Abstract

The shallow water model is one of the important models in dynamical systems. This paper investigates the adaptive chaos control and synchronization of the shallow water model. First, adaptive control laws are designed to stabilize the shallow water model. Then adaptive control laws are derived to chaos synchronization of the shallow water model. The sufficient conditions for the adaptive control and synchronization have been analyzed theoretically, and the results are proved using a Barbalat's Lemma.

1. Introduction

A dynamical system is a system that changes over time. Chaotic systems are dynamical systems that are highly sensitive to initial conditions. Chaos phenomena in weather models were first observed by Lorenz equation; a large number of chaos phenomena and chaos behavior have been discovered in physical, social, economical, biological, and electrical systems.

Atmosphere is a dynamical system. An atmospheric model is a set of equations that describes behavior of the atmosphere. The shallow water model is simple model for the atmosphere. Shallow water model is the set of the equations of motion that describes the evolution of a horizontal structure, hydrostatic homogeneous, and incompressible flow on the sphere [1].

The control of chaotic systems is to design state feedback control laws that stabilize the chaotic systems. Control theory is an interdisciplinary branch of engineering and mathematics that deals with the behavior of dynamical systems. The usual objective of control theory is to calculate solutions for the proper corrective action from the controller that result in system stability.

Synchronization of chaotic systems is phenomena that may occur when two or more chaotic oscillators are coupled or when a chaotic oscillator drives another chaotic oscillator, because of the butterfly effect, which causes the exponential divergence of the trajectories of two identical chaotic systems started with nearby the same initial conditions. Synchronizing two chaotic systems is seemingly a very challenging problem in chaos literature [2–6].

In 1990, Pecora and Caroll [7] introduced a method to synchronize two identical chaotic systems and showed that it was possible for some chaotic systems to be completely synchronized. From then on, chaos synchronization has been widely explored in variety of fields including physical system [8], chemical systems [9], ecological systems [10], secure communications [11, 12], and so forth.

In most of the chaos synchronization approaches, the drive-response formalism has been used. If a particular chaotic system is called the drive system and another chaotic system is called the response system, then the idea of synchronization is to use the output of the drive system to control the response system so that the output of the response system tracks the output of drive system asymptotically stable.

This paper is organized as follows. Section 2 gives notations and definitions of the stability in the chaotic system. Section 3 presents the adaptive control chaos of the shallow water model. Section 4 presents adaptive synchronization of the shallow water model. The conclusion discussion is in Section 5.

2. Notations and Definitions

𝑋 denotes an infinite dimensional Banach Space with the corresponding norm β€–β€–, 𝑅 denotes the real line.

Consider a nonlinear nonautonomous differential equation of the general form Μ‡π‘₯(𝑑)=𝑓(𝑑,π‘₯(𝑑)),𝑑β‰₯𝑑0π‘₯ξ€·π‘‘βˆˆπ‘…,0ξ€Έ=π‘₯0,(2.1) where the state π‘₯(𝑑) take values in 𝑋, 𝑓(𝑑,π‘₯)βˆΆπ‘…Γ—π‘‹β†’π‘‹ is a given nonlinear function and 𝑓(𝑑,0)=0, for all π‘‘βˆˆπ‘…. The stability conditions were proposed and presented in [13].

Definition 2.1. The zero solution of (2.1) is said to be stable if for every πœ€>0,𝑑0βˆˆπ‘…, there exists a number 𝛿>0 (depending uponπœ€ and𝑑0) such that for any solution π‘₯(𝑑) of (2.1) with β€–π‘₯0β€–<𝛿 implies β€–π‘₯(𝑑)β€–<πœ€, for all𝑑β‰₯𝑑0.

Definition 2.2. The zero solution of (2.1) is said to be asymptotically stable if it is stable and there is a number 𝛿>0 such that any solution π‘₯(𝑑) with β€–π‘₯0β€–<𝛿 satisfies limπ‘‘β†’βˆžβ€–π‘₯(𝑑)β€–=0.
Consider the control system Μ‡π‘₯(𝑑)=𝑓(𝑑,π‘₯(𝑑),𝑒(𝑑)),𝑑β‰₯0,(2.2) where 𝑒(𝑑) is the external control input. The adaptive control is the control method to design state feedback control laws that stabilize the chaotic systems.

Definition 2.3. The control system (2.2) is stabilizable if there exists feedback control 𝑒(𝑑)=π‘˜(π‘₯(𝑑)) such that the system Μ‡π‘₯(𝑑)=𝑓(𝑑,π‘₯(𝑑),π‘˜(π‘₯(𝑑))),𝑑β‰₯0,(2.3) is asymptotically stable.
Consider two nonlinear systems Μ‡π‘₯=𝑓(𝑑,π‘₯(𝑑)),(2.4)̇𝑦=𝑔(𝑑,𝑦(𝑑))+𝑒(𝑑,π‘₯(𝑑),𝑦(𝑑)),(2.5) where π‘₯,π‘¦βˆˆπ‘…,𝑓,π‘”βˆˆπΆπ‘Ÿ[𝑅×𝑅,𝑅],π‘’βˆˆπΆπ‘Ÿ[𝑅×𝑅×𝑅,𝑅],π‘Ÿβ‰₯1,𝑅 is the set  of nonnegative real number. Assume that (2.4) is the drive system, (2.5) is the response system, and 𝑒(𝑑,π‘₯(𝑑),𝑦(𝑑))is the control vector.

Definition 2.4. Response system and drive system are said to be synchronic if for any initial conditionsπ‘₯(𝑑0),𝑦(𝑑0)βˆˆπ‘…,  limπ‘‘β†’βˆžβ€–π‘₯(𝑑)βˆ’π‘¦(𝑑)β€–=0.

Lemma 2.5 (Barbalat’s lemma as used in stability). For nonautonomous system, Μ‡π‘₯(𝑑)=𝑓(𝑑,π‘₯(𝑑))(2.6) If there exists a scalar function 𝑉(π‘₯,𝑑) such that (1)𝑉 has a lower bound,(2)̇𝑉≀0,(3)̇𝑉(π‘₯,𝑑) is uniformly continuous in time,then limπ‘‘β†’βˆžΜ‡π‘‰(π‘₯,𝑑)=0 by applying the Barbalat’s Lemma to stabilize the chaotic systems.

3. Adaptive Control Chaos of the Shallow Water Model

A chaotic system has complex dynamical behaviors; those posses some special features, such as being extremely sensitive to tiny variations of initial conditions. In this section, adaptive control method is applied to control chaos shallow water model.

Shallow water model is the set of the equations of motion that describes the evolution of a horizontal structure, hydrostatic homogeneous, and incompressible flow on the sphere. Euler’s equations of motion of an ideal fluid are as follows: 𝐷𝑒1𝐷𝑑=βˆ’πœŒπœ•π‘πœ•π‘₯+𝑓𝑣,𝐷𝑣1𝐷𝑑=βˆ’πœŒπœ•π‘πœ•π‘¦βˆ’π‘“π‘’,𝐷𝑀1𝐷𝑑=βˆ’πœŒπœ•π‘πœ•π‘§βˆ’π‘”,(3.1) where 𝜌 is the density of the fluid, 𝑝 is the pressure, 𝑔 is the gravity, and 𝑓 is coliolis parameter. Using the hydrostatic approximation, πœ•π‘πœ•π‘§=βˆ’πœŒπ‘”.(3.2) This implies𝐷𝑀/𝐷𝑑=0. Assume the pressure 𝑝 of fluid is constant, this implies that πœ•π‘/πœ•π‘‘=0 and consider the continuity equation (or the incompressibility condition), πœ•π‘’+πœ•π‘₯πœ•π‘£+πœ•π‘¦πœ•π‘€πœ•π‘§=0.(3.3) By solving for πœ•π‘€/πœ•π‘§ and integrating with respect to 𝑧, then 𝑀 can be expressed as πœ•π‘€ξ‚΅πœ•π‘§=βˆ’πœ•π‘’+πœ•π‘₯πœ•π‘£ξ‚Ά,ξ€œπœ•π‘¦π‘€=β„Ž0βˆ’ξ‚΅πœ•π‘’+πœ•π‘₯πœ•π‘£ξ‚Άξ‚΅πœ•π‘¦π‘‘π‘§=βˆ’β„Žπœ•π‘’+πœ•π‘₯πœ•π‘£ξ‚Ά.πœ•π‘¦(3.4) The surface (of the fluid) boundary condition on 𝑀 is that the fluid particles follow the surface(i.e.,π·β„Ž/𝐷𝑑=𝑀|surface). Thus π·β„Žξ‚΅π·π‘‘=βˆ’β„Žπœ•π‘’+πœ•π‘₯πœ•π‘£ξ‚Άπœ•π‘¦.(3.5) To get an expression for the pressure in the fluid, integrate the hydrostatic equation (3.2) from 𝑝=0 at the top downward, ξ€œπ‘(π‘₯,𝑦,𝑧)=π‘§β„Žβˆ’π‘”πœŒπ‘‘π‘§=(β„Žβˆ’π‘§)πœŒπ‘”.(3.6) Take the partial derivatives of 𝑝 (at the surface) with respect to π‘₯ and 𝑦, πœ•π‘=πœ•πœ•π‘₯πœ•π‘₯((β„Žβˆ’π‘§)πœŒπ‘”)=πœŒπ‘”πœ•β„Ž1πœ•π‘₯βŸΉβˆ’πœŒπœ•π‘πœ•π‘₯=βˆ’π‘”πœ•β„Ž,πœ•π‘₯πœ•π‘=πœ•πœ•π‘¦πœ•π‘¦((β„Žβˆ’π‘§)πœŒπ‘”)=πœŒπ‘”πœ•β„Ž1πœ•π‘¦βŸΉβˆ’πœŒπœ•π‘πœ•π‘¦=βˆ’π‘”πœ•β„Ž.πœ•π‘¦(3.7) Taking (3.2)–(3.7) into (3.1), so the shallow water model in Cartesian coordinates is as follows: 𝐷𝑒𝐷𝑑=βˆ’π‘”πœ•β„Žπœ•π‘₯+𝑓𝑣,𝐷𝑣𝐷𝑑=βˆ’π‘”πœ•β„Žπœ•π‘¦βˆ’π‘“π‘’,𝐷𝑀𝐷𝑑=βˆ’β„Žπœ•π‘’+πœ•π‘₯πœ•π‘£ξ‚Ή.πœ•π‘¦(3.8) In the vector form, the shallow water model is as follows: ̇̇𝑉=βˆ’π‘“π‘˜Γ—π‘‰βˆ’βˆ‡Ξ¦,Ξ¦=βˆ’Ξ¦βˆ‡π‘‰,(3.9) where 𝐕=𝑒⇀𝑖+𝑣⇀𝑗 is the horizontal velocity, Ξ¦=π‘”β„Ž is the geopotential height.

Consider the controlled system of (3.9) which has the forṁ𝑉=βˆ’π‘“π‘˜Γ—π‘‰βˆ’βˆ‡Ξ¦+𝑒1,Μ‡Ξ¦=βˆ’Ξ¦βˆ‡π‘‰+𝑒2,(3.10) where 𝑒1,𝑒2 is external control input which will drag the chaotic trajectory (𝑉,Ξ¦) of the shallow water model to equilibrium point 𝐸=(𝑉,Ξ¦) which is one of two steady states 𝐸0,𝐸1.

In this case the control law is 𝑒1ξ‚€=βˆ’π‘”π‘‰βˆ’π‘‰ξ‚,𝑒2ξ‚€=βˆ’π‘˜Ξ¦βˆ’Ξ¦ξ‚,(3.11) where π‘˜,𝑔 (estimate of π‘˜βˆ—, π‘”βˆ—, resp.) are updated according to the following adaptive algorithm: ̇𝑔=πœ‡π‘‰βˆ’π‘‰ξ‚2,Μ‡ξ‚€π‘˜=πœŒΞ¦βˆ’Ξ¦ξ‚2,(3.12) where πœ‡,𝜌 is adaption gains. Then the controlled systems have the following form: ̇𝑉=βˆ’π‘“π‘˜Γ—π‘‰βˆ’βˆ‡Ξ¦βˆ’π‘”π‘‰βˆ’π‘‰ξ‚Μ‡ξ‚€,(3.13)Ξ¦=βˆ’Ξ¦βˆ‡π‘‰βˆ’π‘˜Ξ¦βˆ’Ξ¦ξ‚.(3.14)

Theorem 3.1. For 𝑔<π‘”βˆ—,π‘˜<π‘˜βˆ—, the equilibrium point 𝐸=(𝑉,Ξ¦) of the system (3.13), (3.14) is asymptotically stable.

Proof. Let us consider the Lyapunov function π‘‰ξ€·πœ‰1,πœ‰2,πœ‰3ξ€Έ=12ξ‚Έξ‚€π‘‰βˆ’π‘‰ξ‚2+ξ‚€Ξ¦βˆ’Ξ¦ξ‚2+1πœ‡ξ€·π‘”βˆ’π‘”βˆ—ξ€Έ2+1πœŒξ€·π‘˜βˆ’π‘˜βˆ—ξ€Έ2ξ‚Ή.(3.15) The time derivative of 𝑉 is ̇𝑉=π‘‰βˆ’π‘‰ξ‚Μ‡ξ‚€π‘‰+Ξ¦βˆ’Ξ¦ξ‚Μ‡1Ξ¦+πœ‡ξ€·π‘”βˆ’π‘”βˆ—ξ€Έ1̇𝑔+πœŒξ€·π‘˜βˆ’π‘˜βˆ—ξ€ΈΜ‡π‘˜.(3.16) By substituting (3.13)-(3.14) in (3.16), ̇𝑉=ξ‚€π‘‰βˆ’π‘‰ξ‚€ξ‚ξ‚ƒβˆ’π‘“π‘˜Γ—π‘‰βˆ’βˆ‡Ξ¦βˆ’π‘”π‘‰βˆ’π‘‰+ξ‚€ξ‚ξ‚„Ξ¦βˆ’Ξ¦ξ‚€ξ‚ξ‚ƒβˆ’Ξ¦βˆ‡π‘‰βˆ’π‘˜Ξ¦βˆ’Ξ¦+1ξ‚ξ‚„πœ‡ξ€·π‘”βˆ’π‘”βˆ—ξ€Έπœ‡ξ‚€π‘‰βˆ’π‘‰ξ‚2+1πœŒξ€·π‘˜βˆ’π‘˜βˆ—ξ€ΈπœŒξ‚€Ξ¦βˆ’Ξ¦ξ‚2.(3.17) Let πœ‚1=(π‘‰βˆ’π‘‰),πœ‚2=(Ξ¦βˆ’Ξ¦). Since (𝑉,Ξ¦) is an equilibrium point of the uncontrolled system (3.9), ̇𝑉 becomes ̇𝑉=πœ‚1ξ‚ƒξ‚€βˆ’π‘“π‘˜Γ—π‘‰βˆ’βˆ‡Ξ¦βˆ’π‘”π‘‰βˆ’π‘‰ξ‚ξ‚„+πœ‚2ξ‚ƒξ‚€βˆ’Ξ¦βˆ‡π‘‰βˆ’π‘˜Ξ¦βˆ’Ξ¦+ξ€·ξ‚ξ‚„π‘”βˆ’π‘”βˆ—ξ€Έπœ‚21+ξ€·π‘˜βˆ’π‘˜βˆ—ξ€Έπœ‚22=(βˆ’π‘“π‘˜Γ—π‘‰)πœ‚1βˆ’βˆ‡Ξ¦πœ‚1βˆ’π‘”πœ‚21βˆ’Ξ¦βˆ‡Vπœ‚2βˆ’π‘˜πœ‚22+ξ€·π‘”βˆ’π‘”βˆ—ξ€Έπœ‚21+ξ€·π‘˜βˆ’π‘˜βˆ—ξ€Έπœ‚22.(3.18) It is clear that if we choose 𝑔<π‘”βˆ— and π‘˜<π‘˜βˆ—, then ̇𝑉 is negative semidefinite. Since 𝑉 is positive definite and ̇𝑉 is negative semidefinite, πœ‚1,πœ‚2,𝑔,π‘˜βˆˆπΏβˆž. From ̇𝑉(𝑑)≀0, we can easily show that the square of πœ‚1,πœ‚2 is integrable with respect to 𝑑, namely, πœ‚1,πœ‚2∈𝐿2. From (3.13)-(3.14), for any initial conditions, we have Μ‡πœ‚1,Μ‡πœ‚2∈𝐿∞. By the well-known Barbalat’s Lemma, we conclude that πœ‚1,πœ‚2β†’(0,0) asπ‘‘β†’βˆž. Therefore, the equilibrium point 𝐸=(𝑉,Ξ¦) of the system (3.13)-(3.14) is asymptotically stable.

4. Adaptive Synchronization of the Shallow Water Model

In this section, the adaptive synchronization is introduced to make two of the shallow water model. The sufficient condition for the synchronization has been analyzed theoretically, and the result is proved using a Barbalat’s Lemma. Assume that there are two shallow water models such that the drive system is to control the response system. The drive and response system are given as ̇𝑉=βˆ’π‘“1π‘˜1×𝑉1βˆ’βˆ‡Ξ¦1,Μ‡Ξ¦=βˆ’Ξ¦1βˆ‡π‘‰1,̇𝑉=βˆ’π‘“2π‘˜2×𝑉2βˆ’βˆ‡Ξ¦2βˆ’π‘’1,Μ‡Ξ¦=βˆ’Ξ¦2βˆ‡π‘‰2βˆ’π‘’2(4.1) where 𝑒=[𝑒1,𝑒2]𝑇 is the controller. We choose 𝑒1=π‘˜β€²1𝑒𝑉,𝑒2=π‘˜ξ…ž2𝑒Φ,(4.2) where π‘˜ξ…ž1,π‘˜ξ…ž2β‰₯0 and 𝑒𝑉,𝑒Φare the error states which are defined as follows 𝑒𝑉=𝑉2βˆ’π‘‰1,𝑒Φ=Ξ¦2βˆ’Ξ¦1.(4.3)

Theorem 4.1. Let π‘˜1,𝑓1,π‘˜ξ…ž1,π‘˜ξ…ž2β‰₯0 be property chosen so that the following matrix inequalities holds: βŽ›βŽœβŽœβŽπ‘˜π‘ƒ=1𝑓1+π‘˜ξ…ž100π‘˜ξ…ž2⎞⎟⎟⎠>0,(4.4) then the two shallow water models (4.1) can be synchronized under the adaptive control (4.2).

Proof. It is easy to see from (4.1) that the error system is ̇𝑒𝑉=βˆ’π‘“2π‘˜2×𝑉2βˆ’βˆ‡Ξ¦2+𝑓1π‘˜1×𝑉1+βˆ‡Ξ¦1βˆ’π‘’1,̇𝑒Φ=βˆ’Ξ¦2βˆ‡π‘‰2+Ξ¦1βˆ‡π‘‰1βˆ’π‘’2.(4.5) Let π‘’π‘˜π‘“=π‘˜2𝑓2βˆ’π‘˜1𝑓1. Choose the Lyapunov function as follows: 1𝑉(𝑑)=2𝑒2𝑉+𝑒2Ξ¦ξ€».(4.6) Then the differentiation of 𝑉 along trajectories of (4.6) is ̇𝑉(𝑑)=𝑒𝑉̇𝑒𝑉+𝑒Φ̇𝑒Φ=π‘’π‘‰ξ€Ίβˆ’π‘“2π‘˜2×𝑉2βˆ’βˆ‡Ξ¦2+𝑓1π‘˜1×𝑉1+βˆ‡Ξ¦1βˆ’π‘’1ξ€»+π‘’Ξ¦ξ€Ίβˆ’Ξ¦2βˆ‡π‘‰2+Ξ¦1βˆ‡π‘‰1βˆ’π‘’2ξ€»=βˆ’π‘’π‘‰ξ€Ίπ‘“2π‘˜2×𝑉2+βˆ‡Ξ¦2βˆ’π‘“1π‘˜1×𝑉1βˆ’βˆ‡Ξ¦1+𝑒1ξ€»βˆ’π‘’Ξ¦ξ€ΊΞ¦2βˆ‡π‘‰2βˆ’Ξ¦1βˆ‡π‘‰1+𝑒2ξ€»=βˆ’π‘’π‘‰ξ€Ίπ‘“2π‘˜2×𝑉2βˆ’π‘“1π‘˜1×𝑉1+𝑓1π‘˜1×𝑉2βˆ’π‘“1π‘˜1×𝑉2ξ€»βˆ’π‘’π‘‰ξ€Ίβˆ‡Ξ¦2βˆ’βˆ‡Ξ¦1ξ€»βˆ’π‘’π‘‰π‘’1βˆ’π‘’Ξ¦ξ€ΊΞ¦2βˆ‡π‘‰2βˆ’Ξ¦1βˆ‡π‘‰1+Ξ¦1βˆ‡π‘‰2βˆ’Ξ¦1βˆ‡π‘‰2ξ€»βˆ’π‘’Ξ¦π‘’2=βˆ’π‘’π‘‰ξ€Ίπ‘’π‘˜π‘“Γ—π‘‰2+𝑓1π‘˜1𝑉2βˆ’π‘‰1ξ€Έξ€»βˆ’π‘’π‘‰βˆ‡ξ€·Ξ¦2βˆ’Ξ¦1ξ€Έβˆ’π‘’π‘‰π‘˜ξ…ž1π‘’π‘‰βˆ’π‘’Ξ¦Ξ¦ξ€Ίξ€·2βˆ’Ξ¦1ξ€Έβˆ‡π‘‰2+Ξ¦1βˆ‡ξ€·π‘‰2βˆ’π‘‰1ξ€Έξ€»βˆ’π‘’Ξ¦π‘˜ξ…ž2𝑒Φ=βˆ’π‘’π‘‰ξ€Ίπ‘’π‘˜π‘“Γ—π‘‰2+𝑓1π‘˜1π‘’π‘‰ξ€»βˆ’π‘’π‘‰βˆ‡π‘’Ξ¦βˆ’π‘’2π‘‰π‘˜ξ…ž1βˆ’π‘’Ξ¦ξ€Ίπ‘’Ξ¦βˆ‡π‘‰2+Ξ¦1βˆ‡π‘’π‘‰ξ€»βˆ’π‘’2Ξ¦π‘˜ξ…ž2=βˆ’π‘’π‘‰π‘’π‘˜π‘“Γ—π‘‰2+𝑓1π‘˜1𝑒2π‘‰βˆ’π‘’π‘‰βˆ‡π‘’Ξ¦βˆ’π‘’2π‘‰π‘˜ξ…ž1βˆ’π‘’2Ξ¦βˆ‡π‘‰2βˆ’π‘’Ξ¦Ξ¦1βˆ‡π‘’π‘‰βˆ’π‘’2Ξ¦π‘˜ξ…ž2β‰€βˆ’π‘“1π‘˜1𝑒2π‘‰βˆ’π‘’2π‘‰π‘˜ξ…ž1βˆ’π‘’2Ξ¦π‘˜ξ…ž2ξ€·π‘“β‰€βˆ’1π‘˜1+π‘˜ξ…ž1𝑒2π‘‰βˆ’π‘˜ξ…ž2𝑒2Ξ¦=βˆ’π‘’π‘‡π‘ƒπ‘’,(4.7) where 𝑃 is as in (4.4). Since 𝑉(𝑑) is positive definite and ̇𝑉(𝑑) is negative semidefinite, it follows that 𝑒𝑉,𝑒Φ,π‘˜1,𝑓1,π‘˜ξ…ž1,π‘˜ξ…ž2∈𝐿∞. Froṁ𝑉(𝑑)β‰€βˆ’π‘’π‘‡π‘ƒπ‘’, we can easily show that the square of 𝑒𝑉,𝑒Φ is integrable with respect to 𝑑, namely, 𝑒𝑉,π‘’Ξ¦βˆˆπΏ2. From (4.5), for any initial conditions, we have ̇𝑒𝑉(𝑑),̇𝑒Φ(𝑑)∈𝐿∞. By the well-known Barbalat’s Lemma, we conclude that (𝑒𝑉,𝑒Φ)β†’(0,0) as π‘‘β†’βˆž. Therefore, in the closed-loop system, 𝑉2(𝑑)→𝑉1(𝑑),Ξ¦2(𝑑)β†’Ξ¦1(𝑑)as π‘‘β†’βˆž. This implies that the two shallow water models have synchronized under the adaptive controls (4.2).

5. Conclusions

In this paper, we applied adaptive control theory for the chaos control and synchronization of the shallow water model. First, we designed adaptive control laws to stabilize the shallow water model based on the adaptive control theory and stability theory. Then, we derived adaptive synchronization to the shallow water model. The sufficient conditions for the adaptive control and synchronization of the shallow water model have been analyzed theoretically, and the results are proved using a Barbalat’s Lemma.