September  2021, 26(9): 4645-4661. doi: 10.3934/dcdsb.2020306

Impulses in driving semigroups of nonautonomous dynamical systems: Application to cascade systems

1. 

Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, São Carlos - SP, 13566-590, Brazil

2. 

Departamento de Matemática, Centro de Ciências Físicas e Matemáticas, Universidade Federal de Santa Catarina, Florianópolis-SC, 88040-900, Brazil

3. 

Faculdade de Matemática, Universidade Federal de Uberlândia, Uberlândia-MG, 38400-902, Brazil

4. 

Departamento de Estatística, Análise Matemática e Optimización & Instituto de Matemáticas, Universidade de Santiago de Compostela, Santiago de Compostela, Spain

* Corresponding author

Received  April 2020 Revised  August 2020 Published  September 2021 Early access  October 2020

Fund Project: The first author is partially supported by FAPESP grant 2016/24711-1 and CNPq grant 310497/2016-7. The second author is partially supported by CNPq, project # 407635/2016-5. The third author is partially supported by FAPEMIG, project # APQ-00371-18.The fourth author is partially supported by the predoctoral contact BES-2017-082334

In this paper we investigate the long time behavior of a nonautonomous dynamical system (cocycle) when its driving semigroup is subjected to impulses. We provide conditions to ensure the existence of global attractors for the associated impulsive skew-product semigroups, uniform attractors for the coupled impulsive cocycle and pullback attractors for the associated evolution processes. Finally, we illustrate the theory with an application to cascade systems.

Citation: Everaldo de Mello Bonotto, Matheus Cheque Bortolan, Rodolfo Collegari, José Manuel Uzal. Impulses in driving semigroups of nonautonomous dynamical systems: Application to cascade systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (9) : 4645-4661. doi: 10.3934/dcdsb.2020306
References:
[1]

N. U. Ahmed, Existence of optimal controls for a general class of impulsive systems on Banach spaces, SIAM J. Control Optim., 42 (2003), 669-685.  doi: 10.1137/S0363012901391299.  Google Scholar

[2]

M. Benchora, J. Henderson and S. Ntouyas, Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications, 2. Hindawi Publishing Corporation, New York, 2006. doi: 10.1155/9789775945501.  Google Scholar

[3]

E. M. BonottoM. C. BortolanT. Caraballo and R. Collegari, Attractors for impulsive non-autonomous dynamical systems and their relations, J. Differential Equations, 262 (2017), 3524-3550.  doi: 10.1016/j.jde.2016.11.036.  Google Scholar

[4]

E. M. BonottoM. C. BortolanT. Caraballo and R. Collegari, Impulsive non-autonomous dynamical systems and impulsive cocycle attractors, Math. Methods in the Appl. Sci., 40 (2017), 1095-1113.  doi: 10.1002/mma.4038.  Google Scholar

[5]

E. M. Bonotto and P. Kalita, On attractors of generalized semiflows with impulses, The Journal of Geometric Analysis, 30 (2020), 1412-1449.  doi: 10.1007/s12220-019-00143-0.  Google Scholar

[6]

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

[7]

E. M. BonottoM. C. BortolanT. Caraballo and R. Collegari, Impulsive surfaces on dynamical systems, Acta Mathematica Hungarica, 150 (2016), 209-216.  doi: 10.1007/s10474-016-0631-0.  Google Scholar

[8]

E. M. Bonotto, M. C. Bortolan, A. N. Carvalho and R. Czaja, Global attractors for impulsive dynamical systems - a precompact approach, J. Differential Equations, (2015), 2602-2625. doi: 10.1016/j.jde.2015.03.033.  Google Scholar

[9]

M. C. BortolanA. N. Carvalho and J. A. Langa, Structure of attractors for skew product semiflows, J. Differential Equations, 257 (2014), 490-522.  doi: 10.1016/j.jde.2014.04.008.  Google Scholar

[10]

M. C. Bortolan and J. M. Uzal, Pullback attractors to impulsive evolution processes: Applications to differential equations and tube conditions, Discrete Contin. Dyn. Syst., 40 (2020), 2791-2826.  doi: 10.3934/dcds.2020150.  Google Scholar

[11]

B. BouchardN.-M. Dang and C.-A. Lehalle, Optimal control of trading algorithms: A general impulse control approach, SIAM J. Finan. Math., 2 (2011), 404-438.  doi: 10.1137/090777293.  Google Scholar

[12]

T. Cardinali and R. Servadei, Periodic solutions of nonlinear impulsive differential inclusions with constraints, Proc. Am. Math. Soc., 132 (2004), 2339-2349.  doi: 10.1090/S0002-9939-04-07343-5.  Google Scholar

[13]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[14]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49. American Mathematical Society, Providence, RI, 2002. doi: 10.1090/coll/049.  Google Scholar

[15]

S. DashkovskiyO. Kapustyan and I. Romaniuk, Global attractors of impulsive parabolic inclusions, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1875-1886.  doi: 10.3934/dcdsb.2017111.  Google Scholar

[16]

M. H. A. DavisX. Guo and G. Wu, Impulse control of multidimensional jump diffusions, SIAM J. Control Optim., 48 (2010), 5276-5293.  doi: 10.1137/090780419.  Google Scholar

[17]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications, Applied Mathematical Sciences, 163. Springer-Verlag London, Ltd., London, 2008. doi: 10.1007/978-1-84628-708-4.  Google Scholar

[18]

J. A. Feroe, Existence and stability of multiple impulse solutions of a nerve equation, SIAM J. Appl. Math., 42 (1982), 235-246.  doi: 10.1137/0142017.  Google Scholar

[19]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications, 18. Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[20]

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

[21]

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

[22]

K. LiC. DingF. Wang and J. Hu, Limit set maps in impulsive semidynamical systems, J. Dyn. Control. Syst., 20 (2014), 47-58.  doi: 10.1007/s10883-013-9204-5.  Google Scholar

[23]

V. F. Rožko, A certain class of almost periodic motions in systems with pulses, Differencial' nye Uravnenja, 8 (1972), 2012-2022.   Google Scholar

[24]

V. F. Rožko, Ljapunov stability in discontinuous dynamical systems, Differencial'nye Uravnenja, 11 (1975), 1005-1012, 1148.  Google Scholar

[25]

V. F. Rožko, The almost recurrent and recurrent motions of discontinuous dynamical systems, Differencial'nye Uravnenja, 9 (1973), 1826-1830, 1925.  Google Scholar

[26]

H. Song and H. Wu, Pullback attractors of nonautonomous reaction-diffusion equations, J. Math. Anal. Appl., 325 (2007), 1200-1215.  doi: 10.1016/j.jmaa.2006.02.041.  Google Scholar

show all references

References:
[1]

N. U. Ahmed, Existence of optimal controls for a general class of impulsive systems on Banach spaces, SIAM J. Control Optim., 42 (2003), 669-685.  doi: 10.1137/S0363012901391299.  Google Scholar

[2]

M. Benchora, J. Henderson and S. Ntouyas, Impulsive Differential Equations and Inclusions, Contemporary Mathematics and Its Applications, 2. Hindawi Publishing Corporation, New York, 2006. doi: 10.1155/9789775945501.  Google Scholar

[3]

E. M. BonottoM. C. BortolanT. Caraballo and R. Collegari, Attractors for impulsive non-autonomous dynamical systems and their relations, J. Differential Equations, 262 (2017), 3524-3550.  doi: 10.1016/j.jde.2016.11.036.  Google Scholar

[4]

E. M. BonottoM. C. BortolanT. Caraballo and R. Collegari, Impulsive non-autonomous dynamical systems and impulsive cocycle attractors, Math. Methods in the Appl. Sci., 40 (2017), 1095-1113.  doi: 10.1002/mma.4038.  Google Scholar

[5]

E. M. Bonotto and P. Kalita, On attractors of generalized semiflows with impulses, The Journal of Geometric Analysis, 30 (2020), 1412-1449.  doi: 10.1007/s12220-019-00143-0.  Google Scholar

[6]

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

[7]

E. M. BonottoM. C. BortolanT. Caraballo and R. Collegari, Impulsive surfaces on dynamical systems, Acta Mathematica Hungarica, 150 (2016), 209-216.  doi: 10.1007/s10474-016-0631-0.  Google Scholar

[8]

E. M. Bonotto, M. C. Bortolan, A. N. Carvalho and R. Czaja, Global attractors for impulsive dynamical systems - a precompact approach, J. Differential Equations, (2015), 2602-2625. doi: 10.1016/j.jde.2015.03.033.  Google Scholar

[9]

M. C. BortolanA. N. Carvalho and J. A. Langa, Structure of attractors for skew product semiflows, J. Differential Equations, 257 (2014), 490-522.  doi: 10.1016/j.jde.2014.04.008.  Google Scholar

[10]

M. C. Bortolan and J. M. Uzal, Pullback attractors to impulsive evolution processes: Applications to differential equations and tube conditions, Discrete Contin. Dyn. Syst., 40 (2020), 2791-2826.  doi: 10.3934/dcds.2020150.  Google Scholar

[11]

B. BouchardN.-M. Dang and C.-A. Lehalle, Optimal control of trading algorithms: A general impulse control approach, SIAM J. Finan. Math., 2 (2011), 404-438.  doi: 10.1137/090777293.  Google Scholar

[12]

T. Cardinali and R. Servadei, Periodic solutions of nonlinear impulsive differential inclusions with constraints, Proc. Am. Math. Soc., 132 (2004), 2339-2349.  doi: 10.1090/S0002-9939-04-07343-5.  Google Scholar

[13]

A. N. Carvalho, J. A. Langa and J. C. Robinson, Attractors for Infinite-Dimensional Non-Autonomous Dynamical Systems, Applied Mathematical Sciences, 182. Springer, New York, 2013. doi: 10.1007/978-1-4614-4581-4.  Google Scholar

[14]

V. V. Chepyzhov and M. I. Vishik, Attractors for Equations of Mathematical Physics, American Mathematical Society Colloquium Publications, 49. American Mathematical Society, Providence, RI, 2002. doi: 10.1090/coll/049.  Google Scholar

[15]

S. DashkovskiyO. Kapustyan and I. Romaniuk, Global attractors of impulsive parabolic inclusions, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1875-1886.  doi: 10.3934/dcdsb.2017111.  Google Scholar

[16]

M. H. A. DavisX. Guo and G. Wu, Impulse control of multidimensional jump diffusions, SIAM J. Control Optim., 48 (2010), 5276-5293.  doi: 10.1137/090780419.  Google Scholar

[17]

M. di Bernardo, C. J. Budd, A. R. Champneys and P. Kowalczyk, Piecewise-Smooth Dynamical Systems: Theory and Applications, Applied Mathematical Sciences, 163. Springer-Verlag London, Ltd., London, 2008. doi: 10.1007/978-1-84628-708-4.  Google Scholar

[18]

J. A. Feroe, Existence and stability of multiple impulse solutions of a nerve equation, SIAM J. Appl. Math., 42 (1982), 235-246.  doi: 10.1137/0142017.  Google Scholar

[19]

A. F. Filippov, Differential Equations with Discontinuous Righthand Sides, Mathematics and its Applications, 18. Kluwer Academic Publishers Group, Dordrecht, 1988. doi: 10.1007/978-94-015-7793-9.  Google Scholar

[20]

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

[21]

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

[22]

K. LiC. DingF. Wang and J. Hu, Limit set maps in impulsive semidynamical systems, J. Dyn. Control. Syst., 20 (2014), 47-58.  doi: 10.1007/s10883-013-9204-5.  Google Scholar

[23]

V. F. Rožko, A certain class of almost periodic motions in systems with pulses, Differencial' nye Uravnenja, 8 (1972), 2012-2022.   Google Scholar

[24]

V. F. Rožko, Ljapunov stability in discontinuous dynamical systems, Differencial'nye Uravnenja, 11 (1975), 1005-1012, 1148.  Google Scholar

[25]

V. F. Rožko, The almost recurrent and recurrent motions of discontinuous dynamical systems, Differencial'nye Uravnenja, 9 (1973), 1826-1830, 1925.  Google Scholar

[26]

H. Song and H. Wu, Pullback attractors of nonautonomous reaction-diffusion equations, J. Math. Anal. Appl., 325 (2007), 1200-1215.  doi: 10.1016/j.jmaa.2006.02.041.  Google Scholar

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