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

Abstract

In this work, input-to-state stability of Lur’e hyperbolic distributed complex-valued parameter control systems has been addressed. Using comparison principle, delay-dependent sufficient conditions for the input-to-state stability in complex Hilbert spaces are established in terms of linear operator inequalities. Finally, numerical computation illustrates our result.

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: Σ₀∶𝜉̇(𝑡)(𝑡)()(𝑡)(𝑡),𝑧(𝑡)(𝑡)()(𝑡),𝜂(𝑡)((𝑡=𝐴𝜉+𝐵𝜉𝑡−ℎ+𝐶𝑢+𝐷𝜂=𝑀𝜉+𝑁𝜉𝑡−ℎ+𝑅𝑢=−𝜑𝑡,𝑧)) (1) with the Neumann boundary condition 𝑤^(𝑖)(0,𝑡)=𝑤^(𝑖)(𝜋,𝑡)=0(𝑖=0,1) and the initial condition 𝑤(𝑥,𝑡)=𝜙(𝑥,𝑡),𝑡∈[−ℎ,0]in complex Hilbert spaces ()()[],|𝑤|ℋ=𝑤∶𝑤𝑥,𝑡=𝜙𝑥,𝑡,𝑡∈−ℎ,0∈𝑊^2,2))()(),((0,𝜋,ℝs.t.boundarycondition𝑤0,𝑡=𝑤𝜋,𝑡=0 (2) where 𝑤(𝑥,𝑡) is the complex-valued state, 𝑖 is the imaginary unit, 𝑎₀>0,𝑎₁<0, and 𝜉(𝑡)𝑤()𝑤∶=𝑥,𝑡_𝑡()(𝑡)𝑢𝑥,𝑡,𝑢∶=¹()𝑢𝑥,𝑡₂(),𝜂(𝑡)𝜂𝑥,𝑡∶=¹()𝜂𝑥,𝑡₂(𝑥,𝑡)𝑎,𝐴∶=01₀∇²−𝑖𝑎₁∇−𝜇₀,𝐵∶=00−𝑎₂−𝜇₁𝐶∶=00−𝑖𝑏₁𝑏₂,𝑐𝐷∶=00₁𝑐₂𝑚,𝑀=¹00𝑚₂,𝑛𝑁=00₁𝑛₂𝑟,𝑅=00₁𝑟₂ (3) and where 𝜑(𝑡,𝑧(𝑡))∶𝑅×𝐻→𝐻 is an abstract nonlinear function satisfying the following sector condition: 𝜑(𝑡,𝑧(𝑡))−𝐾₁𝑧(𝑡),𝜑(𝑡,𝑧(𝑡))−𝐾₂𝑧(𝑡)≤0 (4) with operators 𝐾₁𝑘∶=¹¹𝑘₁₂𝑝₁₁∇²+𝑝₁₂𝑘₁₃,𝐾₂𝑘∶=²¹𝑘₂₂𝑝₁₁∇²+𝑝₂₂𝑘₂₃. (5) 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 ‖𝑢‖_sup∶=sup{‖𝑢(𝑡)‖∶𝑡≥−ℎ}<∞ is denoted by ℒ_∞.

For each 𝜙∈𝐶([−ℎ,0],ℋ) and 𝑢∈ℒ_∞, we denote by 𝑤(𝑡,𝜙,𝑢) the solution trajectory of systems (1) with initial state 𝜙 and control input 𝑢.

Definition 1. A function 𝛾∶𝐑₊→𝐑₊ is said to be a class 𝒦-function if it is continuous, zero at zero and strictly increasing. A function 𝛽∶𝐑₊×𝐑₊→𝐑₊ is said to be a class 𝒦ℒ-function if for each fixed 𝑡≥0, the function 𝛽(⋅,𝑡) is a class 𝒦-function and for each fixed 𝑠≥0, the function 𝛽(𝑠,⋅) is decreasing and 𝛽(𝑠,𝑡)→0 as 𝑡→∞.
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 𝛽∶𝐑₊×𝐑₊→𝐑₊ and a class 𝒦-function 𝛾∶𝐑₊→𝐑₊ such that for any initial state 𝜙∈𝐶([−ℎ,0],ℋ) and any bounded control input 𝑢∈ℒ_∞, it holds that 𝑥𝑡,𝜙,𝑢≤𝛽𝜙_ℎ,𝑡+𝛾‖𝑢‖_sup, (6) where ‖𝜙‖_ℎ∶=sup{‖𝜙(𝜃)‖∶−ℎ≤𝜃≤0}.

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: ⟨𝑤()(≤1𝑥,𝑡,𝑖∇𝑤𝑥,𝑡)⟩2⟨𝑤()(−𝑤()𝑥,𝑡,𝑤𝑥,𝑡)⟩𝑥,𝑡,∇²𝑤(𝑥,𝑡). (7)

Lemma 4 (see [5]). The following inequality holds: 𝑤(𝑥,𝑡),∇²𝑤(1𝑥,𝑡)≤−2⟨𝑤(𝑥,𝑡),𝑤(𝑥,𝑡)⟩. (8)

Lemma 5 (see comparison principle [6]). If the function 𝑔(𝑥,𝑦) is continuous and satisfies a Lipschitz condition, then the implication 𝐷₊𝑚(𝑥)((𝑥𝐷≤𝑔𝑥,𝑚))₊𝑢(𝑥)((𝑥𝑚𝑥≥𝑔𝑥,𝑢))₀≤𝑢𝑥₀⇒𝑚(𝑥)≤𝑢(𝑥)𝑓𝑜𝑟𝑡≥𝑡₀ (9) is true for continuous functions 𝑚(𝑥) and 𝑢(𝑥).

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

3. Main Results

Theorem 6. Given a scalar 𝛽>0, if there exist scalars 𝑞₀₁,𝑞₀₂,𝑞₀₃>0,𝑝₁>0,𝜀>0 and positive definite real-valued matrices 𝑄>0 and 𝑃₁>0 such that the following LMIs hold: −𝑎₀−ℎ₃−12𝑎₁<0,𝑎₀>0,𝑎₁𝑞<0,₀₂−𝑞₀₃𝛽>0,2𝑝₁𝛽+𝑎₁𝑞₀₂−𝑞₀₃𝛽𝑎>0,₀−ℎ₃𝑞₀₂−𝑞₀₃𝛽−𝑝₁1𝛽>0,Γ∶=2𝑎₀−ℎ₃𝑞₀₃+34𝑎₁𝑞₀₃0+𝑞00⁰¹𝑞₀₂𝑞₀₂𝑞₀₃>0, (10)(11)(12)(13)ℜ𝑄𝐴+𝑄𝐴^𝑇+𝑄𝑄𝐵−𝐷𝐾_1𝑚𝑁∗−𝑒^−2𝛽ℎ𝑄∗∗∗∗𝑄𝐷−𝜀𝑀^𝑇𝐾_2𝑚−𝐾_1𝑚^𝑇𝑄𝐶−𝐷𝐾_1𝑚𝑅−𝜀𝑁^𝑇𝐾_2𝑚−𝐾_1𝑚^𝑇0−2𝜀𝐼−𝜀𝐾_2𝑚−𝐾_1𝑚𝑅∗−2𝑃₁<0, (14) where 𝑞𝑄∶=⁰¹+12𝑎₀−ℎ₃𝑞₀₃+34𝑎₁𝑞₀₃𝑞₀₂𝑞₀₂𝑞₀₃,𝐴∶=𝛽1−ℎ₁−12𝑎₀−ℎ₃−34𝑎₁𝛽−𝜇₀−ℎ₂,𝐾_1𝑚𝑘∶=¹¹𝑘₁₂𝑝₁₂𝑘₁₃,𝐾_2𝑚𝑘∶=²¹𝑘₂₂𝑝₂₂𝑘₂₃,ℎ₁∶=𝑐₁𝑘₁₁𝑚₁+𝑐₂𝑝₁₂𝑚₁,ℎ₂∶=𝑐₁𝑘₁₂𝑚₂+𝑐₂𝑘₁₃𝑚₂,ℎ₃∶=𝑐₂𝑝₁₁𝑚₁>0. (15) Then system (1) is input-to-state stable with decay rate 𝛽>0.

Proof. Using the loop transformation technique [7], it comes to conclude that the absolute input-to-state stability of system (1) in the sector [𝐾₁,𝐾₂] is equivalent to that of the following system: 𝜉̇(𝑡)=𝐴−𝐷𝐾₁𝑀𝜉(𝑡)+𝐵−𝐷𝐾₁𝑁𝜉()𝜂(𝑡)+𝑡−ℎ+𝐷𝐶−𝐷𝐾₁𝑅𝑢(𝑡),𝑧(𝑡)(𝑡)()(𝑡),𝜂(𝑡)𝜑((𝑡,𝜉(𝑡)(𝑡)[]=𝑀𝜉+𝑁𝜉𝑡−ℎ+𝑅𝑢=−𝑡,𝑧))=𝜙,for𝑡∈−ℎ,0 (16) in the sector [0,𝐾₂−𝐾₁], where abstract nonlinear function 𝜑(𝑡,𝑧(𝑡)) satisfies 𝜑(𝑡,𝑧(𝑡)),𝜑(𝑡,𝑧(𝑡))−𝐾₂−𝐾₁𝑧(𝑡)≤0. (17) Choose the following Lyapunov-Krasovskii functional in complex Hilbert spaces: 𝑉(𝑡)=𝜉(𝑡),𝑃𝜉(𝑡)+⁰_−ℎ𝑒^2𝛽𝜃𝜉(𝑡+𝜃),𝑄𝜉(𝑡+𝜃)d𝜃. (18)
It follows from (18) that 𝑉̇(𝑡)+2𝛽𝑉(𝑡)−2𝑢(𝑡),𝑃₁𝑢(𝑡𝜂(𝑡),𝜂(𝑡𝜂(𝑡),𝐾)−2𝜀)−2𝜀₂−𝐾₁((𝑡)()(𝑡=𝜒(𝑡)(𝑡,𝑀𝜉+𝑁𝜉𝑡−ℎ+𝑅𝑢)),Ξ𝜒) (19) where 𝜒(𝑡)𝜉(𝑡)𝜉()𝜂(𝑡)𝑢(𝑡)∶=𝑡−ℎ, (20)Ξ∶=𝐴−𝐷𝐾₁𝑀+𝛽𝐼^∗𝑃+𝑃𝐴−𝐷𝐾₁∗∗∗𝑃𝑀+𝛽𝐼+𝑄𝐵−𝐷𝐾₁𝑁𝑃𝐷−𝜀𝑀^∗𝐾₂−𝐾₁^∗𝑃𝐶−𝐷𝐾₁𝑅−𝑒^−2𝛽ℎ𝑄−𝜀𝑁^∗𝐾₂−𝐾₁^∗0∗−2𝜀𝐼−𝜀𝐾₂−𝐾₁𝑅∗∗−2𝑃₁. (21)
Taking the operators 𝑞𝑃∶=⁰¹−𝑎₀𝑞₀₃+𝑝₁∇²+𝑖𝑎₁𝑞₀₃∇+𝑝₁∇²+ℎ₃𝑞₀₃∇²𝑞₀₂𝑞₀₂𝑞₀₃,𝑄>0 (22)
the proof is given in the following steps.
Step  1. To prove that operator 𝑃 is self-adjoint positive definite operator:
Using Lemmas 3 and 4 and inequality (10), we have that 𝑤(𝑥,𝑡),−𝑎₀−ℎ₃𝑞₀₃∇²𝑤(+𝑤()𝑥,𝑡)𝑥,𝑡,𝑖𝑎₁𝑞₀₃(≥−𝑎∇𝑤𝑥,𝑡)₀−ℎ₃𝑞₀₃−𝑎₁𝑞₀₃2𝑤(𝑥,𝑡),∇²𝑤(+𝑎𝑥,𝑡)₁𝑞₀₃2⟨𝑤()(≥−1𝑥,𝑡,𝑤𝑥,𝑡)⟩2−𝑎₀−ℎ₃𝑞₀₃−𝑎₁𝑞₀₃2⟨𝑤()(+𝑎𝑥,𝑡,𝑤𝑥,𝑡)⟩₁𝑞₀₃2⟨𝑤(𝑥,𝑡),𝑤(𝑥,𝑡)⟩ (23) from which it follows that 𝜉(𝑡)(𝑡≥𝑤()𝑤,𝑃𝜉)𝑥,𝑡_𝑡(),1𝑥,𝑡2𝑎₀−ℎ₃𝑞₀₃+34𝑎₁𝑞₀₃0+𝑞00₀₁𝑞₀₂𝑞₀₂𝑞₀₃𝑤()𝑤𝑥,𝑡_𝑡().𝑥,𝑡 (24) In view of LMI (13), positive definiteness of operator 𝑃 is verified.
Step  2. In view of Lemmas 3 and 4 and inequalities (11)-(12), direct computation can obtain that 𝑤(𝑥,𝑡),−𝑖𝑎₁𝑞₀₂−𝑞₀₃𝛽∇+𝑝₁𝛽∇²𝑤(=𝑤(),𝑥,𝑡)𝑥,𝑡−𝑖𝑎₁𝑞₀₂−𝑞₀₃𝛽∇𝑤(+𝑤()𝑥,𝑡)𝑥,𝑡,𝑝₁𝛽∇²𝑤(≤1𝑥,𝑡)2−𝑎₁𝑞₀₂−𝑞₀₃𝛽𝑤()(+⟨𝑥,𝑡,𝑤𝑥,𝑡)⟩2𝑝₁𝛽+𝑎₁𝑞₀₂−𝑞₀₃𝛽𝑤(𝑥,𝑡),∇²𝑤(≤1𝑥,𝑡)2−𝑝₁3𝛽−2𝑎₁𝑞₀₂−𝑞₀₃𝛽⟨𝑤(𝑥,𝑡),𝑤(𝑥,𝑡)⟩ (25) from which it is easy to obtain, in view of LMI (14), that 𝜒(𝑡)(𝑡≤𝜒(𝑡),𝑄𝑄𝐴,Ξ𝜒)𝐴+^𝐻+𝑄𝑄𝐵−𝐷𝐾_1𝑚𝑁∗−𝑒^−2𝛽ℎ𝑄∗∗∗∗𝑄𝐷−𝜀𝑀^𝑇𝐾_2𝑚−𝐾_1𝑚^𝑇𝑄𝐶−𝐷𝐾_1𝑚𝑅−𝜀𝑁^𝑇𝐾_2𝑚−𝐾_1𝑚^𝑇0−2𝜀𝐼−𝜀𝐾_2𝑚−𝐾_1𝑚𝑅∗−2𝑃₁𝜒(𝑡) (26) which implies that the inequality 𝑉̇(𝑡)+2𝛽𝑉(𝑡)−2⟨𝑢(𝑡),𝑃₁𝑢(𝑡)⟩≤0 holds for any 𝜒(𝑡) satisfying (17) and hence along the solution trajectories of system (16), by virtue of Lemma 5, we have that 𝜆_min(Γ)‖𝑤(𝑥,𝑡)‖²≤𝑉(𝑡)≤𝑒^−2𝛽𝑡𝑉(0)+1𝛽sup_𝑡≥0𝑢(𝑡),𝑃₁𝑢(𝑡)=𝑒^−2𝛽𝑡𝑉(0)+1𝛽𝜆_max𝑃₁sup_𝑡≥0‖𝑢(𝑡)‖²≤𝑒^−𝛽𝑡𝑉(0)+𝜆_max𝑃₁𝛽sup_𝑡≥0‖𝑢(𝑡)‖². (27) It follows from (27) that ‖𝑤(≤1𝑥,𝑡)‖𝜆_min(Γ)𝑒^−𝛽𝑡𝑉(0)+𝜆_max𝑃₁𝛽‖𝑢‖_sup. (28) 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 𝑎₀=30,𝑎₁=−0.12,𝑎₂=3,𝜇₀=20,𝜇₁=−0.2,𝐶=⁰⁰_−4𝑖−2,𝐷=⁰⁰_−0.30.8,𝑀=¹⁰₀₂,𝑁=_001.20.3,𝑅=⁰⁰_0.70.4,𝐾_1𝑚=¹⁰_00.2,𝐾_2𝑚=²⁰_00.9,𝑝₁₁=1,ℎ₁=−0.30,ℎ₂=0.32, andℎ₃=0.80 yields that system (1) is input-to-state stable with decay rate 𝛽=0.30 and maximum delay ℎ_max=2.2421.

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).