, 1) and the initial condition (, ) = (, ),  ∈ [−ℎ, 0] in complex Hilbert spaces where (, ) is the complex-valued state,  is the imaginary unit,  0 > 0,   ] . ( 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 U with the supremum norm For each  ∈ ([−ℎ, 0], H) and  ∈ L ∞ , we denote by (, , ) the solution trajectory of systems (1) with initial state  and control input .Definition 1.A function  : R + → R + is said to be a class K-function if it is continuous, zero at zero and strictly increasing.A function  : R + × R + → R + is said to be a class KL-function if for each fixed  ≥ 0, the function (⋅, ) is a class K-function and for each fixed  ≥ 0, the function (, ⋅) is decreasing and (, ) → 0 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 KL-function  : R + × R + → R + and a class K-function  : R + → R + such that for any initial state  ∈ ([−ℎ, 0], H) and any bounded control input  ∈ L ∞ , 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.
In the sequel, we shall give our main results using Lemmas 3, 4, and 5.
Proof.Using the loop transformation technique [7], it comes to conclude that the absolute input-to-state stability of system (1) in the sector [ 1 ,  2 ] is equivalent to that of the following system: where Taking the operators the proof is given in the following steps.