American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Forwards dynamics of non-autonomous dynamical systems: Driving semigroups without backwards uniqueness and structure of the attractor
  • CPAA Home
  • This Issue
  • Next Article
    Controllability of the one-dimensional fractional heat equation under positivity constraints
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 & 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

E. M. Bonotto and D. P. Demuner, Autonomous dissipative semidynamical systems with impulses, Topological Methods in Nonlinear Analysis, 41 (2013), 1-38.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

K. Ciesielski, On semicontinuity in impulsive dynamical systems, Bull. Polish Acad. Sci. Math., 52 (2004), 71-80.  doi: 10.4064/ba52-1-8.  Google Scholar

[14]

K. Ciesielski, On stability in impulsive dynamical systems, Bull. Polish Acad. Sci. Math., 52 (2004), 81-91.  doi: 10.4064/ba52-1-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

D. Husemoller, Fibre Bundles, Springer-Verlag, Berlin-Heidelberg-New York, 1994. doi: 10.1007/978-1-4757-2261-1.  Google Scholar

[19]

S. K. Kaul, On impulsive semidynamical systems, J. Math. Anal. Appl., 150 (1990), 120-128.  doi: 10.1016/0022-247X(90)90199-P.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. doi: 10.1142/0906.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[6]

E. M. Bonotto and D. P. Demuner, Autonomous dissipative semidynamical systems with impulses, Topological Methods in Nonlinear Analysis, 41 (2013), 1-38.   Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

K. Ciesielski, On semicontinuity in impulsive dynamical systems, Bull. Polish Acad. Sci. Math., 52 (2004), 71-80.  doi: 10.4064/ba52-1-8.  Google Scholar

[14]

K. Ciesielski, On stability in impulsive dynamical systems, Bull. Polish Acad. Sci. Math., 52 (2004), 81-91.  doi: 10.4064/ba52-1-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

D. Husemoller, Fibre Bundles, Springer-Verlag, Berlin-Heidelberg-New York, 1994. doi: 10.1007/978-1-4757-2261-1.  Google Scholar

[19]

S. K. Kaul, On impulsive semidynamical systems, J. Math. Anal. Appl., 150 (1990), 120-128.  doi: 10.1016/0022-247X(90)90199-P.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

V. Lakshmikantham, D. D. Bainov and P. S. Simeonov, Theory of Impulsive Differential Equations, World Scientific, Singapore, 1989. doi: 10.1142/0906.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Tomás Caraballo, David Cheban. On the structure of the global attractor for non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2012, 11 (2) : 809-828. doi: 10.3934/cpaa.2012.11.809
[2]

Tomás Caraballo, David Cheban. On the structure of the global attractor for infinite-dimensional non-autonomous dynamical systems with weak convergence. Communications on Pure & Applied Analysis, 2013, 12 (1) : 281-302. doi: 10.3934/cpaa.2013.12.281
[3]

Michael Zgurovsky, Mark Gluzman, Nataliia Gorban, Pavlo Kasyanov, Liliia Paliichuk, Olha Khomenko. Uniform global attractors for non-autonomous dissipative dynamical systems. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 2053-2065. doi: 10.3934/dcdsb.2017120
[4]

Chunyou Sun, Daomin Cao, Jinqiao Duan. Non-autonomous wave dynamics with memory --- asymptotic regularity and uniform attractor. Discrete & Continuous Dynamical Systems - B, 2008, 9 (3&4, May) : 743-761. doi: 10.3934/dcdsb.2008.9.743
[5]

Olivier Goubet, Wided Kechiche. Uniform attractor for non-autonomous nonlinear Schrödinger equation. Communications on Pure & Applied Analysis, 2011, 10 (2) : 639-651. doi: 10.3934/cpaa.2011.10.639
[6]

Xiaolin Jia, Caidi Zhao, Juan Cao. Uniform attractor of the non-autonomous discrete Selkov model. Discrete & Continuous Dynamical Systems - A, 2014, 34 (1) : 229-248. doi: 10.3934/dcds.2014.34.229
[7]

Alexandre N. Carvalho, José A. Langa, James C. Robinson. Forwards dynamics of non-autonomous dynamical systems: Driving semigroups without backwards uniqueness and structure of the attractor. Communications on Pure & Applied Analysis, 2020, 19 (4) : 1997-2013. doi: 10.3934/cpaa.2020088
[8]

Xinyuan Liao, Caidi Zhao, Shengfan Zhou. Compact uniform attractors for dissipative non-autonomous lattice dynamical systems. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1087-1111. doi: 10.3934/cpaa.2007.6.1087
[9]

Wen Tan. The regularity of pullback attractor for a non-autonomous p-Laplacian equation with dynamical boundary condition. Discrete & Continuous Dynamical Systems - B, 2019, 24 (2) : 529-546. doi: 10.3934/dcdsb.2018194
[10]

Alexandre N. Carvalho, José A. Langa, James C. Robinson. Non-autonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 703-747. doi: 10.3934/dcdsb.2015.20.703
[11]

V. V. Chepyzhov, M. I. Vishik, W. L. Wendland. On non-autonomous sine-Gordon type equations with a simple global attractor and some averaging. Discrete & Continuous Dynamical Systems - A, 2005, 12 (1) : 27-38. doi: 10.3934/dcds.2005.12.27
[12]

T. Tachim Medjo. Non-autonomous 3D primitive equations with oscillating external force and its global attractor. Discrete & Continuous Dynamical Systems - A, 2012, 32 (1) : 265-291. doi: 10.3934/dcds.2012.32.265
[13]

Rodrigo Samprogna, Tomás Caraballo. Pullback attractor for a dynamic boundary non-autonomous problem with Infinite Delay. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 509-523. doi: 10.3934/dcdsb.2017195
[14]

T. Caraballo, J. A. Langa, J. Valero. Structure of the pullback attractor for a non-autonomous scalar differential inclusion. Discrete & Continuous Dynamical Systems - S, 2016, 9 (4) : 979-994. doi: 10.3934/dcdss.2016037
[15]

T. Tachim Medjo. A non-autonomous 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external force and its global attractor. Communications on Pure & Applied Analysis, 2011, 10 (2) : 415-433. doi: 10.3934/cpaa.2011.10.415
[16]

Pablo G. Barrientos, Abbas Fakhari. Ergodicity of non-autonomous discrete systems with non-uniform expansion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (4) : 1361-1382. doi: 10.3934/dcdsb.2019231
[17]

Ahmed Y. Abdallah, Rania T. Wannan. Second order non-autonomous lattice systems and their uniform attractors. Communications on Pure & Applied Analysis, 2019, 18 (4) : 1827-1846. doi: 10.3934/cpaa.2019085
[18]

Noriaki Yamazaki. Global attractors for non-autonomous multivalued dynamical systems associated with double obstacle problems. Conference Publications, 2003, 2003 (Special) : 935-944. doi: 10.3934/proc.2003.2003.935
[19]

Zhaojuan Wang, Shengfan Zhou. Random attractor and random exponential attractor for stochastic non-autonomous damped cubic wave equation with linear multiplicative white noise. Discrete & Continuous Dynamical Systems - A, 2018, 38 (9) : 4767-4817. doi: 10.3934/dcds.2018210
[20]

Zhaojuan Wang, Shengfan Zhou. Random attractor for stochastic non-autonomous damped wave equation with critical exponent. Discrete & Continuous Dynamical Systems - A, 2017, 37 (1) : 545-573. doi: 10.3934/dcds.2017022

2019 Impact Factor: 1.105

Tools

Metrics

  • PDF downloads (128)
  • HTML views (112)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences