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 [2ā€“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

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

Consider a nonlinear nonautonomous differential equation of the general form Ģ‡š‘„(š‘”)=š‘“(š‘”,š‘„(š‘”)),š‘”ā‰„š‘”0š‘„ī€·š‘”āˆˆš‘…,0ī€ø=š‘„0,(2.1) where the state š‘„(š‘”) take values in š‘‹, š‘“(š‘”,š‘„)āˆ¶š‘…Ć—š‘‹ā†’š‘‹ is a given nonlinear function and š‘“(š‘”,0)=0, for all š‘”āˆˆš‘…. 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 šœ€>0,š‘”0āˆˆš‘…, there exists a number š›æ>0 (depending uponšœ€ andš‘”0) such that for any solution š‘„(š‘”) of (2.1) with ā€–š‘„0ā€–<š›æ implies ā€–š‘„(š‘”)ā€–<šœ€, for allš‘”ā‰„š‘”0.

Definition 2.2. The zero solution of (2.1) is said to be asymptotically stable if it is stable and there is a number š›æ>0 such that any solution š‘„(š‘”) with ā€–š‘„0ā€–<š›æ satisfies limš‘”ā†’āˆžā€–š‘„(š‘”)ā€–=0.
Consider the control system Ģ‡š‘„(š‘”)=š‘“(š‘”,š‘„(š‘”),š‘¢(š‘”)),š‘”ā‰„0,(2.2) where š‘¢(š‘”) 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 š‘¢(š‘”)=š‘˜(š‘„(š‘”)) such that the system Ģ‡š‘„(š‘”)=š‘“(š‘”,š‘„(š‘”),š‘˜(š‘„(š‘”))),š‘”ā‰„0,(2.3) is asymptotically stable.
Consider two nonlinear systems Ģ‡š‘„=š‘“(š‘”,š‘„(š‘”)),(2.4)Ģ‡š‘¦=š‘”(š‘”,š‘¦(š‘”))+š‘¢(š‘”,š‘„(š‘”),š‘¦(š‘”)),(2.5) where š‘„,š‘¦āˆˆš‘…,š‘“,š‘”āˆˆš¶š‘Ÿ[š‘…Ć—š‘…,š‘…],š‘¢āˆˆš¶š‘Ÿ[š‘…Ć—š‘…Ć—š‘…,š‘…],š‘Ÿā‰„1,š‘… is the setā€‰ā€‰of nonnegative real number. Assume that (2.4) is the drive system, (2.5) is the response system, and š‘¢(š‘”,š‘„(š‘”),š‘¦(š‘”))is the control vector.

Definition 2.4. Response system and drive system are said to be synchronic if for any initial conditionsš‘„(š‘”0),š‘¦(š‘”0)āˆˆš‘…,ā€‰ā€‰limš‘”ā†’āˆžā€–š‘„(š‘”)āˆ’š‘¦(š‘”)ā€–=0.

Lemma 2.5 (Barbalatā€™s lemma as used in stability). For nonautonomous system, Ģ‡š‘„(š‘”)=š‘“(š‘”,š‘„(š‘”))(2.6) If there exists a scalar function š‘‰(š‘„,š‘”) such that (1)š‘‰ has a lower bound,(2)Ģ‡š‘‰ā‰¤0,(3)Ģ‡š‘‰(š‘„,š‘”) is uniformly continuous in time,then limš‘”ā†’āˆžĢ‡š‘‰(š‘„,š‘”)=0 by applying the Barbalatā€™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. Eulerā€™s equations of motion of an ideal fluid are as follows: š·š‘¢1š·š‘”=āˆ’šœŒšœ•š‘šœ•š‘„+š‘“š‘£,š·š‘£1š·š‘”=āˆ’šœŒšœ•š‘šœ•š‘¦āˆ’š‘“š‘¢,š·š‘¤1š·š‘”=āˆ’šœŒšœ•š‘šœ•š‘§āˆ’š‘”,(3.1) where šœŒ is the density of the fluid, š‘ is the pressure, š‘” is the gravity, and š‘“ is coliolis parameter. Using the hydrostatic approximation, šœ•š‘šœ•š‘§=āˆ’šœŒš‘”.(3.2) This impliesš·š‘¤/š·š‘”=0. Assume the pressure š‘ of fluid is constant, this implies that šœ•š‘/šœ•š‘”=0 and consider the continuity equation (or the incompressibility condition), šœ•š‘¢+šœ•š‘„šœ•š‘£+šœ•š‘¦šœ•š‘¤šœ•š‘§=0.(3.3) By solving for šœ•š‘¤/šœ•š‘§ and integrating with respect to š‘§, then š‘¤ can be expressed as šœ•š‘¤ī‚µšœ•š‘§=āˆ’šœ•š‘¢+šœ•š‘„šœ•š‘£ī‚¶,ī€œšœ•š‘¦š‘¤=ā„Ž0āˆ’ī‚µšœ•š‘¢+šœ•š‘„šœ•š‘£ī‚¶ī‚µšœ•š‘¦š‘‘š‘§=āˆ’ā„Žšœ•š‘¢+šœ•š‘„šœ•š‘£ī‚¶.šœ•š‘¦(3.4) The surface (of the fluid) boundary condition on š‘¤ is that the fluid particles follow the surface(i.e.,š·ā„Ž/š·š‘”=š‘¤|surface). Thus š·ā„Žī‚µš·š‘”=āˆ’ā„Žšœ•š‘¢+šœ•š‘„šœ•š‘£ī‚¶šœ•š‘¦.(3.5) To get an expression for the pressure in the fluid, integrate the hydrostatic equation (3.2) from š‘=0 at the top downward, ī€œš‘(š‘„,š‘¦,š‘§)=š‘§ā„Žāˆ’š‘”šœŒš‘‘š‘§=(ā„Žāˆ’š‘§)šœŒš‘”.(3.6) Take the partial derivatives of š‘ (at the surface) with respect to š‘„ and š‘¦, šœ•š‘=šœ•šœ•š‘„šœ•š‘„((ā„Žāˆ’š‘§)šœŒš‘”)=šœŒš‘”šœ•ā„Ž1šœ•š‘„āŸ¹āˆ’šœŒšœ•š‘šœ•š‘„=āˆ’š‘”šœ•ā„Ž,šœ•š‘„šœ•š‘=šœ•šœ•š‘¦šœ•š‘¦((ā„Žāˆ’š‘§)šœŒš‘”)=šœŒš‘”šœ•ā„Ž1šœ•š‘¦āŸ¹āˆ’šœŒšœ•š‘šœ•š‘¦=āˆ’š‘”šœ•ā„Ž.šœ•š‘¦(3.7) Taking (3.2)ā€“(3.7) into (3.1), so the shallow water model in Cartesian coordinates is as follows: š·š‘¢š·š‘”=āˆ’š‘”šœ•ā„Žšœ•š‘„+š‘“š‘£,š·š‘£š·š‘”=āˆ’š‘”šœ•ā„Žšœ•š‘¦āˆ’š‘“š‘¢,š·š‘¤ī‚øš·š‘”=āˆ’ā„Žšœ•š‘¢+šœ•š‘„šœ•š‘£ī‚¹.šœ•š‘¦(3.8) In the vector form, the shallow water model is as follows: Ģ‡Ģ‡š‘‰=āˆ’š‘“š‘˜Ć—š‘‰āˆ’āˆ‡Ī¦,Ī¦=āˆ’Ī¦āˆ‡š‘‰,(3.9) where š•=š‘¢ā‡€š‘–+š‘£ā‡€š‘— is the horizontal velocity, Ī¦=š‘”ā„Ž is the geopotential height.

Consider the controlled system of (3.9) which has the formĢ‡š‘‰=āˆ’š‘“š‘˜Ć—š‘‰āˆ’āˆ‡Ī¦+š‘¢1,Ģ‡Ī¦=āˆ’Ī¦āˆ‡š‘‰+š‘¢2,(3.10) where š‘¢1,š‘¢2 is external control input which will drag the chaotic trajectory (š‘‰,Ī¦) of the shallow water model to equilibrium point šø=(š‘‰,Ī¦) which is one of two steady states šø0,šø1.

In this case the control law is š‘¢1ī‚€=āˆ’š‘”š‘‰āˆ’š‘‰ī‚,š‘¢2ī‚€=āˆ’š‘˜Ī¦āˆ’Ī¦ī‚,(3.11) where š‘˜,š‘” (estimate of š‘˜āˆ—, š‘”āˆ—, resp.) are updated according to the following adaptive algorithm: ī‚€Ģ‡š‘”=šœ‡š‘‰āˆ’š‘‰ī‚2,Ģ‡ī‚€š‘˜=šœŒĪ¦āˆ’Ī¦ī‚2,(3.12) where šœ‡,šœŒ is adaption gains. Then the controlled systems have the following form: Ģ‡ī‚€š‘‰=āˆ’š‘“š‘˜Ć—š‘‰āˆ’āˆ‡Ī¦āˆ’š‘”š‘‰āˆ’š‘‰ī‚Ģ‡ī‚€,(3.13)Ī¦=āˆ’Ī¦āˆ‡š‘‰āˆ’š‘˜Ī¦āˆ’Ī¦ī‚.(3.14)

Theorem 3.1. For š‘”<š‘”āˆ—,š‘˜<š‘˜āˆ—, the equilibrium point šø=(š‘‰,Ī¦) of the system (3.13), (3.14) is asymptotically stable.

Proof. Let us consider the Lyapunov function š‘‰ī€·šœ‰1,šœ‰2,šœ‰3ī€ø=12ī‚øī‚€š‘‰āˆ’š‘‰ī‚2+ī‚€Ī¦āˆ’Ī¦ī‚2+1šœ‡ī€·š‘”āˆ’š‘”āˆ—ī€ø2+1šœŒī€·š‘˜āˆ’š‘˜āˆ—ī€ø2ī‚¹.(3.15) The time derivative of š‘‰ is Ģ‡ī‚€š‘‰=š‘‰āˆ’š‘‰ī‚Ģ‡ī‚€š‘‰+Ī¦āˆ’Ī¦ī‚Ģ‡1Ī¦+šœ‡ī€·š‘”āˆ’š‘”āˆ—ī€ø1Ģ‡š‘”+šœŒī€·š‘˜āˆ’š‘˜āˆ—ī€øĢ‡š‘˜.(3.16) By substituting (3.13)-(3.14) in (3.16), Ģ‡š‘‰=ī‚€š‘‰āˆ’š‘‰ī‚€ī‚ī‚ƒāˆ’š‘“š‘˜Ć—š‘‰āˆ’āˆ‡Ī¦āˆ’š‘”š‘‰āˆ’š‘‰+ī‚€ī‚ī‚„Ī¦āˆ’Ī¦ī‚€ī‚ī‚ƒāˆ’Ī¦āˆ‡š‘‰āˆ’š‘˜Ī¦āˆ’Ī¦+1ī‚ī‚„šœ‡ī€·š‘”āˆ’š‘”āˆ—ī€øšœ‡ī‚€š‘‰āˆ’š‘‰ī‚2+1šœŒī€·š‘˜āˆ’š‘˜āˆ—ī€øšœŒī‚€Ī¦āˆ’Ī¦ī‚2.(3.17) Let šœ‚1=(š‘‰āˆ’š‘‰),šœ‚2=(Ī¦āˆ’Ī¦). Since (š‘‰,Ī¦) is an equilibrium point of the uncontrolled system (3.9), Ģ‡š‘‰ becomes Ģ‡š‘‰=šœ‚1ī‚ƒī‚€āˆ’š‘“š‘˜Ć—š‘‰āˆ’āˆ‡Ī¦āˆ’š‘”š‘‰āˆ’š‘‰ī‚ī‚„+šœ‚2ī‚ƒī‚€āˆ’Ī¦āˆ‡š‘‰āˆ’š‘˜Ī¦āˆ’Ī¦+ī€·ī‚ī‚„š‘”āˆ’š‘”āˆ—ī€øšœ‚21+ī€·š‘˜āˆ’š‘˜āˆ—ī€øšœ‚22=(āˆ’š‘“š‘˜Ć—š‘‰)šœ‚1āˆ’āˆ‡Ī¦šœ‚1āˆ’š‘”šœ‚21āˆ’Ī¦āˆ‡Všœ‚2āˆ’š‘˜šœ‚22+ī€·š‘”āˆ’š‘”āˆ—ī€øšœ‚21+ī€·š‘˜āˆ’š‘˜āˆ—ī€øšœ‚22.(3.18) It is clear that if we choose š‘”<š‘”āˆ— and š‘˜<š‘˜āˆ—, then Ģ‡š‘‰ is negative semidefinite. Since š‘‰ is positive definite and Ģ‡š‘‰ is negative semidefinite, šœ‚1,šœ‚2,š‘”,š‘˜āˆˆšæāˆž. From Ģ‡š‘‰(š‘”)ā‰¤0, we can easily show that the square of šœ‚1,šœ‚2 is integrable with respect to š‘”, namely, šœ‚1,šœ‚2āˆˆšæ2. From (3.13)-(3.14), for any initial conditions, we have Ģ‡šœ‚1,Ģ‡šœ‚2āˆˆšæāˆž. By the well-known Barbalatā€™s Lemma, we conclude that šœ‚1,šœ‚2ā†’(0,0) asš‘”ā†’āˆž. Therefore, the equilibrium point šø=(š‘‰,Ī¦) 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 Barbalatā€™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 Ģ‡š‘‰=āˆ’š‘“1š‘˜1Ɨš‘‰1āˆ’āˆ‡Ī¦1,Ģ‡Ī¦=āˆ’Ī¦1āˆ‡š‘‰1,Ģ‡š‘‰=āˆ’š‘“2š‘˜2Ɨš‘‰2āˆ’āˆ‡Ī¦2āˆ’š‘¢1,Ģ‡Ī¦=āˆ’Ī¦2āˆ‡š‘‰2āˆ’š‘¢2(4.1) where š‘¢=[š‘¢1,š‘¢2]š‘‡ is the controller. We choose š‘¢1=š‘˜ā€²1š‘’š‘‰,š‘¢2=š‘˜ī…ž2š‘’Ī¦,(4.2) where š‘˜ī…ž1,š‘˜ī…ž2ā‰„0 and š‘’š‘‰,š‘’Ī¦are the error states which are defined as follows š‘’š‘‰=š‘‰2āˆ’š‘‰1,š‘’Ī¦=Ī¦2āˆ’Ī¦1.(4.3)

Theorem 4.1. Let š‘˜1,š‘“1,š‘˜ī…ž1,š‘˜ī…ž2ā‰„0 be property chosen so that the following matrix inequalities holds: āŽ›āŽœāŽœāŽš‘˜š‘ƒ=1š‘“1+š‘˜ī…ž100š‘˜ī…ž2āŽžāŽŸāŽŸāŽ >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 Ģ‡š‘’š‘‰=āˆ’š‘“2š‘˜2Ɨš‘‰2āˆ’āˆ‡Ī¦2+š‘“1š‘˜1Ɨš‘‰1+āˆ‡Ī¦1āˆ’š‘¢1,Ģ‡š‘’Ī¦=āˆ’Ī¦2āˆ‡š‘‰2+Ī¦1āˆ‡š‘‰1āˆ’š‘¢2.(4.5) Let š‘’š‘˜š‘“=š‘˜2š‘“2āˆ’š‘˜1š‘“1. Choose the Lyapunov function as follows: 1š‘‰(š‘”)=2ī€ŗš‘’2š‘‰+š‘’2Ī¦ī€».(4.6) Then the differentiation of š‘‰ along trajectories of (4.6) is Ģ‡š‘‰(š‘”)=š‘’š‘‰Ģ‡š‘’š‘‰+š‘’Ī¦Ģ‡š‘’Ī¦=š‘’š‘‰ī€ŗāˆ’š‘“2š‘˜2Ɨš‘‰2āˆ’āˆ‡Ī¦2+š‘“1š‘˜1Ɨš‘‰1+āˆ‡Ī¦1āˆ’š‘¢1ī€»+š‘’Ī¦ī€ŗāˆ’Ī¦2āˆ‡š‘‰2+Ī¦1āˆ‡š‘‰1āˆ’š‘¢2ī€»=āˆ’š‘’š‘‰ī€ŗš‘“2š‘˜2Ɨš‘‰2+āˆ‡Ī¦2āˆ’š‘“1š‘˜1Ɨš‘‰1āˆ’āˆ‡Ī¦1+š‘¢1ī€»āˆ’š‘’Ī¦ī€ŗĪ¦2āˆ‡š‘‰2āˆ’Ī¦1āˆ‡š‘‰1+š‘¢2ī€»=āˆ’š‘’š‘‰ī€ŗš‘“2š‘˜2Ɨš‘‰2āˆ’š‘“1š‘˜1Ɨš‘‰1+š‘“1š‘˜1Ɨš‘‰2āˆ’š‘“1š‘˜1Ɨš‘‰2ī€»āˆ’š‘’š‘‰ī€ŗāˆ‡Ī¦2āˆ’āˆ‡Ī¦1ī€»āˆ’š‘’š‘‰š‘¢1āˆ’š‘’Ī¦ī€ŗĪ¦2āˆ‡š‘‰2āˆ’Ī¦1āˆ‡š‘‰1+Ī¦1āˆ‡š‘‰2āˆ’Ī¦1āˆ‡š‘‰2ī€»āˆ’š‘’Ī¦š‘¢2=āˆ’š‘’š‘‰ī€ŗš‘’š‘˜š‘“Ć—š‘‰2+š‘“1š‘˜1ī€·š‘‰2āˆ’š‘‰1ī€øī€»āˆ’š‘’š‘‰āˆ‡ī€·Ī¦2āˆ’Ī¦1ī€øāˆ’š‘’š‘‰š‘˜ī…ž1š‘’š‘‰āˆ’š‘’Ī¦Ī¦ī€ŗī€·2āˆ’Ī¦1ī€øāˆ‡š‘‰2+Ī¦1āˆ‡ī€·š‘‰2āˆ’š‘‰1ī€øī€»āˆ’š‘’Ī¦š‘˜ī…ž2š‘’Ī¦=āˆ’š‘’š‘‰ī€ŗš‘’š‘˜š‘“Ć—š‘‰2+š‘“1š‘˜1š‘’š‘‰ī€»āˆ’š‘’š‘‰āˆ‡š‘’Ī¦āˆ’š‘’2š‘‰š‘˜ī…ž1āˆ’š‘’Ī¦ī€ŗš‘’Ī¦āˆ‡š‘‰2+Ī¦1āˆ‡š‘’š‘‰ī€»āˆ’š‘’2Ī¦š‘˜ī…ž2=āˆ’š‘’š‘‰š‘’š‘˜š‘“Ć—š‘‰2+š‘“1š‘˜1š‘’2š‘‰āˆ’š‘’š‘‰āˆ‡š‘’Ī¦āˆ’š‘’2š‘‰š‘˜ī…ž1āˆ’š‘’2Ī¦āˆ‡š‘‰2āˆ’š‘’Ī¦Ī¦1āˆ‡š‘’š‘‰āˆ’š‘’2Ī¦š‘˜ī…ž2ā‰¤āˆ’š‘“1š‘˜1š‘’2š‘‰āˆ’š‘’2š‘‰š‘˜ī…ž1āˆ’š‘’2Ī¦š‘˜ī…ž2ī€·š‘“ā‰¤āˆ’1š‘˜1+š‘˜ī…ž1ī€øš‘’2š‘‰āˆ’š‘˜ī…ž2š‘’2Ī¦=āˆ’š‘’š‘‡š‘ƒš‘’,(4.7) where š‘ƒ is as in (4.4). Since š‘‰(š‘”) is positive definite and Ģ‡š‘‰(š‘”) is negative semidefinite, it follows that š‘’š‘‰,š‘’Ī¦,š‘˜1,š‘“1,š‘˜ī…ž1,š‘˜ī…ž2āˆˆšæāˆž. FromĢ‡š‘‰(š‘”)ā‰¤āˆ’š‘’š‘‡š‘ƒš‘’, we can easily show that the square of š‘’š‘‰,š‘’Ī¦ is integrable with respect to š‘”, namely, š‘’š‘‰,š‘’Ī¦āˆˆšæ2. From (4.5), for any initial conditions, we have Ģ‡š‘’š‘‰(š‘”),Ģ‡š‘’Ī¦(š‘”)āˆˆšæāˆž. By the well-known Barbalatā€™s Lemma, we conclude that (š‘’š‘‰,š‘’Ī¦)ā†’(0,0) as š‘”ā†’āˆž. Therefore, in the closed-loop system, š‘‰2(š‘”)ā†’š‘‰1(š‘”),Ī¦2(š‘”)ā†’Ī¦1(š‘”)as š‘”ā†’āˆž. 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 Barbalatā€™s Lemma.