Stability of Fuzzy Cellular Neural Networks with Impulses
Stability of Fuzzy Cellular Neural Networks with Impulses Tingwen Huang1 and Marco Roque-Sol2 1
2
Texas A&M University at Qatar, Doha, P. O. Box 5825, Qatar [email protected] Mathematics Department, Texas A&M University, College Station, TX77843, USA [email protected]
Abstract. In this paper, we study impulsive fuzzy cellular neural networks. Criteria are obtained for the existence and exponential stability of a unique equilibrium of fuzzy cellular neural networks impulsive state displacements at fixed instants of time.
1
Introduction
Fuzzy cellular neural networks (FCNN) is a generalization of cellular neural networks (CNN) by using fuzzy operations in the synaptic law calculation allowing us to combine the low level information processing capability of CNN’s with the high level information processing capability, such as image understanding, of fuzzy systems. Yang el al. in [13]-[15] introduced FCNN and investigated existence and stability of equilibrium point of FCNN. After the introduction of FCNN, some researchers studied stability of FCNN with constant time delays and time-varying delays (see [9]), others considered stability of FCNN with distributed delays (see [4], [6]) and exponential stability of FCNN with diffusion effect (see [5]). However, in real world, many evolutionary processes are characterized by abrupt changes at certain time. These changes are called to be impulsive phenomena, which are encountered in many fields such as physics, chemistry, population dynamics, optimal control, etc. In this paper, we will study FCNN model incorporating impulses: n n n dxi (t) = −di xi (t) + bij μj + Ii + αij fj (xj (t)) + Tij μj dt j=1 j=1 j=1
+
n j=1
βij fj (xj (t)) +
n
Hij μj ,
t = tk ,
j=1
n x(t+ 0 ) = x0 ∈ R , − Δxi (tk ) = xi (t+ k ) − xi (tk ) = −γik xi (tk ),
i = 1, · · · , n,
k = 1, 2, · · · , (1)
where αij , βij , Tij and Hij are elements of fuzzy feedback MIN template, fuzzy feedback MAX template, fuzzy feed-forward MIN template and fuzzy feedJ. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 243–248, 2006. c Springer-Verlag Berlin Heidelberg 2006
244
T. Huang and M. Roque-Sol
forward MAX template respectively; bij are elements of feed-forward template; and denote the fuzzy AND and fuzzy OR operation respectively; xi , μi and Ii denote state, input and bias of the ith neurons respectively; fi is the activation − function; Δxi (tk ) = xi (t+ k ) − xi (tk ), k = 1, 2, · · · , are the impulses at moments tk and t1 < t2 < · · · is a strictly increasing sequence such that limk→∞ tk = ∞. As usual in the theory of impulsive differential equations, at the points of discontinuity tk of the solution t → xi (t) we assume that xi (tk ) = xi (t− k ). According to (1), there exist the limits x˙ i (t− ˙ i (t+ ˙ i (tk ) = x˙ i (t− k ) and x k ), we assume x k ). In this paper, we assume that H: fi is a bounded function defined on R and satisfies |fi (x) − fi (y)| ≤ li |x − y|,
i = 1, · · · , n,
(2)
for any x, y ∈ R.
2
Main Results
In order to obtain the main results on the existence and stability of the equilibrium point of FCNN with impulses, we would like to cite the following lemma first. Lemma 1. ([14]). For any aij ∈ R, xj , yj ∈ R, i, j = 1, · · · , n, we have the following estimations, |
n
aij xj −
j=1
and |
n
n
aij yj | ≤
j=1
aij xj −
j=1
n
(|aij | · |xj − yj |)
(3)
(|aij | · |xj − yj |)
(4)
1≤j≤n
aij yj | ≤
j=1
1≤j≤n
Now, we are ready to state and prove the main results. First, we establish sufficient conditions to guarantee the existence and uniqueness of the equilibrium point for the system (1). Theorem 1. Suppose that assumption H is satisfied. Suppose further that (i) di > 0; (ii) the following inequalities hold: di − li
n j=1
|αji | − li
n
|βji | > 0,
i = 1, · · · , n;
j=1
(iii) γik = 0, i = 1, · · · , n, k = 1, 2, · · · ; Then impulsive FCNN (1) exists equilibrium point x∗ = (x∗1 , · · · , x∗n ).
(5)
Stability of Fuzzy Cellular Neural Networks with Impulses
245
We can similarly prove this theorem by imitating the proof of Theorem 2.1 in [2] through constructing a contraction map, then by the contraction mapping principle, there exists a unique fixed point of the map. Existence of a unique solution of system (1) follows. Theorem 2. Assume H is satisfied and the two conditions: (i) and (ii) in Theorem 1 are satisfied too. Further assume that the impulsive operators Ii (xi (t)) satisfy Ii (xi (t)) = −γik (xi (tk ) − x∗i ),
0 < γik < 2,
i = 1, · · · , n,
k ∈ Z +.
(6)
where x∗ is the unique equilibrium point of system (1). The proof of the theorem is similar to the proof of Theorem 1. An additional difference is the consideration of the impulse effect. From the imposed condition on impulsive operator, it is clear that constructing mapping is still contracted on those discontinuous points. Thus, the results follow. We omit the rigorous proofs for Theorem 1 and Theorem 2. Theorem 3. Assume that all conditions of Theorem 2 hold. Then there exists a positive constant λ such that all solutions of system (1) satisfy the following inequality: n n |xi (t) − x∗i | ≤ e−λt |xi (0) − x∗i | t > 0. (7) i=1 ∗
where x =
(x∗1 , · · · , x∗n )T
i=1
denotes the equilibrium.
Proof. By x∗ is the equilibrium of (1), we have the following: n d (xi (t) − x∗ ) = −di (xi (t) − x∗ ) + αij (fj (xj (t)) − x∗ ) dt j=1
+
n
βij (fj (xj (t)) − x∗ )
j=1
(8) for i = 1, 2, · · · , n, t > 0, t = tk , k ∈ Z + , and hence by assumption H and Lemma 1, n d+ |xi (t) − x∗ | ≤ −di |xi (t) − x∗ | + αij lj |xj (t) − x∗ | dt j=1
+
n
βij lj |xj (t) − x∗ |
j=1
≤ −di |xi (t) − x∗ | +
n j=1
|αij |lj |xj (t) − x∗ |
246
T. Huang and M. Roque-Sol
+
n
|βij |lj |xj (t) − x∗ |
j=1
(9) d+ dt
for i = 1, 2, · · · , n, t > 0, t = tk , k ∈ Z + , where Also, from the conditions, we have
so
is the upper right derivative.
∗ ∗ ∗ xi (t+ k ) − xi = xi (tk ) + Ii (x(tk )) − xi = (1 − γik )(xi (tk ) − xi ),
(10)
∗ ∗ ∗ |xi (t+ k ) − xi | = |(1 − γik )(xi (tk ) − xi )| ≤ |xi (tk ) − xi |,
(11)
for i = 1, · · · , n, k ∈ Z . Let us define the Lyapunov function V (·) by +
V (t) = V (x1 , x2 , · · · , xm )(t) =
n
|xi (t) − x∗i |,
(12)
i=1
for t > 0 and by (9), we can obtain d+ d+ V (t) = |xi (t) − x∗i | dt dt i=1 n
≤
n
{−di |xi (t) − x∗ | +
i=1
+
=
n
|αij |lj |xj (t) − x∗ |
j=1
n
|βij |lj |xj (t) − x∗ |}
j=1 n
n
i=1
j=1
{−di + li
|αji | + li
n
|βji |}|xi (t) − x∗ |
j=1
(13) By the condition: di − li
n
|αji | − li
j=1
n
|βji | > 0,
i = 1, · · · , n;
(14)
i = 1, · · · , n.
(15)
j=1
there exists a positive number λ, such that di − li
n j=1
|αij | − li
n
|βij | ≥ λ,
j=1
Thus, by (13) and (15), we have the following: d+ V (t) ≤ −λV (t), dt
t > 0, t = tk .
(16)
Stability of Fuzzy Cellular Neural Networks with Impulses
247
Also, we have the following: V (t+ k) =
n
∗ |xi (t+ k ) − xi | ≤
i=1
n
|xi (tk ) − x∗i | = V (tk ),
k ∈ Z +.
(17)
i=1
Now, using the stability theorem in [11] and (16), (17), we obtain d+ V (t) ≤ −λV (t), dt Therefore, we have
V (t) ≤ e−λt V (0),
t > 0.
(18)
t > 0.
(19)
i.e., n
|xi (t) − x∗i | ≤ e−λt
i=1
n
|xi (0) − x∗i |,
t > 0.
(20)
i=1
Thus, we have completed the proof of this theorem.
3
Conclusion
In this paper, we discuss the existence and exponential stability of the equilibrium of FCNN with impulses. Several sufficient conditions set up here are easily verified.
Acknowledgments The first author is grateful for the support of Texas A&M University at Qatar.
References 1. Cao, J., Wang, J., Liao, X.: Novel Stability Criteria of Delayed Cellular Neural Networks. International Journal of Neural Systems 13(2003)365-375 2. Gopalsamy, K.: Stability of Artificial Neural Networks with Impulses. Applied Mathematics and Computation 154(2004)783-813 3. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis, Cambridge University Press, Cambridge (1999) 4. Huang, T., Zhang, L.: Exponential Stability of Fuzzy Cellular Neural Networks. In: Wang, J., Liao X., Yi Z. (eds.): Advances in Neural Networks. Lecture Notes in Computer Science, Vol. 3496. Springer-Verlag, Berlin Heidelberg, New York (2005)168-173 5. Huang, T.: Exponential Stability of Delayed Fuzzy Cellular Neural Networks with Diffusion, to appear in Chaos, Solitons and Fractrals. 6. Huang, T.: Exponential Stability of Fuzzy Cellular Neural Networks with Unbounded Distributed Delay, to appear in Physics Letters A.
248
T. Huang and M. Roque-Sol
7. Li, Y.: Global Exponential Stability of BAM Neural Networks with Delays and Impulses. Chaos, Solitons and Fractals 24(2005)279-285 8. Li, C., Liao, X., Zhang, R.: Impulsive Synchronization of Nonlinear Coupled Chaotic Systems. Physics Letters A 328(2004)47-50 9. Liu, Y., Tang, W.: Exponential Stability of Fuzzy Cellular Neural Networks with Constant and Time-varying Delays. Physics Letters A 323(2004)224-233 10. Liao, X.F., Wong, K, Li, C: Global Exponential Stability for a Class of Generalized Neural Networks with Distributed Delays. Nonlinear Analysis: Real World Applications 5(2004)527-547 11. Samoilenko A., Perestyuk N.: Impulsive Differential Equations, in: World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, Vol. 14. World Scientific, Singapore (1995) 12. Yang, T., Yang, L.B., Wu, C.W., Chua, L.O.: Fuzzy Cellular Neural Networks: Theory. In Proc. of IEEE International Workshop on Cellular Neural networks and Applications (1996)181-186 13. Yang, T. , Yang, L.B., Wu, C.W., Chua, L.O.: Fuzzy Cellular Neural Networks: Applications. In Proc. of IEEE International Workshop on Cellular Neural Networks and Applications (1996)225-230 14. Yang, T., Yang, L.B.: The Global Stability of Fuzzy Cellular Neural Network. Circuits and Systems I: Fundamental Theory and Applications 43(1996)880 - 883 15. Yang, X., Liao, X., Evans, D., Tnag, Y.: Existence and Stability of Periodic Solution in Impulsive Hopfield. Neural Networks with Finite Distributed Delays 343(2005)108-116
2
Texas A&M University at Qatar, Doha, P. O. Box 5825, Qatar [email protected] Mathematics Department, Texas A&M University, College Station, TX77843, USA [email protected]
Abstract. In this paper, we study impulsive fuzzy cellular neural networks. Criteria are obtained for the existence and exponential stability of a unique equilibrium of fuzzy cellular neural networks impulsive state displacements at fixed instants of time.
1
Introduction
Fuzzy cellular neural networks (FCNN) is a generalization of cellular neural networks (CNN) by using fuzzy operations in the synaptic law calculation allowing us to combine the low level information processing capability of CNN’s with the high level information processing capability, such as image understanding, of fuzzy systems. Yang el al. in [13]-[15] introduced FCNN and investigated existence and stability of equilibrium point of FCNN. After the introduction of FCNN, some researchers studied stability of FCNN with constant time delays and time-varying delays (see [9]), others considered stability of FCNN with distributed delays (see [4], [6]) and exponential stability of FCNN with diffusion effect (see [5]). However, in real world, many evolutionary processes are characterized by abrupt changes at certain time. These changes are called to be impulsive phenomena, which are encountered in many fields such as physics, chemistry, population dynamics, optimal control, etc. In this paper, we will study FCNN model incorporating impulses: n n n dxi (t) = −di xi (t) + bij μj + Ii + αij fj (xj (t)) + Tij μj dt j=1 j=1 j=1
+
n j=1
βij fj (xj (t)) +
n
Hij μj ,
t = tk ,
j=1
n x(t+ 0 ) = x0 ∈ R , − Δxi (tk ) = xi (t+ k ) − xi (tk ) = −γik xi (tk ),
i = 1, · · · , n,
k = 1, 2, · · · , (1)
where αij , βij , Tij and Hij are elements of fuzzy feedback MIN template, fuzzy feedback MAX template, fuzzy feed-forward MIN template and fuzzy feedJ. Wang et al. (Eds.): ISNN 2006, LNCS 3971, pp. 243–248, 2006. c Springer-Verlag Berlin Heidelberg 2006
244
T. Huang and M. Roque-Sol
forward MAX template respectively; bij are elements of feed-forward template; and denote the fuzzy AND and fuzzy OR operation respectively; xi , μi and Ii denote state, input and bias of the ith neurons respectively; fi is the activation − function; Δxi (tk ) = xi (t+ k ) − xi (tk ), k = 1, 2, · · · , are the impulses at moments tk and t1 < t2 < · · · is a strictly increasing sequence such that limk→∞ tk = ∞. As usual in the theory of impulsive differential equations, at the points of discontinuity tk of the solution t → xi (t) we assume that xi (tk ) = xi (t− k ). According to (1), there exist the limits x˙ i (t− ˙ i (t+ ˙ i (tk ) = x˙ i (t− k ) and x k ), we assume x k ). In this paper, we assume that H: fi is a bounded function defined on R and satisfies |fi (x) − fi (y)| ≤ li |x − y|,
i = 1, · · · , n,
(2)
for any x, y ∈ R.
2
Main Results
In order to obtain the main results on the existence and stability of the equilibrium point of FCNN with impulses, we would like to cite the following lemma first. Lemma 1. ([14]). For any aij ∈ R, xj , yj ∈ R, i, j = 1, · · · , n, we have the following estimations, |
n
aij xj −
j=1
and |
n
n
aij yj | ≤
j=1
aij xj −
j=1
n
(|aij | · |xj − yj |)
(3)
(|aij | · |xj − yj |)
(4)
1≤j≤n
aij yj | ≤
j=1
1≤j≤n
Now, we are ready to state and prove the main results. First, we establish sufficient conditions to guarantee the existence and uniqueness of the equilibrium point for the system (1). Theorem 1. Suppose that assumption H is satisfied. Suppose further that (i) di > 0; (ii) the following inequalities hold: di − li
n j=1
|αji | − li
n
|βji | > 0,
i = 1, · · · , n;
j=1
(iii) γik = 0, i = 1, · · · , n, k = 1, 2, · · · ; Then impulsive FCNN (1) exists equilibrium point x∗ = (x∗1 , · · · , x∗n ).
(5)
Stability of Fuzzy Cellular Neural Networks with Impulses
245
We can similarly prove this theorem by imitating the proof of Theorem 2.1 in [2] through constructing a contraction map, then by the contraction mapping principle, there exists a unique fixed point of the map. Existence of a unique solution of system (1) follows. Theorem 2. Assume H is satisfied and the two conditions: (i) and (ii) in Theorem 1 are satisfied too. Further assume that the impulsive operators Ii (xi (t)) satisfy Ii (xi (t)) = −γik (xi (tk ) − x∗i ),
0 < γik < 2,
i = 1, · · · , n,
k ∈ Z +.
(6)
where x∗ is the unique equilibrium point of system (1). The proof of the theorem is similar to the proof of Theorem 1. An additional difference is the consideration of the impulse effect. From the imposed condition on impulsive operator, it is clear that constructing mapping is still contracted on those discontinuous points. Thus, the results follow. We omit the rigorous proofs for Theorem 1 and Theorem 2. Theorem 3. Assume that all conditions of Theorem 2 hold. Then there exists a positive constant λ such that all solutions of system (1) satisfy the following inequality: n n |xi (t) − x∗i | ≤ e−λt |xi (0) − x∗i | t > 0. (7) i=1 ∗
where x =
(x∗1 , · · · , x∗n )T
i=1
denotes the equilibrium.
Proof. By x∗ is the equilibrium of (1), we have the following: n d (xi (t) − x∗ ) = −di (xi (t) − x∗ ) + αij (fj (xj (t)) − x∗ ) dt j=1
+
n
βij (fj (xj (t)) − x∗ )
j=1
(8) for i = 1, 2, · · · , n, t > 0, t = tk , k ∈ Z + , and hence by assumption H and Lemma 1, n d+ |xi (t) − x∗ | ≤ −di |xi (t) − x∗ | + αij lj |xj (t) − x∗ | dt j=1
+
n
βij lj |xj (t) − x∗ |
j=1
≤ −di |xi (t) − x∗ | +
n j=1
|αij |lj |xj (t) − x∗ |
246
T. Huang and M. Roque-Sol
+
n
|βij |lj |xj (t) − x∗ |
j=1
(9) d+ dt
for i = 1, 2, · · · , n, t > 0, t = tk , k ∈ Z + , where Also, from the conditions, we have
so
is the upper right derivative.
∗ ∗ ∗ xi (t+ k ) − xi = xi (tk ) + Ii (x(tk )) − xi = (1 − γik )(xi (tk ) − xi ),
(10)
∗ ∗ ∗ |xi (t+ k ) − xi | = |(1 − γik )(xi (tk ) − xi )| ≤ |xi (tk ) − xi |,
(11)
for i = 1, · · · , n, k ∈ Z . Let us define the Lyapunov function V (·) by +
V (t) = V (x1 , x2 , · · · , xm )(t) =
n
|xi (t) − x∗i |,
(12)
i=1
for t > 0 and by (9), we can obtain d+ d+ V (t) = |xi (t) − x∗i | dt dt i=1 n
≤
n
{−di |xi (t) − x∗ | +
i=1
+
=
n
|αij |lj |xj (t) − x∗ |
j=1
n
|βij |lj |xj (t) − x∗ |}
j=1 n
n
i=1
j=1
{−di + li
|αji | + li
n
|βji |}|xi (t) − x∗ |
j=1
(13) By the condition: di − li
n
|αji | − li
j=1
n
|βji | > 0,
i = 1, · · · , n;
(14)
i = 1, · · · , n.
(15)
j=1
there exists a positive number λ, such that di − li
n j=1
|αij | − li
n
|βij | ≥ λ,
j=1
Thus, by (13) and (15), we have the following: d+ V (t) ≤ −λV (t), dt
t > 0, t = tk .
(16)
Stability of Fuzzy Cellular Neural Networks with Impulses
247
Also, we have the following: V (t+ k) =
n
∗ |xi (t+ k ) − xi | ≤
i=1
n
|xi (tk ) − x∗i | = V (tk ),
k ∈ Z +.
(17)
i=1
Now, using the stability theorem in [11] and (16), (17), we obtain d+ V (t) ≤ −λV (t), dt Therefore, we have
V (t) ≤ e−λt V (0),
t > 0.
(18)
t > 0.
(19)
i.e., n
|xi (t) − x∗i | ≤ e−λt
i=1
n
|xi (0) − x∗i |,
t > 0.
(20)
i=1
Thus, we have completed the proof of this theorem.
3
Conclusion
In this paper, we discuss the existence and exponential stability of the equilibrium of FCNN with impulses. Several sufficient conditions set up here are easily verified.
Acknowledgments The first author is grateful for the support of Texas A&M University at Qatar.
References 1. Cao, J., Wang, J., Liao, X.: Novel Stability Criteria of Delayed Cellular Neural Networks. International Journal of Neural Systems 13(2003)365-375 2. Gopalsamy, K.: Stability of Artificial Neural Networks with Impulses. Applied Mathematics and Computation 154(2004)783-813 3. Horn, R.A., Johnson, C.R.: Topics in Matrix Analysis, Cambridge University Press, Cambridge (1999) 4. Huang, T., Zhang, L.: Exponential Stability of Fuzzy Cellular Neural Networks. In: Wang, J., Liao X., Yi Z. (eds.): Advances in Neural Networks. Lecture Notes in Computer Science, Vol. 3496. Springer-Verlag, Berlin Heidelberg, New York (2005)168-173 5. Huang, T.: Exponential Stability of Delayed Fuzzy Cellular Neural Networks with Diffusion, to appear in Chaos, Solitons and Fractrals. 6. Huang, T.: Exponential Stability of Fuzzy Cellular Neural Networks with Unbounded Distributed Delay, to appear in Physics Letters A.
248
T. Huang and M. Roque-Sol
7. Li, Y.: Global Exponential Stability of BAM Neural Networks with Delays and Impulses. Chaos, Solitons and Fractals 24(2005)279-285 8. Li, C., Liao, X., Zhang, R.: Impulsive Synchronization of Nonlinear Coupled Chaotic Systems. Physics Letters A 328(2004)47-50 9. Liu, Y., Tang, W.: Exponential Stability of Fuzzy Cellular Neural Networks with Constant and Time-varying Delays. Physics Letters A 323(2004)224-233 10. Liao, X.F., Wong, K, Li, C: Global Exponential Stability for a Class of Generalized Neural Networks with Distributed Delays. Nonlinear Analysis: Real World Applications 5(2004)527-547 11. Samoilenko A., Perestyuk N.: Impulsive Differential Equations, in: World Scientific Series on Nonlinear Science. Series A: Monographs and Treatises, Vol. 14. World Scientific, Singapore (1995) 12. Yang, T., Yang, L.B., Wu, C.W., Chua, L.O.: Fuzzy Cellular Neural Networks: Theory. In Proc. of IEEE International Workshop on Cellular Neural networks and Applications (1996)181-186 13. Yang, T. , Yang, L.B., Wu, C.W., Chua, L.O.: Fuzzy Cellular Neural Networks: Applications. In Proc. of IEEE International Workshop on Cellular Neural Networks and Applications (1996)225-230 14. Yang, T., Yang, L.B.: The Global Stability of Fuzzy Cellular Neural Network. Circuits and Systems I: Fundamental Theory and Applications 43(1996)880 - 883 15. Yang, X., Liao, X., Evans, D., Tnag, Y.: Existence and Stability of Periodic Solution in Impulsive Hopfield. Neural Networks with Finite Distributed Delays 343(2005)108-116
