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Volume 2012 |Article ID 529251 | https://doi.org/10.1155/2012/529251

P. Sangapate, "Adaptive Control and Synchronization of the Shallow Water Model", Mathematical Problems in Engineering, vol. 2012, Article ID 529251, 9 pages, 2012. https://doi.org/10.1155/2012/529251

Adaptive Control and Synchronization of the Shallow Water Model

Academic Editor: Carlo Cattani
Received06 Jul 2011
Revised24 Sep 2011
Accepted26 Sep 2011
Published19 Feb 2012

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 [2โ€“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

๐‘‹ denotes an infinite dimensional Banach Space with the corresponding norm โ€–โ€–, ๐‘… denotes the real line.

Consider a nonlinear nonautonomous differential equation of the general form ฬ‡๐‘ฅ(๐‘ก)=๐‘“(๐‘ก,๐‘ฅ(๐‘ก)),๐‘กโ‰ฅ๐‘ก0๐‘ฅ๎€ท๐‘กโˆˆ๐‘…,0๎€ธ=๐‘ฅ0,(2.1) where the state ๐‘ฅ(๐‘ก) take values in ๐‘‹, ๐‘“(๐‘ก,๐‘ฅ)โˆถ๐‘…ร—๐‘‹โ†’๐‘‹ is a given nonlinear function and ๐‘“(๐‘ก,0)=0, for all ๐‘กโˆˆ๐‘…. 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 ๐œ€>0,๐‘ก0โˆˆ๐‘…, there exists a number ๐›ฟ>0 (depending upon๐œ€ and๐‘ก0) such that for any solution ๐‘ฅ(๐‘ก) of (2.1) with โ€–๐‘ฅ0โ€–<๐›ฟ implies โ€–๐‘ฅ(๐‘ก)โ€–<๐œ€, for all๐‘กโ‰ฅ๐‘ก0.

Definition 2.2. The zero solution of (2.1) is said to be asymptotically stable if it is stable and there is a number ๐›ฟ>0 such that any solution ๐‘ฅ(๐‘ก) with โ€–๐‘ฅ0โ€–<๐›ฟ satisfies lim๐‘กโ†’โˆžโ€–๐‘ฅ(๐‘ก)โ€–=0.
Consider the control system ฬ‡๐‘ฅ(๐‘ก)=๐‘“(๐‘ก,๐‘ฅ(๐‘ก),๐‘ข(๐‘ก)),๐‘กโ‰ฅ0,(2.2) where ๐‘ข(๐‘ก) 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 ๐‘ข(๐‘ก)=๐‘˜(๐‘ฅ(๐‘ก)) such that the system ฬ‡๐‘ฅ(๐‘ก)=๐‘“(๐‘ก,๐‘ฅ(๐‘ก),๐‘˜(๐‘ฅ(๐‘ก))),๐‘กโ‰ฅ0,(2.3) is asymptotically stable.
Consider two nonlinear systems ฬ‡๐‘ฅ=๐‘“(๐‘ก,๐‘ฅ(๐‘ก)),(2.4)ฬ‡๐‘ฆ=๐‘”(๐‘ก,๐‘ฆ(๐‘ก))+๐‘ข(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฆ(๐‘ก)),(2.5) where ๐‘ฅ,๐‘ฆโˆˆ๐‘…,๐‘“,๐‘”โˆˆ๐ถ๐‘Ÿ[๐‘…ร—๐‘…,๐‘…],๐‘ขโˆˆ๐ถ๐‘Ÿ[๐‘…ร—๐‘…ร—๐‘…,๐‘…],๐‘Ÿโ‰ฅ1,๐‘… is the setโ€‰โ€‰of nonnegative real number. Assume that (2.4) is the drive system, (2.5) is the response system, and ๐‘ข(๐‘ก,๐‘ฅ(๐‘ก),๐‘ฆ(๐‘ก))is the control vector.

Definition 2.4. Response system and drive system are said to be synchronic if for any initial conditions๐‘ฅ(๐‘ก0),๐‘ฆ(๐‘ก0)โˆˆ๐‘…,โ€‰โ€‰lim๐‘กโ†’โˆžโ€–๐‘ฅ(๐‘ก)โˆ’๐‘ฆ(๐‘ก)โ€–=0.

Lemma 2.5 (Barbalatโ€™s lemma as used in stability). For nonautonomous system, ฬ‡๐‘ฅ(๐‘ก)=๐‘“(๐‘ก,๐‘ฅ(๐‘ก))(2.6) If there exists a scalar function ๐‘‰(๐‘ฅ,๐‘ก) such that (1)๐‘‰ has a lower bound,(2)ฬ‡๐‘‰โ‰ค0,(3)ฬ‡๐‘‰(๐‘ฅ,๐‘ก) is uniformly continuous in time,then lim๐‘กโ†’โˆžฬ‡๐‘‰(๐‘ฅ,๐‘ก)=0 by applying the Barbalatโ€™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. Eulerโ€™s equations of motion of an ideal fluid are as follows: ๐ท๐‘ข1๐ท๐‘ก=โˆ’๐œŒ๐œ•๐‘๐œ•๐‘ฅ+๐‘“๐‘ฃ,๐ท๐‘ฃ1๐ท๐‘ก=โˆ’๐œŒ๐œ•๐‘๐œ•๐‘ฆโˆ’๐‘“๐‘ข,๐ท๐‘ค1๐ท๐‘ก=โˆ’๐œŒ๐œ•๐‘๐œ•๐‘งโˆ’๐‘”,(3.1) where ๐œŒ is the density of the fluid, ๐‘ is the pressure, ๐‘” is the gravity, and ๐‘“ is coliolis parameter. Using the hydrostatic approximation, ๐œ•๐‘๐œ•๐‘ง=โˆ’๐œŒ๐‘”.(3.2) This implies๐ท๐‘ค/๐ท๐‘ก=0. Assume the pressure ๐‘ of fluid is constant, this implies that ๐œ•๐‘/๐œ•๐‘ก=0 and consider the continuity equation (or the incompressibility condition), ๐œ•๐‘ข+๐œ•๐‘ฅ๐œ•๐‘ฃ+๐œ•๐‘ฆ๐œ•๐‘ค๐œ•๐‘ง=0.(3.3) By solving for ๐œ•๐‘ค/๐œ•๐‘ง and integrating with respect to ๐‘ง, then ๐‘ค can be expressed as ๐œ•๐‘ค๎‚ต๐œ•๐‘ง=โˆ’๐œ•๐‘ข+๐œ•๐‘ฅ๐œ•๐‘ฃ๎‚ถ,๎€œ๐œ•๐‘ฆ๐‘ค=โ„Ž0โˆ’๎‚ต๐œ•๐‘ข+๐œ•๐‘ฅ๐œ•๐‘ฃ๎‚ถ๎‚ต๐œ•๐‘ฆ๐‘‘๐‘ง=โˆ’โ„Ž๐œ•๐‘ข+๐œ•๐‘ฅ๐œ•๐‘ฃ๎‚ถ.๐œ•๐‘ฆ(3.4) The surface (of the fluid) boundary condition on ๐‘ค is that the fluid particles follow the surface(i.e.,๐ทโ„Ž/๐ท๐‘ก=๐‘ค|surface). Thus ๐ทโ„Ž๎‚ต๐ท๐‘ก=โˆ’โ„Ž๐œ•๐‘ข+๐œ•๐‘ฅ๐œ•๐‘ฃ๎‚ถ๐œ•๐‘ฆ.(3.5) To get an expression for the pressure in the fluid, integrate the hydrostatic equation (3.2) from ๐‘=0 at the top downward, ๎€œ๐‘(๐‘ฅ,๐‘ฆ,๐‘ง)=๐‘งโ„Žโˆ’๐‘”๐œŒ๐‘‘๐‘ง=(โ„Žโˆ’๐‘ง)๐œŒ๐‘”.(3.6) Take the partial derivatives of ๐‘ (at the surface) with respect to ๐‘ฅ and ๐‘ฆ, ๐œ•๐‘=๐œ•๐œ•๐‘ฅ๐œ•๐‘ฅ((โ„Žโˆ’๐‘ง)๐œŒ๐‘”)=๐œŒ๐‘”๐œ•โ„Ž1๐œ•๐‘ฅโŸนโˆ’๐œŒ๐œ•๐‘๐œ•๐‘ฅ=โˆ’๐‘”๐œ•โ„Ž,๐œ•๐‘ฅ๐œ•๐‘=๐œ•๐œ•๐‘ฆ๐œ•๐‘ฆ((โ„Žโˆ’๐‘ง)๐œŒ๐‘”)=๐œŒ๐‘”๐œ•โ„Ž1๐œ•๐‘ฆโŸนโˆ’๐œŒ๐œ•๐‘๐œ•๐‘ฆ=โˆ’๐‘”๐œ•โ„Ž.๐œ•๐‘ฆ(3.7) Taking (3.2)โ€“(3.7) into (3.1), so the shallow water model in Cartesian coordinates is as follows: ๐ท๐‘ข๐ท๐‘ก=โˆ’๐‘”๐œ•โ„Ž๐œ•๐‘ฅ+๐‘“๐‘ฃ,๐ท๐‘ฃ๐ท๐‘ก=โˆ’๐‘”๐œ•โ„Ž๐œ•๐‘ฆโˆ’๐‘“๐‘ข,๐ท๐‘ค๎‚ธ๐ท๐‘ก=โˆ’โ„Ž๐œ•๐‘ข+๐œ•๐‘ฅ๐œ•๐‘ฃ๎‚น.๐œ•๐‘ฆ(3.8) In the vector form, the shallow water model is as follows: ฬ‡ฬ‡๐‘‰=โˆ’๐‘“๐‘˜ร—๐‘‰โˆ’โˆ‡ฮฆ,ฮฆ=โˆ’ฮฆโˆ‡๐‘‰,(3.9) where ๐•=๐‘ขโ‡€๐‘–+๐‘ฃโ‡€๐‘— is the horizontal velocity, ฮฆ=๐‘”โ„Ž is the geopotential height.

Consider the controlled system of (3.9) which has the formฬ‡๐‘‰=โˆ’๐‘“๐‘˜ร—๐‘‰โˆ’โˆ‡ฮฆ+๐‘ข1,ฬ‡ฮฆ=โˆ’ฮฆโˆ‡๐‘‰+๐‘ข2,(3.10) where ๐‘ข1,๐‘ข2 is external control input which will drag the chaotic trajectory (๐‘‰,ฮฆ) of the shallow water model to equilibrium point ๐ธ=(๐‘‰,ฮฆ) which is one of two steady states ๐ธ0,๐ธ1.

In this case the control law is ๐‘ข1๎‚€=โˆ’๐‘”๐‘‰โˆ’๐‘‰๎‚,๐‘ข2๎‚€=โˆ’๐‘˜ฮฆโˆ’ฮฆ๎‚,(3.11) where ๐‘˜,๐‘” (estimate of ๐‘˜โˆ—, ๐‘”โˆ—, resp.) are updated according to the following adaptive algorithm: ๎‚€ฬ‡๐‘”=๐œ‡๐‘‰โˆ’๐‘‰๎‚2,ฬ‡๎‚€๐‘˜=๐œŒฮฆโˆ’ฮฆ๎‚2,(3.12) where ๐œ‡,๐œŒ is adaption gains. Then the controlled systems have the following form: ฬ‡๎‚€๐‘‰=โˆ’๐‘“๐‘˜ร—๐‘‰โˆ’โˆ‡ฮฆโˆ’๐‘”๐‘‰โˆ’๐‘‰๎‚ฬ‡๎‚€,(3.13)ฮฆ=โˆ’ฮฆโˆ‡๐‘‰โˆ’๐‘˜ฮฆโˆ’ฮฆ๎‚.(3.14)

Theorem 3.1. For ๐‘”<๐‘”โˆ—,๐‘˜<๐‘˜โˆ—, the equilibrium point ๐ธ=(๐‘‰,ฮฆ) of the system (3.13), (3.14) is asymptotically stable.

Proof. Let us consider the Lyapunov function ๐‘‰๎€ท๐œ‰1,๐œ‰2,๐œ‰3๎€ธ=12๎‚ธ๎‚€๐‘‰โˆ’๐‘‰๎‚2+๎‚€ฮฆโˆ’ฮฆ๎‚2+1๐œ‡๎€ท๐‘”โˆ’๐‘”โˆ—๎€ธ2+1๐œŒ๎€ท๐‘˜โˆ’๐‘˜โˆ—๎€ธ2๎‚น.(3.15) The time derivative of ๐‘‰ is ฬ‡๎‚€๐‘‰=๐‘‰โˆ’๐‘‰๎‚ฬ‡๎‚€๐‘‰+ฮฆโˆ’ฮฆ๎‚ฬ‡1ฮฆ+๐œ‡๎€ท๐‘”โˆ’๐‘”โˆ—๎€ธ1ฬ‡๐‘”+๐œŒ๎€ท๐‘˜โˆ’๐‘˜โˆ—๎€ธฬ‡๐‘˜.(3.16) By substituting (3.13)-(3.14) in (3.16), ฬ‡๐‘‰=๎‚€๐‘‰โˆ’๐‘‰๎‚€๎‚๎‚ƒโˆ’๐‘“๐‘˜ร—๐‘‰โˆ’โˆ‡ฮฆโˆ’๐‘”๐‘‰โˆ’๐‘‰+๎‚€๎‚๎‚„ฮฆโˆ’ฮฆ๎‚€๎‚๎‚ƒโˆ’ฮฆโˆ‡๐‘‰โˆ’๐‘˜ฮฆโˆ’ฮฆ+1๎‚๎‚„๐œ‡๎€ท๐‘”โˆ’๐‘”โˆ—๎€ธ๐œ‡๎‚€๐‘‰โˆ’๐‘‰๎‚2+1๐œŒ๎€ท๐‘˜โˆ’๐‘˜โˆ—๎€ธ๐œŒ๎‚€ฮฆโˆ’ฮฆ๎‚2.(3.17) Let ๐œ‚1=(๐‘‰โˆ’๐‘‰),๐œ‚2=(ฮฆโˆ’ฮฆ). Since (๐‘‰,ฮฆ) is an equilibrium point of the uncontrolled system (3.9), ฬ‡๐‘‰ becomes ฬ‡๐‘‰=๐œ‚1๎‚ƒ๎‚€โˆ’๐‘“๐‘˜ร—๐‘‰โˆ’โˆ‡ฮฆโˆ’๐‘”๐‘‰โˆ’๐‘‰๎‚๎‚„+๐œ‚2๎‚ƒ๎‚€โˆ’ฮฆโˆ‡๐‘‰โˆ’๐‘˜ฮฆโˆ’ฮฆ+๎€ท๎‚๎‚„๐‘”โˆ’๐‘”โˆ—๎€ธ๐œ‚21+๎€ท๐‘˜โˆ’๐‘˜โˆ—๎€ธ๐œ‚22=(โˆ’๐‘“๐‘˜ร—๐‘‰)๐œ‚1โˆ’โˆ‡ฮฆ๐œ‚1โˆ’๐‘”๐œ‚21โˆ’ฮฆโˆ‡V๐œ‚2โˆ’๐‘˜๐œ‚22+๎€ท๐‘”โˆ’๐‘”โˆ—๎€ธ๐œ‚21+๎€ท๐‘˜โˆ’๐‘˜โˆ—๎€ธ๐œ‚22.(3.18) It is clear that if we choose ๐‘”<๐‘”โˆ— and ๐‘˜<๐‘˜โˆ—, then ฬ‡๐‘‰ is negative semidefinite. Since ๐‘‰ is positive definite and ฬ‡๐‘‰ is negative semidefinite, ๐œ‚1,๐œ‚2,๐‘”,๐‘˜โˆˆ๐ฟโˆž. From ฬ‡๐‘‰(๐‘ก)โ‰ค0, we can easily show that the square of ๐œ‚1,๐œ‚2 is integrable with respect to ๐‘ก, namely, ๐œ‚1,๐œ‚2โˆˆ๐ฟ2. From (3.13)-(3.14), for any initial conditions, we have ฬ‡๐œ‚1,ฬ‡๐œ‚2โˆˆ๐ฟโˆž. By the well-known Barbalatโ€™s Lemma, we conclude that ๐œ‚1,๐œ‚2โ†’(0,0) as๐‘กโ†’โˆž. Therefore, the equilibrium point ๐ธ=(๐‘‰,ฮฆ) 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 Barbalatโ€™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 ฬ‡๐‘‰=โˆ’๐‘“1๐‘˜1ร—๐‘‰1โˆ’โˆ‡ฮฆ1,ฬ‡ฮฆ=โˆ’ฮฆ1โˆ‡๐‘‰1,ฬ‡๐‘‰=โˆ’๐‘“2๐‘˜2ร—๐‘‰2โˆ’โˆ‡ฮฆ2โˆ’๐‘ข1,ฬ‡ฮฆ=โˆ’ฮฆ2โˆ‡๐‘‰2โˆ’๐‘ข2(4.1) where ๐‘ข=[๐‘ข1,๐‘ข2]๐‘‡ is the controller. We choose ๐‘ข1=๐‘˜โ€ฒ1๐‘’๐‘‰,๐‘ข2=๐‘˜๎…ž2๐‘’ฮฆ,(4.2) where ๐‘˜๎…ž1,๐‘˜๎…ž2โ‰ฅ0 and ๐‘’๐‘‰,๐‘’ฮฆare the error states which are defined as follows ๐‘’๐‘‰=๐‘‰2โˆ’๐‘‰1,๐‘’ฮฆ=ฮฆ2โˆ’ฮฆ1.(4.3)

Theorem 4.1. Let ๐‘˜1,๐‘“1,๐‘˜๎…ž1,๐‘˜๎…ž2โ‰ฅ0 be property chosen so that the following matrix inequalities holds: โŽ›โŽœโŽœโŽ๐‘˜๐‘ƒ=1๐‘“1+๐‘˜๎…ž100๐‘˜๎…ž2โŽžโŽŸโŽŸโŽ >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 ฬ‡๐‘’๐‘‰=โˆ’๐‘“2๐‘˜2ร—๐‘‰2โˆ’โˆ‡ฮฆ2+๐‘“1๐‘˜1ร—๐‘‰1+โˆ‡ฮฆ1โˆ’๐‘ข1,ฬ‡๐‘’ฮฆ=โˆ’ฮฆ2โˆ‡๐‘‰2+ฮฆ1โˆ‡๐‘‰1โˆ’๐‘ข2.(4.5) Let ๐‘’๐‘˜๐‘“=๐‘˜2๐‘“2โˆ’๐‘˜1๐‘“1. Choose the Lyapunov function as follows: 1๐‘‰(๐‘ก)=2๎€บ๐‘’2๐‘‰+๐‘’2ฮฆ๎€ป.(4.6) Then the differentiation of ๐‘‰ along trajectories of (4.6) is ฬ‡๐‘‰(๐‘ก)=๐‘’๐‘‰ฬ‡๐‘’๐‘‰+๐‘’ฮฆฬ‡๐‘’ฮฆ=๐‘’๐‘‰๎€บโˆ’๐‘“2๐‘˜2ร—๐‘‰2โˆ’โˆ‡ฮฆ2+๐‘“1๐‘˜1ร—๐‘‰1+โˆ‡ฮฆ1โˆ’๐‘ข1๎€ป+๐‘’ฮฆ๎€บโˆ’ฮฆ2โˆ‡๐‘‰2+ฮฆ1โˆ‡๐‘‰1โˆ’๐‘ข2๎€ป=โˆ’๐‘’๐‘‰๎€บ๐‘“2๐‘˜2ร—๐‘‰2+โˆ‡ฮฆ2โˆ’๐‘“1๐‘˜1ร—๐‘‰1โˆ’โˆ‡ฮฆ1+๐‘ข1๎€ปโˆ’๐‘’ฮฆ๎€บฮฆ2โˆ‡๐‘‰2โˆ’ฮฆ1โˆ‡๐‘‰1+๐‘ข2๎€ป=โˆ’๐‘’๐‘‰๎€บ๐‘“2๐‘˜2ร—๐‘‰2โˆ’๐‘“1๐‘˜1ร—๐‘‰1+๐‘“1๐‘˜1ร—๐‘‰2โˆ’๐‘“1๐‘˜1ร—๐‘‰2๎€ปโˆ’๐‘’๐‘‰๎€บโˆ‡ฮฆ2โˆ’โˆ‡ฮฆ1๎€ปโˆ’๐‘’๐‘‰๐‘ข1โˆ’๐‘’ฮฆ๎€บฮฆ2โˆ‡๐‘‰2โˆ’ฮฆ1โˆ‡๐‘‰1+ฮฆ1โˆ‡๐‘‰2โˆ’ฮฆ1โˆ‡๐‘‰2๎€ปโˆ’๐‘’ฮฆ๐‘ข2=โˆ’๐‘’๐‘‰๎€บ๐‘’๐‘˜๐‘“ร—๐‘‰2+๐‘“1๐‘˜1๎€ท๐‘‰2โˆ’๐‘‰1๎€ธ๎€ปโˆ’๐‘’๐‘‰โˆ‡๎€ทฮฆ2โˆ’ฮฆ1๎€ธโˆ’๐‘’๐‘‰๐‘˜๎…ž1๐‘’๐‘‰โˆ’๐‘’ฮฆฮฆ๎€บ๎€ท2โˆ’ฮฆ1๎€ธโˆ‡๐‘‰2+ฮฆ1โˆ‡๎€ท๐‘‰2โˆ’๐‘‰1๎€ธ๎€ปโˆ’๐‘’ฮฆ๐‘˜๎…ž2๐‘’ฮฆ=โˆ’๐‘’๐‘‰๎€บ๐‘’๐‘˜๐‘“ร—๐‘‰2+๐‘“1๐‘˜1๐‘’๐‘‰๎€ปโˆ’๐‘’๐‘‰โˆ‡๐‘’ฮฆโˆ’๐‘’2๐‘‰๐‘˜๎…ž1โˆ’๐‘’ฮฆ๎€บ๐‘’ฮฆโˆ‡๐‘‰2+ฮฆ1โˆ‡๐‘’๐‘‰๎€ปโˆ’๐‘’2ฮฆ๐‘˜๎…ž2=โˆ’๐‘’๐‘‰๐‘’๐‘˜๐‘“ร—๐‘‰2+๐‘“1๐‘˜1๐‘’2๐‘‰โˆ’๐‘’๐‘‰โˆ‡๐‘’ฮฆโˆ’๐‘’2๐‘‰๐‘˜๎…ž1โˆ’๐‘’2ฮฆโˆ‡๐‘‰2โˆ’๐‘’ฮฆฮฆ1โˆ‡๐‘’๐‘‰โˆ’๐‘’2ฮฆ๐‘˜๎…ž2โ‰คโˆ’๐‘“1๐‘˜1๐‘’2๐‘‰โˆ’๐‘’2๐‘‰๐‘˜๎…ž1โˆ’๐‘’2ฮฆ๐‘˜๎…ž2๎€ท๐‘“โ‰คโˆ’1๐‘˜1+๐‘˜๎…ž1๎€ธ๐‘’2๐‘‰โˆ’๐‘˜๎…ž2๐‘’2ฮฆ=โˆ’๐‘’๐‘‡๐‘ƒ๐‘’,(4.7) where ๐‘ƒ is as in (4.4). Since ๐‘‰(๐‘ก) is positive definite and ฬ‡๐‘‰(๐‘ก) is negative semidefinite, it follows that ๐‘’๐‘‰,๐‘’ฮฆ,๐‘˜1,๐‘“1,๐‘˜๎…ž1,๐‘˜๎…ž2โˆˆ๐ฟโˆž. Fromฬ‡๐‘‰(๐‘ก)โ‰คโˆ’๐‘’๐‘‡๐‘ƒ๐‘’, we can easily show that the square of ๐‘’๐‘‰,๐‘’ฮฆ is integrable with respect to ๐‘ก, namely, ๐‘’๐‘‰,๐‘’ฮฆโˆˆ๐ฟ2. From (4.5), for any initial conditions, we have ฬ‡๐‘’๐‘‰(๐‘ก),ฬ‡๐‘’ฮฆ(๐‘ก)โˆˆ๐ฟโˆž. By the well-known Barbalatโ€™s Lemma, we conclude that (๐‘’๐‘‰,๐‘’ฮฆ)โ†’(0,0) as ๐‘กโ†’โˆž. Therefore, in the closed-loop system, ๐‘‰2(๐‘ก)โ†’๐‘‰1(๐‘ก),ฮฆ2(๐‘ก)โ†’ฮฆ1(๐‘ก)as ๐‘กโ†’โˆž. 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 Barbalatโ€™s Lemma.

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Copyright © 2012 P. Sangapate. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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