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The optimal control of Chen chaotic dynamical system
Applied Mathematics and Computation 131 (2002) 171–180 www.elsevier.com/locate/amc
The optimal control of Chen chaotic dynamical system M.T. Yassen Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
Abstract This paper is devoted to study the problem of the optimal control for the equilibrium point of a new chaotic system introduced by Chen in a simple three-dimensional autonomous system, which resembles some familiar features from both the Lorenz and Rossler systems. The stability of the equilibrium points is studied. The conditions of the asymptotic stability of the equilibrium point of this system are used to obtain the optimal control functions. The general solution of the dynamical system of the perturbed state is obtained as a function of time. Ó 2002 Elsevier Science Inc. All rights reserved.
1. Introduction In the last few decades optimal control theory has grown rapidly. The essential properties of an optimal control problem are that we have a system which evolves in time according to certain laws. These laws are embodied in equations that contain elements which can be adjusted from outside the system, known as the controls. By suitable choice of these controls it may be possible to force the system into a desired target state [1]. Based on the Lyapunov stability theory, the adaptive control and synchronization problem of two Lorenz systems are investigated in [2], the problem of synchronization of Rossler and Chen chaotic system using active control has been studied in [3]. The problem of optimal control and synchronization of Lotka–Volterra model is studied in [4]. The objective of the present paper is to study the optimal stabilization of the equilibrium points of Chen dynamical system, which has the chaotic attractor
E-mail address: [email protected] (M.T. Yassen). 0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 1 ) 0 0 1 3 7 - 0
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M.T. Yassen / Appl. Math. Comput. 131 (2002) 171–180
when a ¼ 35, b ¼ 3, c ¼ 28 [5]. The equilibrium points of Chen dynamical system are determined. The stability of the equilibrium points is studied. The control functions ensuring the optimal stabilization of these points are obtained as a function of the state variables. The general solution of the equations of the perturbed state is obtained as a function of time. Numerical simulations of the obtained results are presented. 2. Chen dynamical system Chen [5] has found a new chaotic attractor in a simple three-dimensional autonomous system, which resembles some familiar features from both the Lorenz and Rossler systems. The chaotic Chen dynamical system is described by the set of ordinary differential equations: x_ ¼ aðy xÞ; y_ ¼ ðc aÞx xz þ cy;
ð1Þ
z_ ¼ xy bz; where x, y and z are the state variables and a, b, and c are three positive real constants. The chaotic attractors of the system (1) are shown in Figs. 1 and 2 in the xy-plane and xz-plane, respectively. The divergence of the flow (1) is given by rF¼
oF1 oF2 oF3 þ þ ¼ a þ c b < 0; ox oy oz
where F ¼ ðF1 ; F2 ; F3 Þ ¼ ðaðy xÞ; ðc aÞx xz þ cy; xy bzÞ:
Fig. 1.
M.T. Yassen / Appl. Math. Comput. 131 (2002) 171–180
173
Fig. 2.
Then, the system (1) is a forced dissipative system similar to Lorenz system [6]. Thus the solutions of the system of equations (1) are bounded as t ! 1 for positive values of a, b, and c. In the present paper, we will study the problem of optimal control of the controlled Chen dynamical system. The controlled Chen dynamical system is presented mathematically by the following system: x_ ¼ aðy xÞ þ u1 ; y_ ¼ ðc aÞx xz þ cy þ u2 ;
ð2Þ
z_ ¼ xy bz þ u3 ; where the functions u1 , u2 , u3 are the control functions which will be determined from the conditions that ensure the asymptotic stabilization of the equilibrium points of this chaotic system.
3. Stability of the equilibrium points of Chen dynamical system The trajectory of the system (1) is specified by ðxðtÞ; yðtÞ; zðtÞÞ. The equilibrium points of the system (1) are X0 ¼ ð0; 0; 0Þ; X1 ¼ ða; a; 2c aÞ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where a ¼ bð2c aÞ.
X2 ¼ ða; a; 2c aÞ;
ð3Þ
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Proposition 1. The equilibrium point X0 ¼ ð0; 0; 0Þ is unstable. Proof. The Jacobian matrix of the system (1) at the equilibrium X0 ¼ ð0; 0; 0Þ is given by 2 3 a a 0 J0 ¼ 4 c a c 0 5: 0 0 b The eigenvalues of the Jacobian J0 are: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ca 1 þ ðc aÞ2 þ 4að2c aÞ; k1 ¼ 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ca 1 ðc aÞ2 þ 4að2c aÞ; k2 ¼ 2 2 k3 ¼ b: Since a ¼ 35, b ¼ 3, c ¼ 28, then k1 > 0, k2 < 0 and k3 < 0. Therefore X0 ¼ ð0; 0; 0Þ is unstable. Proposition 2. The equilibrium point X1 ¼ ða; a; 2c aÞ; where a ¼ is unstable.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bð2c aÞ
Proof. The Jacobian matrix of the system (1) at the equilibrium point X1 ¼ ða; a; 2c aÞ is given by 2 3 a a 0 J1 ¼ 4 c c a 5: a a b The characteristic equation of the Jacobian J1 has the form k3 þ ða þ b cÞk2 þ bck þ 2abð2c aÞ ¼ 0: We suppose that a1 ¼ a þ b c, a2 ¼ bc and a3 ¼ 2abð2c aÞ. Since a ¼ 35, b ¼ 3, c ¼ 28, then a1 a2 < a3 . According to Routh–Hurwitz criteria [7] the equilibrium point X1 ¼ ða; a; 2c aÞ is unstable. By the same way we can prove that X2 ¼ ða; a; 2c aÞ is unstable. Since the equilibrium points of the system (1) are unstable, then the optimal control problem takes place.
4. Equilibrium points of the controlled Chen dynamical system The equilibrium points of the controlled Chen dynamical system (2) are determined from the solution of the following system:
M.T. Yassen / Appl. Math. Comput. 131 (2002) 171–180
aðy xÞ þ u1 ¼ 0; ðc aÞx x z þ cy þ u2 ¼ 0; x y bz þ u 3 ¼ 0;
175
ð4Þ
i ; i ¼ 1; 2; 3 are the state variables and control functions at where x, y , z and u the equilibrium points of controlled Chen dynamical system. Now, we proceed to obtain the perturbed equations of the controlled system (2) about its equilibrium points ðx; y ; zÞ and ui ; i ¼ 1; 2; 3. For this purpose, we introduce the following variables: g1 ¼ x x;
g2 ¼ y y ;
g3 ¼ z z;
and
vi ¼ ui ui ;
i ¼ 1; 2; 3:
ð5Þ
Substituting Eq. (5) into (2) we obtain the perturbed equations about the equilibrium points of the system (2) in the following form: g_ 1 ¼ aðg2 g1 Þ þ v1 ; g_ 2 ¼ ðc aÞg1 g1 g3 ðg3x þ g1zÞ þ cg2 þ v2 ; g_ 3 ¼ g1 g2 þ g2x þ g1 y bg3 þ v3 :
ð6Þ
The special solution gi ¼ 0, vi ¼ 0, i ¼ 1; 2; 3 of (6) describes, the equilibrium points of the controlled Chen dynamical system. If the equilibrium point is unstable, then the control can be made such that this state becomes asymptotically stable.
5. Optimal control of the controlled Chen dynamical system We shall derive the stabilizing control functions vi , using a new developed technique [8]. In such a technique, the requirements to ensure optimal asymptotic stabilization of the equilibrium points (3) are adopted. To formulate the problem of optimal stabilization of equilibrium points, we assume that the feedback control vi are functions of gi , i ¼ 1; 2; 3. Then according to Krasovskii theorem [8], the asymptotic stability of the equilibrium points requires the minimization of the functional 1 J¼ 2
Z
1 1h 2 ½v1 þ aðg2 g1 Þ þ v2 þ ðc aÞg1 g1 g3 c1 c2 0 i¼1 ! i2 1 h i2 ðg3x þ g1zÞ þ cg2 þ v3 þ g1 g2 þ xg2 þ y g1 bg3 dt ð7Þ c3 1
and expression
3 X
ci g2i þ
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M.T. Yassen / Appl. Math. Comput. 131 (2002) 171–180
Bðg1 ; g2 ; g3 ; v1 ; v2 ; v3 Þ ¼ g1 ðv1 þ aðg2 g1 ÞÞ þ g2 ðv2 þ ðc aÞg1 g1 g3 ðxg3 þ zg1 Þ þ cg2 Þ þ g3 ðv3 þ g1 g2 þ xg2 þ yg1 bg3 Þ þ
3 1X 1 2 ci g2i þ ½v1 þ aðg2 g1 Þ 2 i¼1 2c1
1 ½v2 þ ðc aÞg1 g1 g3 ðg3x þ g1zÞ þ cg2 2 2c2 1 2 þ ½v3 þ g1 g2 þ xg2 þ yg1 bg3 ; 2c3 þ
ð8Þ
where ci ; i ¼ 1; 2; 3 are positive constants, must be greater than or equal to zero. We choose the Lyapunov function as follows: V ¼
3 1X g2 : 2 i¼1 i
ð9Þ
From the conditions of optimal stabilization, we suppose that the expression (8) has a zero value when vi ¼ v0i , where v0i are the optimal control functions which are chosen as follows: v01 ¼ ðc1 þ aÞg1 ag2 ; v02 ¼ ðc2 þ cÞg2 ðc aÞg1 þ g1 g3 þ g3x þ zg1 ; v03
ð10Þ
¼ ðc3 þ bÞg3 g1 g2 xg2 y g1 :
Substituting from (10) into (6) we get a system of differential equations, which forms a closed loop system with asymptotically stable of the solution gi ¼ 0; i ¼ 1; 2; 3. To prove the above statement, we consider the Lyapunov function (9). This function is positive definite, has a quadratic form, and its time derivative using (6) and (10) is given by V_ ¼
3 X
ci g2i :
ð11Þ
i¼1
Thus under the effect of the control functions (10) the equilibrium points gi ¼ 0; i ¼ 1; 2; 3 will be asymptotically stable.
6. The general solution of the control problem In this section, we obtain for the controlling functions (10) the general solution of the system (6). Substituting the controlling functions (10) into the system (6) of perturbed motion of the system (2), we get g_ i þ ci gi ¼ 0;
i ¼ 1; 2; 3:
ð12Þ
M.T. Yassen / Appl. Math. Comput. 131 (2002) 171–180
177
The solution of this system is given by gi ¼ g0i eci ðtt0 Þ ;
t t0 ; i ¼ 1; 2; 3;
ð13Þ
where g0i ¼ gi ðt0 Þ, i ¼ 1; 2; 3 are the initial perturbations. Therefore the optimal control functions (10) using (13) can be expressed as the functions of time in the form v01 ¼ ðc1 þ aÞg01 ec1 ðtt0 Þ ag02 ec2 ðtt0 Þ ; v02 ¼ ðc2 þ cÞg02 ec2 ðtt0 Þ ðc aÞg01 ec1 ðtt0 Þ þ g01 g03 eðc1 þc3 Þðtt0 Þ þ g03 ec3 ðtt0 Þx þ zg01 ec1 ðtt0 Þ ; v03
¼ ðc3 þ
bÞg03 ec3 ðtt0 Þ
g01 g02 eðc1 þc2 Þðtt0 Þ
ð14Þ
xg02 ec2 ðtt0 Þ
y g01 ec1 ðtt0 Þ :
6.1. Special cases In this section some of the special cases are studied. The first case, when u1 ¼ u2 ¼ u3 ¼ 0. In this case, the controlled system has the following three equilibrium points: ð0; 0; 0Þ; ða; a; 2c aÞ and ða; a; 2c aÞ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where a ¼ bð2c aÞ. The second case, when u1 ¼ k1x, u2 ¼ k2 y , u3 ¼ k3z. In this case, the controlled system (2) is unstable by the linear approximation for a > 0 or c > 0. Therefore the control problem takes place, and the equilibrium points are ð0; 0; 0Þ;
ðb1 ; b2 ; cÞ
and
ðb1 ; b2 ; cÞ;
where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ðc aÞ þ ðc þ k2 Þð1 ðk1 =aÞÞ ðb k3 Þ ; b1 ¼ 1 ðk1 =aÞ k1 b2 ¼ 1 b1 ; a ð1 ðk1 =aÞÞb21 : c¼ b k3
7. Numerical simulation In this section, some numerical examples are introduced. Different values for the constants ci , g0i , ui , x, y , z and ki are considered. The results are shown in Figs. 3–7.
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M.T. Yassen / Appl. Math. Comput. 131 (2002) 171–180
Fig. 3.
Fig. 4.
Fig. 5.
1. Fig. 1 shows the chaotic attractor for Chen system in xy-plane. 2. Fig. 2 shows the chaotic attractor for Chen system in xz-plane. 3. Fig. 3 shows numerical results for the controlling functions vi against the time with c1 ¼ 2c2 ¼ 4c3 ¼ 1, g01 ¼ g02 ¼ g03 ¼ 1, k1 ¼ k2 ¼ k3 ¼ 1, u1 ¼ u2 ¼ u3 ¼ 0, ðx; y ; zÞ ¼ ð0; 0; 0Þ, t0 ¼ 0: 4. Fig. 4 shows numerical results for the controlling functions vi against the time with c1 ¼ 2c2 ¼ 4c3 p ¼ffiffiffiffiffi1, g01 ¼ g02 ¼ g03 ¼ 1, k1 ¼ k2 ¼ k3 ¼ 1, u1 ¼ pffiffiffiffiffi u2 ¼ u3 ¼ 0, ðx; y ; zÞ ¼ ð 63; 63; 21Þ, t0 ¼ 0: 5. Fig. 5 shows numerical results for the controlling functions vi against the time with c1 ¼ 2c2 ¼ p 4cffiffiffiffiffi g01 ¼ g02 ¼ g03 ¼ 1, k1 ¼ k2 ¼ k3 ¼ 1, u1 ¼ 3 ¼ 1, pffiffiffiffiffi u2 ¼ u3 ¼ 0, ðx; y ; zÞ ¼ ð 63; 63; 21Þ, t0 ¼ 0:
M.T. Yassen / Appl. Math. Comput. 131 (2002) 171–180
Fig. 6.
179
Fig. 7.
6. Fig. 6 shows numerical results for the controlling functions vi against the 0 time with c1 ¼ 2c2 ¼ 4c3 p ¼ffiffiffiffiffiffiffiffiffiffiffiffi 1, g01ffi ¼ g02 ¼pgffiffiffiffiffiffiffiffiffiffiffiffi u1 ¼ x; 3 ¼ 1,ffi k1 ¼ k2 ¼ k3 ¼ 1, u2 ¼ y ; u3 ¼ z ðx; y ; zÞ ¼ ð 105=4; 28=35 105=4; 21Þ, t0 ¼ 0. 7. Fig. 7 shows numerical results for the controlling functions vi against the time with c1 ¼ 2c2 ¼ 4c3 ¼p1,ffiffiffiffiffiffiffiffiffiffiffiffi g01 ¼ffi g02 ¼ g03 p ¼ ffiffiffiffiffiffiffiffiffiffiffiffi 1, k1ffi ¼ k2 ¼ k3 ¼ 1, u1 ¼ x; u2 ¼ y ; u3 ¼ z ðx; y ; zÞ ¼ ð 105=4; 28=35 105=4; 21Þ, t0 ¼ 0. 8. Conclusion The problem of optimal control for the equilibrium state of Chen dynamical system has been studied. The optimal feedback control law is obtained in a closed form. The stability conditions of the equilibrium points are obtained in a closed form. The conditions of asymptotic stability of the equilibrium state of Chen dynamical system are used to obtain the control law as a function of state variables.
References [1] L.M. Hocking, Optimal Control an Introduction to the Theory with Applications, Clarendon Press, Oxford, 1991. [2] T.-L. Liao, S.-H. Lin, Adaptive control and synchronization of Lorenz systems, J. Franklin Inst. 336 (1999) 925–937. [3] H.N. Agiza, M.T. Yassen, Synchronization of Rossler and Chen chaotic dynamical systems using active control, Phys. Lett. A 278 (2001) 191–197. [4] A. El Gohary, M.T. Yassen, Optimal control and synchronization of Lotka–Volterra model, Chaos Solitons & Fractals 12 (11) (2001). [5] G. Chen, T. Ueta, Yet another chaotic attractor, Int. J. Bifurcation Chaos 9 (7) (1999) 1465– 1466.
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M.T. Yassen / Appl. Math. Comput. 131 (2002) 171–180
[6] E.-W. Bai, K.E. Lonngren, Synchronization of two Lorenz systems using active control, Chaos, Solitons & Fractals 8 (1) (1997) 51–58. [7] L.E. Keshet, Mathematical Models in Biology, Random House, New York, 1988. [8] N. Krasovskii, Problems in the stabilization of controlled motion, in: I.G. Malkin (Ed.), Theory of Stability of Motion, Nauka, Moscow, 1969.
The optimal control of Chen chaotic dynamical system M.T. Yassen Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt
Abstract This paper is devoted to study the problem of the optimal control for the equilibrium point of a new chaotic system introduced by Chen in a simple three-dimensional autonomous system, which resembles some familiar features from both the Lorenz and Rossler systems. The stability of the equilibrium points is studied. The conditions of the asymptotic stability of the equilibrium point of this system are used to obtain the optimal control functions. The general solution of the dynamical system of the perturbed state is obtained as a function of time. Ó 2002 Elsevier Science Inc. All rights reserved.
1. Introduction In the last few decades optimal control theory has grown rapidly. The essential properties of an optimal control problem are that we have a system which evolves in time according to certain laws. These laws are embodied in equations that contain elements which can be adjusted from outside the system, known as the controls. By suitable choice of these controls it may be possible to force the system into a desired target state [1]. Based on the Lyapunov stability theory, the adaptive control and synchronization problem of two Lorenz systems are investigated in [2], the problem of synchronization of Rossler and Chen chaotic system using active control has been studied in [3]. The problem of optimal control and synchronization of Lotka–Volterra model is studied in [4]. The objective of the present paper is to study the optimal stabilization of the equilibrium points of Chen dynamical system, which has the chaotic attractor
E-mail address: [email protected] (M.T. Yassen). 0096-3003/02/$ - see front matter Ó 2002 Elsevier Science Inc. All rights reserved. PII: S 0 0 9 6 - 3 0 0 3 ( 0 1 ) 0 0 1 3 7 - 0
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M.T. Yassen / Appl. Math. Comput. 131 (2002) 171–180
when a ¼ 35, b ¼ 3, c ¼ 28 [5]. The equilibrium points of Chen dynamical system are determined. The stability of the equilibrium points is studied. The control functions ensuring the optimal stabilization of these points are obtained as a function of the state variables. The general solution of the equations of the perturbed state is obtained as a function of time. Numerical simulations of the obtained results are presented. 2. Chen dynamical system Chen [5] has found a new chaotic attractor in a simple three-dimensional autonomous system, which resembles some familiar features from both the Lorenz and Rossler systems. The chaotic Chen dynamical system is described by the set of ordinary differential equations: x_ ¼ aðy xÞ; y_ ¼ ðc aÞx xz þ cy;
ð1Þ
z_ ¼ xy bz; where x, y and z are the state variables and a, b, and c are three positive real constants. The chaotic attractors of the system (1) are shown in Figs. 1 and 2 in the xy-plane and xz-plane, respectively. The divergence of the flow (1) is given by rF¼
oF1 oF2 oF3 þ þ ¼ a þ c b < 0; ox oy oz
where F ¼ ðF1 ; F2 ; F3 Þ ¼ ðaðy xÞ; ðc aÞx xz þ cy; xy bzÞ:
Fig. 1.
M.T. Yassen / Appl. Math. Comput. 131 (2002) 171–180
173
Fig. 2.
Then, the system (1) is a forced dissipative system similar to Lorenz system [6]. Thus the solutions of the system of equations (1) are bounded as t ! 1 for positive values of a, b, and c. In the present paper, we will study the problem of optimal control of the controlled Chen dynamical system. The controlled Chen dynamical system is presented mathematically by the following system: x_ ¼ aðy xÞ þ u1 ; y_ ¼ ðc aÞx xz þ cy þ u2 ;
ð2Þ
z_ ¼ xy bz þ u3 ; where the functions u1 , u2 , u3 are the control functions which will be determined from the conditions that ensure the asymptotic stabilization of the equilibrium points of this chaotic system.
3. Stability of the equilibrium points of Chen dynamical system The trajectory of the system (1) is specified by ðxðtÞ; yðtÞ; zðtÞÞ. The equilibrium points of the system (1) are X0 ¼ ð0; 0; 0Þ; X1 ¼ ða; a; 2c aÞ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where a ¼ bð2c aÞ.
X2 ¼ ða; a; 2c aÞ;
ð3Þ
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M.T. Yassen / Appl. Math. Comput. 131 (2002) 171–180
Proposition 1. The equilibrium point X0 ¼ ð0; 0; 0Þ is unstable. Proof. The Jacobian matrix of the system (1) at the equilibrium X0 ¼ ð0; 0; 0Þ is given by 2 3 a a 0 J0 ¼ 4 c a c 0 5: 0 0 b The eigenvalues of the Jacobian J0 are: qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ca 1 þ ðc aÞ2 þ 4að2c aÞ; k1 ¼ 2 2 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ca 1 ðc aÞ2 þ 4að2c aÞ; k2 ¼ 2 2 k3 ¼ b: Since a ¼ 35, b ¼ 3, c ¼ 28, then k1 > 0, k2 < 0 and k3 < 0. Therefore X0 ¼ ð0; 0; 0Þ is unstable. Proposition 2. The equilibrium point X1 ¼ ða; a; 2c aÞ; where a ¼ is unstable.
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi bð2c aÞ
Proof. The Jacobian matrix of the system (1) at the equilibrium point X1 ¼ ða; a; 2c aÞ is given by 2 3 a a 0 J1 ¼ 4 c c a 5: a a b The characteristic equation of the Jacobian J1 has the form k3 þ ða þ b cÞk2 þ bck þ 2abð2c aÞ ¼ 0: We suppose that a1 ¼ a þ b c, a2 ¼ bc and a3 ¼ 2abð2c aÞ. Since a ¼ 35, b ¼ 3, c ¼ 28, then a1 a2 < a3 . According to Routh–Hurwitz criteria [7] the equilibrium point X1 ¼ ða; a; 2c aÞ is unstable. By the same way we can prove that X2 ¼ ða; a; 2c aÞ is unstable. Since the equilibrium points of the system (1) are unstable, then the optimal control problem takes place.
4. Equilibrium points of the controlled Chen dynamical system The equilibrium points of the controlled Chen dynamical system (2) are determined from the solution of the following system:
M.T. Yassen / Appl. Math. Comput. 131 (2002) 171–180
aðy xÞ þ u1 ¼ 0; ðc aÞx x z þ cy þ u2 ¼ 0; x y bz þ u 3 ¼ 0;
175
ð4Þ
i ; i ¼ 1; 2; 3 are the state variables and control functions at where x, y , z and u the equilibrium points of controlled Chen dynamical system. Now, we proceed to obtain the perturbed equations of the controlled system (2) about its equilibrium points ðx; y ; zÞ and ui ; i ¼ 1; 2; 3. For this purpose, we introduce the following variables: g1 ¼ x x;
g2 ¼ y y ;
g3 ¼ z z;
and
vi ¼ ui ui ;
i ¼ 1; 2; 3:
ð5Þ
Substituting Eq. (5) into (2) we obtain the perturbed equations about the equilibrium points of the system (2) in the following form: g_ 1 ¼ aðg2 g1 Þ þ v1 ; g_ 2 ¼ ðc aÞg1 g1 g3 ðg3x þ g1zÞ þ cg2 þ v2 ; g_ 3 ¼ g1 g2 þ g2x þ g1 y bg3 þ v3 :
ð6Þ
The special solution gi ¼ 0, vi ¼ 0, i ¼ 1; 2; 3 of (6) describes, the equilibrium points of the controlled Chen dynamical system. If the equilibrium point is unstable, then the control can be made such that this state becomes asymptotically stable.
5. Optimal control of the controlled Chen dynamical system We shall derive the stabilizing control functions vi , using a new developed technique [8]. In such a technique, the requirements to ensure optimal asymptotic stabilization of the equilibrium points (3) are adopted. To formulate the problem of optimal stabilization of equilibrium points, we assume that the feedback control vi are functions of gi , i ¼ 1; 2; 3. Then according to Krasovskii theorem [8], the asymptotic stability of the equilibrium points requires the minimization of the functional 1 J¼ 2
Z
1 1h 2 ½v1 þ aðg2 g1 Þ þ v2 þ ðc aÞg1 g1 g3 c1 c2 0 i¼1 ! i2 1 h i2 ðg3x þ g1zÞ þ cg2 þ v3 þ g1 g2 þ xg2 þ y g1 bg3 dt ð7Þ c3 1
and expression
3 X
ci g2i þ
176
M.T. Yassen / Appl. Math. Comput. 131 (2002) 171–180
Bðg1 ; g2 ; g3 ; v1 ; v2 ; v3 Þ ¼ g1 ðv1 þ aðg2 g1 ÞÞ þ g2 ðv2 þ ðc aÞg1 g1 g3 ðxg3 þ zg1 Þ þ cg2 Þ þ g3 ðv3 þ g1 g2 þ xg2 þ yg1 bg3 Þ þ
3 1X 1 2 ci g2i þ ½v1 þ aðg2 g1 Þ 2 i¼1 2c1
1 ½v2 þ ðc aÞg1 g1 g3 ðg3x þ g1zÞ þ cg2 2 2c2 1 2 þ ½v3 þ g1 g2 þ xg2 þ yg1 bg3 ; 2c3 þ
ð8Þ
where ci ; i ¼ 1; 2; 3 are positive constants, must be greater than or equal to zero. We choose the Lyapunov function as follows: V ¼
3 1X g2 : 2 i¼1 i
ð9Þ
From the conditions of optimal stabilization, we suppose that the expression (8) has a zero value when vi ¼ v0i , where v0i are the optimal control functions which are chosen as follows: v01 ¼ ðc1 þ aÞg1 ag2 ; v02 ¼ ðc2 þ cÞg2 ðc aÞg1 þ g1 g3 þ g3x þ zg1 ; v03
ð10Þ
¼ ðc3 þ bÞg3 g1 g2 xg2 y g1 :
Substituting from (10) into (6) we get a system of differential equations, which forms a closed loop system with asymptotically stable of the solution gi ¼ 0; i ¼ 1; 2; 3. To prove the above statement, we consider the Lyapunov function (9). This function is positive definite, has a quadratic form, and its time derivative using (6) and (10) is given by V_ ¼
3 X
ci g2i :
ð11Þ
i¼1
Thus under the effect of the control functions (10) the equilibrium points gi ¼ 0; i ¼ 1; 2; 3 will be asymptotically stable.
6. The general solution of the control problem In this section, we obtain for the controlling functions (10) the general solution of the system (6). Substituting the controlling functions (10) into the system (6) of perturbed motion of the system (2), we get g_ i þ ci gi ¼ 0;
i ¼ 1; 2; 3:
ð12Þ
M.T. Yassen / Appl. Math. Comput. 131 (2002) 171–180
177
The solution of this system is given by gi ¼ g0i eci ðtt0 Þ ;
t t0 ; i ¼ 1; 2; 3;
ð13Þ
where g0i ¼ gi ðt0 Þ, i ¼ 1; 2; 3 are the initial perturbations. Therefore the optimal control functions (10) using (13) can be expressed as the functions of time in the form v01 ¼ ðc1 þ aÞg01 ec1 ðtt0 Þ ag02 ec2 ðtt0 Þ ; v02 ¼ ðc2 þ cÞg02 ec2 ðtt0 Þ ðc aÞg01 ec1 ðtt0 Þ þ g01 g03 eðc1 þc3 Þðtt0 Þ þ g03 ec3 ðtt0 Þx þ zg01 ec1 ðtt0 Þ ; v03
¼ ðc3 þ
bÞg03 ec3 ðtt0 Þ
g01 g02 eðc1 þc2 Þðtt0 Þ
ð14Þ
xg02 ec2 ðtt0 Þ
y g01 ec1 ðtt0 Þ :
6.1. Special cases In this section some of the special cases are studied. The first case, when u1 ¼ u2 ¼ u3 ¼ 0. In this case, the controlled system has the following three equilibrium points: ð0; 0; 0Þ; ða; a; 2c aÞ and ða; a; 2c aÞ; pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi where a ¼ bð2c aÞ. The second case, when u1 ¼ k1x, u2 ¼ k2 y , u3 ¼ k3z. In this case, the controlled system (2) is unstable by the linear approximation for a > 0 or c > 0. Therefore the control problem takes place, and the equilibrium points are ð0; 0; 0Þ;
ðb1 ; b2 ; cÞ
and
ðb1 ; b2 ; cÞ;
where sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ½ðc aÞ þ ðc þ k2 Þð1 ðk1 =aÞÞ ðb k3 Þ ; b1 ¼ 1 ðk1 =aÞ k1 b2 ¼ 1 b1 ; a ð1 ðk1 =aÞÞb21 : c¼ b k3
7. Numerical simulation In this section, some numerical examples are introduced. Different values for the constants ci , g0i , ui , x, y , z and ki are considered. The results are shown in Figs. 3–7.
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Fig. 3.
Fig. 4.
Fig. 5.
1. Fig. 1 shows the chaotic attractor for Chen system in xy-plane. 2. Fig. 2 shows the chaotic attractor for Chen system in xz-plane. 3. Fig. 3 shows numerical results for the controlling functions vi against the time with c1 ¼ 2c2 ¼ 4c3 ¼ 1, g01 ¼ g02 ¼ g03 ¼ 1, k1 ¼ k2 ¼ k3 ¼ 1, u1 ¼ u2 ¼ u3 ¼ 0, ðx; y ; zÞ ¼ ð0; 0; 0Þ, t0 ¼ 0: 4. Fig. 4 shows numerical results for the controlling functions vi against the time with c1 ¼ 2c2 ¼ 4c3 p ¼ffiffiffiffiffi1, g01 ¼ g02 ¼ g03 ¼ 1, k1 ¼ k2 ¼ k3 ¼ 1, u1 ¼ pffiffiffiffiffi u2 ¼ u3 ¼ 0, ðx; y ; zÞ ¼ ð 63; 63; 21Þ, t0 ¼ 0: 5. Fig. 5 shows numerical results for the controlling functions vi against the time with c1 ¼ 2c2 ¼ p 4cffiffiffiffiffi g01 ¼ g02 ¼ g03 ¼ 1, k1 ¼ k2 ¼ k3 ¼ 1, u1 ¼ 3 ¼ 1, pffiffiffiffiffi u2 ¼ u3 ¼ 0, ðx; y ; zÞ ¼ ð 63; 63; 21Þ, t0 ¼ 0:
M.T. Yassen / Appl. Math. Comput. 131 (2002) 171–180
Fig. 6.
179
Fig. 7.
6. Fig. 6 shows numerical results for the controlling functions vi against the 0 time with c1 ¼ 2c2 ¼ 4c3 p ¼ffiffiffiffiffiffiffiffiffiffiffiffi 1, g01ffi ¼ g02 ¼pgffiffiffiffiffiffiffiffiffiffiffiffi u1 ¼ x; 3 ¼ 1,ffi k1 ¼ k2 ¼ k3 ¼ 1, u2 ¼ y ; u3 ¼ z ðx; y ; zÞ ¼ ð 105=4; 28=35 105=4; 21Þ, t0 ¼ 0. 7. Fig. 7 shows numerical results for the controlling functions vi against the time with c1 ¼ 2c2 ¼ 4c3 ¼p1,ffiffiffiffiffiffiffiffiffiffiffiffi g01 ¼ffi g02 ¼ g03 p ¼ ffiffiffiffiffiffiffiffiffiffiffiffi 1, k1ffi ¼ k2 ¼ k3 ¼ 1, u1 ¼ x; u2 ¼ y ; u3 ¼ z ðx; y ; zÞ ¼ ð 105=4; 28=35 105=4; 21Þ, t0 ¼ 0. 8. Conclusion The problem of optimal control for the equilibrium state of Chen dynamical system has been studied. The optimal feedback control law is obtained in a closed form. The stability conditions of the equilibrium points are obtained in a closed form. The conditions of asymptotic stability of the equilibrium state of Chen dynamical system are used to obtain the control law as a function of state variables.
References [1] L.M. Hocking, Optimal Control an Introduction to the Theory with Applications, Clarendon Press, Oxford, 1991. [2] T.-L. Liao, S.-H. Lin, Adaptive control and synchronization of Lorenz systems, J. Franklin Inst. 336 (1999) 925–937. [3] H.N. Agiza, M.T. Yassen, Synchronization of Rossler and Chen chaotic dynamical systems using active control, Phys. Lett. A 278 (2001) 191–197. [4] A. El Gohary, M.T. Yassen, Optimal control and synchronization of Lotka–Volterra model, Chaos Solitons & Fractals 12 (11) (2001). [5] G. Chen, T. Ueta, Yet another chaotic attractor, Int. J. Bifurcation Chaos 9 (7) (1999) 1465– 1466.
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[6] E.-W. Bai, K.E. Lonngren, Synchronization of two Lorenz systems using active control, Chaos, Solitons & Fractals 8 (1) (1997) 51–58. [7] L.E. Keshet, Mathematical Models in Biology, Random House, New York, 1988. [8] N. Krasovskii, Problems in the stabilization of controlled motion, in: I.G. Malkin (Ed.), Theory of Stability of Motion, Nauka, Moscow, 1969.