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April  2020, 19(4): 1979-1996. doi: 10.3934/cpaa.2020087

Stability and forward attractors for non-autonomous impulsive semidynamical systems

Everaldo de Mello Bonotto 1,, and  Daniela Paula Demuner 2,

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

* Corresponding author.

Dedicated to Tomás Caraballo on his 60th birthday

Received  April 2019 Revised  October 2019 Published  January 2020

Fund Project: The first author is supported partially by FAPESP grant 2016/24711-1 and CNPq grant 310497/2016-7.

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.

Keywords: Non-autonomous dynamical systems, impulses, global attractor, uniform attractor, stability.
Mathematics Subject Classification: Primary: 34D20, 34D45; Secondary: 34A37.
Citation: Everaldo de Mello Bonotto, Daniela Paula Demuner. Stability and forward attractors for non-autonomous impulsive semidynamical systems. Communications on Pure and Applied Analysis, 2020, 19 (4) : 1979-1996. doi: 10.3934/cpaa.2020087
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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