Combining Methods of Lyapunov for Exponential Stability of Linear Dynamic Systems on Time Scales
Affiliation(s)
1 Department of Mathematics, Vietnam Water Resource University, Hanoi, Vietnam. 2 Department of Mathematics, Vietnam National University of Science, Hanoi, Vietnam.
1 Department of Mathematics, Vietnam Water Resource University, Hanoi, Vietnam. 2 Department of Mathematics, Vietnam National University of Science, Hanoi, Vietnam.
ABSTRACT
Consider the linear dynamic equation on time scales
(1)
where
,
,
is a rd-continuous function, T is a time scales. In this paper, we shall investigate some results for the exponential stability of the dynamic Equation (1) by combinating the first approximate method and the second method of Lyapunov.
Consider the linear dynamic equation on time scales
(1)
where , , is a rd-continuous function, T is a time scales. In this paper, we shall investigate some results for the exponential stability of the dynamic Equation (1) by combinating the first approximate method and the second method of Lyapunov.
Cite this paper
Huy, N. and Chau, D. (2014) Combining Methods of Lyapunov for Exponential Stability of Linear Dynamic Systems on Time Scales. Applied Mathematics, 5, 3452-3459. doi: 10.4236/am.2014.521323.
Huy, N. and Chau, D. (2014) Combining Methods of Lyapunov for Exponential Stability of Linear Dynamic Systems on Time Scales. Applied Mathematics, 5, 3452-3459. doi: 10.4236/am.2014.521323.
References
[1] Hilger, S. (1990) Analysis on Measure Chains—A Unified Approach to Continuous and Discrete Calculus. Results in Mathematics, 18, 19-56.
http://dx.doi.org/10.1007/BF03323153 [2] Bohner, M. and Peterson, A. (2001) Dynamic Equation on Time Scales: An Introduction with Applications. Birkhauser, Boston.
http://dx.doi.org/10.1007/978-1-4612-0201-1 [3] Kaymakacalan, B., Lakshmikantham, V. and Sivasundaram, S. (1996) Dynamic Systems on Measure Chains. Kluwer, Dordrecht. [4] Potzsche, C., Siegmund, S. and Wirth, F. (2003) A Spectral Characterization of Exponential Stability for Linear Time-Invariant Systems on Time Scales. Discrete and Continuous Dynamical Systems, 9, 1223-1241.
http://dx.doi.org/10.3934/dcds.2003.9.1223 [5] Agarwal, R., Bohner, M., O’Regan, D. and Peterson, A. (2002) Dynamic Equation on Time Scales: A Survey. Journal of Computational and Applied Mathematics, 141, 1-26.
http://dx.doi.org/10.1016/S0377-0427(01)00432-0 [6] Aulback, B. and Hilger, S. (1971) Linear Dynamic Processes with Inhomogeneous Time Scale. In: Nonlinear Dynamics and Quantum Dynamical Systems, 3rd Edition, Akademie-Verlag, Berlin, Mathematical Research Bd. 59. [7] Akin-Bohner, E., Bohner, M. and Akin, F. (2005) Pachpatte Inequalities on Time Scales. Journal of Inequalities in Pure and Applied Mathematics, 6, 23.
[1] Hilger, S. (1990) Analysis on Measure Chains—A Unified Approach to Continuous and Discrete Calculus. Results in Mathematics, 18, 19-56.
http://dx.doi.org/10.1007/BF03323153 [2] Bohner, M. and Peterson, A. (2001) Dynamic Equation on Time Scales: An Introduction with Applications. Birkhauser, Boston.
http://dx.doi.org/10.1007/978-1-4612-0201-1 [3] Kaymakacalan, B., Lakshmikantham, V. and Sivasundaram, S. (1996) Dynamic Systems on Measure Chains. Kluwer, Dordrecht. [4] Potzsche, C., Siegmund, S. and Wirth, F. (2003) A Spectral Characterization of Exponential Stability for Linear Time-Invariant Systems on Time Scales. Discrete and Continuous Dynamical Systems, 9, 1223-1241.
http://dx.doi.org/10.3934/dcds.2003.9.1223 [5] Agarwal, R., Bohner, M., O’Regan, D. and Peterson, A. (2002) Dynamic Equation on Time Scales: A Survey. Journal of Computational and Applied Mathematics, 141, 1-26.
http://dx.doi.org/10.1016/S0377-0427(01)00432-0 [6] Aulback, B. and Hilger, S. (1971) Linear Dynamic Processes with Inhomogeneous Time Scale. In: Nonlinear Dynamics and Quantum Dynamical Systems, 3rd Edition, Akademie-Verlag, Berlin, Mathematical Research Bd. 59. [7] Akin-Bohner, E., Bohner, M. and Akin, F. (2005) Pachpatte Inequalities on Time Scales. Journal of Inequalities in Pure and Applied Mathematics, 6, 23.