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Forwards dynamics of non-autonomous dynamical systems: Driving semigroups without backwards uniqueness and structure of the attractor
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Stability and forward attractors for non-autonomous impulsive semidynamical systems
1. | Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo-Campus de São Carlos, Caixa Postal 668, 13560-970, São Carlos SP, Brazil |
2. | Universidade Federal do Espírito Santo, Vitória ES, Brazil |
In this paper, we study the theory of forward attractors for non-autonomous impulsive semidynamical systems. Moreover, we investigate some types of stability of the global attractor as orbital stability, asymptotic stability and stability in the sense of Lyapunov-Barbashin. We present an example to illustrate the theory.
References:
[1] |
R. Ambrosino, F. Calabrese, C. Cosentino and G. De Tommasi,
Sufficient conditions for finite-time stability of impulsive dynamical systems, IEEE Trans. Automat. Control, 54 (2009), 861-865.
doi: 10.1109/TAC.2008.2010965. |
[2] |
L. Barreira and C. Valls,
Lyapunov regularity of impulsive differential equations, J. Diff. Equations, 249 (2010), 1596-1619.
doi: 10.1016/j.jde.2010.07.016. |
[3] |
B. Bouchard, Dang Ngoc-Minh and Lehalle Charles-Albert,
Optimal control of trading algorithms: A general impulse control approach, SIAM J. Finan. Math., 2 (2011), 404-438.
doi: 10.1137/090777293. |
[4] |
E. M. Bonotto and M. Federson,
Topological conjugation and asymptotic stability in impulsive semidynamical systems, J. Math. Anal. Appl., 326 (2007), 869-881.
doi: 10.1016/j.jmaa.2006.03.042. |
[5] |
E. M. Bonotto,
Flows of characteristic 0+ in impulsive semidynamical systems, J. Math. Anal. Appl., 332 (2007), 81-96.
doi: 10.1016/j.jmaa.2006.09.076. |
[6] |
E. M. Bonotto and D. P. Demuner,
Autonomous dissipative semidynamical systems with impulses, Topological Methods in Nonlinear Analysis, 41 (2013), 1-38.
|
[7] |
E. M. Bonotto, D. P. Demuner and M. Z. Jimenez,
Convergence for non-autonomous semidynamical systems with impulses, J. Diff. Equations, 266 (2019), 227-256.
doi: 10.1016/j.jde.2018.07.035. |
[8] |
E. M. Bonotto, J. Costa Ferreira and M. Federson,
Uniform asymptotic stability of a discontinuous predator-prey model under control via non-autonomous systems theory, Differential and Integral Equations, 31 (2018), 519-546.
|
[9] |
E. M. Bonotto, M. C. Bortolan, A. N. Carvalho and R. Czaja,
Global attractors for impulsive dynamical systems - a precompact approach, J. Diff. Equations, 259 (2015), 2602-2625.
doi: 10.1016/j.jde.2015.03.033. |
[10] |
E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari,
Impulsive non-autonomous dynamical systems and impulsive cocycle attractors, Math. Meth. Appl. Sci., 40 (2017), 1095-1113.
doi: 10.1002/mma.4038. |
[11] |
E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari, Attractors for impulsive non-autonomous dynamical systems and their relations, J. Diff. Equations, (2017), 3524–3550.
doi: 10.1016/j.jde.2016.11.036. |
[12] |
D. N. Cheban, Global Attractors of Non-autonomous Dissipative Dynamical Systems, Interdiscip. Math. Sci., vol. 1, World Scientific Publishing, Hackensack, NJ, 2004.
doi: 10.1142/9789812563088. |
[13] |
K. Ciesielski,
On semicontinuity in impulsive dynamical systems, Bull. Polish Acad. Sci. Math., 52 (2004), 71-80.
doi: 10.4064/ba52-1-8. |
[14] |
K. Ciesielski,
On stability in impulsive dynamical systems, Bull. Polish Acad. Sci. Math., 52 (2004), 81-91.
doi: 10.4064/ba52-1-9. |
[15] |
M. H. A. Davis, X. Guo and Gu oliang Wu,
Impulse control of multidimensional jump diffusions, SIAM J. Control Optim., 48 (2010), 5276-5293.
doi: 10.1137/090780419. |
[16] |
A. El-Gohary and A. S. Al-Ruzaiza,
Chaos and adaptive control in two prey, one predator system with nonlinear feedback, Chaos Solitons and Fractals, 34 (2007), 443-453.
doi: 10.1016/j.chaos.2006.03.101. |
[17] |
W. M. Haddad and Q. Hui,
Energy dissipating hybrid control for impulsive dynamical systems, Nonlinear Anal., 69 (2008), 3232-3248.
doi: 10.1016/j.na.2005.10.052. |
[18] |
D. Husemoller, Fibre Bundles, Springer-Verlag, Berlin-Heidelberg-New York, 1994.
doi: 10.1007/978-1-4757-2261-1. |
[19] |
S. K. Kaul,
On impulsive semidynamical systems, J. Math. Anal. Appl., 150 (1990), 120-128.
doi: 10.1016/0022-247X(90)90199-P. |
[20] |
S. K. Kaul,
On impulsive semidynamical systems Ⅱ - recursive properties, Nonlinear Analysis: Theory, Methods & Applications, 7-8 (1991), 635-645.
doi: 10.1016/0362-546X(91)90171-V. |
[21] |
S. K. Kaul,
Stability and asymptotic stability in impulsive semidynamical systems, J. Applied Math. and Stochastic Analysis, 7 (1994), 509-523.
doi: 10.1155/S1048953394000390. |
[22] |
V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
doi: 10.1142/0906. |
[23] |
Yang Jin and Zhao Min, Complex behavior in a fish algae consumption model with impulsive control strategy, Discrete Dyn. Nat. Soc. Art., ID 163541, (2011).
doi: 10.1155/2011/163541. |
[24] |
H. Ye, A. N. Michel and L. Hou,
Stability analysis of systems with impulse effects, IEEE Transactions on Automatic Control, 43 (1998), 1719-1723.
doi: 10.1109/9.736069. |
[25] |
Zhao L., Chen L. and Zhang Q., The geometrical analysis of a predator-prey model with two state impulses, Mathematical Biosciences, 23 (2012), 55–64.
doi: 10.1016/j.mbs.2012.03.011. |
show all references
References:
[1] |
R. Ambrosino, F. Calabrese, C. Cosentino and G. De Tommasi,
Sufficient conditions for finite-time stability of impulsive dynamical systems, IEEE Trans. Automat. Control, 54 (2009), 861-865.
doi: 10.1109/TAC.2008.2010965. |
[2] |
L. Barreira and C. Valls,
Lyapunov regularity of impulsive differential equations, J. Diff. Equations, 249 (2010), 1596-1619.
doi: 10.1016/j.jde.2010.07.016. |
[3] |
B. Bouchard, Dang Ngoc-Minh and Lehalle Charles-Albert,
Optimal control of trading algorithms: A general impulse control approach, SIAM J. Finan. Math., 2 (2011), 404-438.
doi: 10.1137/090777293. |
[4] |
E. M. Bonotto and M. Federson,
Topological conjugation and asymptotic stability in impulsive semidynamical systems, J. Math. Anal. Appl., 326 (2007), 869-881.
doi: 10.1016/j.jmaa.2006.03.042. |
[5] |
E. M. Bonotto,
Flows of characteristic 0+ in impulsive semidynamical systems, J. Math. Anal. Appl., 332 (2007), 81-96.
doi: 10.1016/j.jmaa.2006.09.076. |
[6] |
E. M. Bonotto and D. P. Demuner,
Autonomous dissipative semidynamical systems with impulses, Topological Methods in Nonlinear Analysis, 41 (2013), 1-38.
|
[7] |
E. M. Bonotto, D. P. Demuner and M. Z. Jimenez,
Convergence for non-autonomous semidynamical systems with impulses, J. Diff. Equations, 266 (2019), 227-256.
doi: 10.1016/j.jde.2018.07.035. |
[8] |
E. M. Bonotto, J. Costa Ferreira and M. Federson,
Uniform asymptotic stability of a discontinuous predator-prey model under control via non-autonomous systems theory, Differential and Integral Equations, 31 (2018), 519-546.
|
[9] |
E. M. Bonotto, M. C. Bortolan, A. N. Carvalho and R. Czaja,
Global attractors for impulsive dynamical systems - a precompact approach, J. Diff. Equations, 259 (2015), 2602-2625.
doi: 10.1016/j.jde.2015.03.033. |
[10] |
E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari,
Impulsive non-autonomous dynamical systems and impulsive cocycle attractors, Math. Meth. Appl. Sci., 40 (2017), 1095-1113.
doi: 10.1002/mma.4038. |
[11] |
E. M. Bonotto, M. C. Bortolan, T. Caraballo and R. Collegari, Attractors for impulsive non-autonomous dynamical systems and their relations, J. Diff. Equations, (2017), 3524–3550.
doi: 10.1016/j.jde.2016.11.036. |
[12] |
D. N. Cheban, Global Attractors of Non-autonomous Dissipative Dynamical Systems, Interdiscip. Math. Sci., vol. 1, World Scientific Publishing, Hackensack, NJ, 2004.
doi: 10.1142/9789812563088. |
[13] |
K. Ciesielski,
On semicontinuity in impulsive dynamical systems, Bull. Polish Acad. Sci. Math., 52 (2004), 71-80.
doi: 10.4064/ba52-1-8. |
[14] |
K. Ciesielski,
On stability in impulsive dynamical systems, Bull. Polish Acad. Sci. Math., 52 (2004), 81-91.
doi: 10.4064/ba52-1-9. |
[15] |
M. H. A. Davis, X. Guo and Gu oliang Wu,
Impulse control of multidimensional jump diffusions, SIAM J. Control Optim., 48 (2010), 5276-5293.
doi: 10.1137/090780419. |
[16] |
A. El-Gohary and A. S. Al-Ruzaiza,
Chaos and adaptive control in two prey, one predator system with nonlinear feedback, Chaos Solitons and Fractals, 34 (2007), 443-453.
doi: 10.1016/j.chaos.2006.03.101. |
[17] |
W. M. Haddad and Q. Hui,
Energy dissipating hybrid control for impulsive dynamical systems, Nonlinear Anal., 69 (2008), 3232-3248.
doi: 10.1016/j.na.2005.10.052. |
[18] |
D. Husemoller, Fibre Bundles, Springer-Verlag, Berlin-Heidelberg-New York, 1994.
doi: 10.1007/978-1-4757-2261-1. |
[19] |
S. K. Kaul,
On impulsive semidynamical systems, J. Math. Anal. Appl., 150 (1990), 120-128.
doi: 10.1016/0022-247X(90)90199-P. |
[20] |
S. K. Kaul,
On impulsive semidynamical systems Ⅱ - recursive properties, Nonlinear Analysis: Theory, Methods & Applications, 7-8 (1991), 635-645.
doi: 10.1016/0362-546X(91)90171-V. |
[21] |
S. K. Kaul,
Stability and asymptotic stability in impulsive semidynamical systems, J. Applied Math. and Stochastic Analysis, 7 (1994), 509-523.
doi: 10.1155/S1048953394000390. |
[22] |
V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989.
doi: 10.1142/0906. |
[23] |
Yang Jin and Zhao Min, Complex behavior in a fish algae consumption model with impulsive control strategy, Discrete Dyn. Nat. Soc. Art., ID 163541, (2011).
doi: 10.1155/2011/163541. |
[24] |
H. Ye, A. N. Michel and L. Hou,
Stability analysis of systems with impulse effects, IEEE Transactions on Automatic Control, 43 (1998), 1719-1723.
doi: 10.1109/9.736069. |
[25] |
Zhao L., Chen L. and Zhang Q., The geometrical analysis of a predator-prey model with two state impulses, Mathematical Biosciences, 23 (2012), 55–64.
doi: 10.1016/j.mbs.2012.03.011. |
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