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doi: 10.3934/dcdsb.2021289
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The existence and properties of the solution of a class of nonlinear differential equations with switching at variable times

School of Mathematics and Statistics, Guizhou University, Guiyang, Guizhou 550025, China

* Corresponding author: Yunfei Peng

Received  August 2021 Early access December 2021

Fund Project: The first author is supported by the Foundation of Postgraduate of Guizhou Province grant 2019032; The second author is supported by the National Natural Science Foundation of China grant 12061021 and 11661020

In this paper, we deal with the qualitative theory for a class of nonlinear differential equations with switching at variable times (SSVT), such as the existence and uniqueness of the solution, the continuous dependence and differentiability of the solution with respect to parameters and the stability. Firstly, we obtain the existence and uniqueness of a global solution by defining a reasonable solution (see Definition 2.1). Secondly, the continuous dependence and differentiability of the solution with respect to the initial state and the switching line are investigated. Finally, the global exponential stability of the system is discussed. Moreover, we give the necessary and sufficient conditions of SSVT just switching $ k\in \mathbb{N} $ times on bounded time intervals.

Citation: Huanting Li, Yunfei Peng, Kuilin Wu. The existence and properties of the solution of a class of nonlinear differential equations with switching at variable times. Discrete and Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2021289
References:
[1]

J. L. Aguilar and R. A. Garcia, An extension of LaSalle's invariance principle for switched systems, Systems Control Lett., 55 (2006), 376-384.  doi: 10.1016/j.sysconle.2005.07.009.

[2]

L. I. Allerhand and U. Shaked, Stability of stochastic nonlinear systems with state-dependent switching, IEEE Transactions on Automatic Control, 58 (2013), 994-1001. doi: 10.1109/TAC.2013.2246094.

[3]

J. BaekC. Kang and W. Kim, Practical approach for developing lateral motion control of autonomous lane change system, Applied Sciences, 10 (2020), 31-43.  doi: 10.3390/app10093143.

[4]

M. S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Automat. Control, 43 (1998), 475-482.  doi: 10.1109/9.664150.

[5]

R. W. Brockett, Hybrid models for motion control systems, in essays on control: Perspectives in the theory and its applications, Progr. Systems Control Theory, 14 (1993), 29-53. 

[6]

C. Cai and A. Teel, Robust input-to-state stability for hybrid systems, SIAM J. Control Optim., 51 (2013), 1651-1678.  doi: 10.1137/110824747.

[7]

I. Capuzzo Dolcetta and L. C. Evans, Optimal switching for ordinary differential equations, SIAM J. Control Optim., 22 (1984), 143-161.  doi: 10.1137/0322011.

[8]

J. H. B. Deane and D. C. Hamill, Instability, subharmonics, and chaos in power electronic systems, 20th Annual IEEE Power Electronics Specialists Conference, 5 (1990), 260-268.  doi: 10.1109/PESC.1989.48470.

[9]

S. EngellS. KowalewskiC. Schulz and O. Stursberg, Continuous-discrete interactions in chemical processing plants, Proceedings of IEEE, 88 (2000), 1050-1068.  doi: 10.1109/5.871308.

[10]

L. C. Evans and A. Friedman, Optimal stochastic switching and the Dirichlet problem for the Bellman equation, Trans. Amer. Math. Soc., 253 (1979), 365-389.  doi: 10.1090/S0002-9947-1979-0536953-4.

[11]

J. Ezzine and A. H. Haddad, On the controllability and observability of hybrid systems, Internat. J. Control, 49 (1989), 2045-2055.  doi: 10.1080/00207178908559761.

[12]

J. P. Hespanha, Uniform stability of switched linear systems: Extensions of Lasalle's invariance principle, IEEE Trans. Automat. Control, 49 (2004), 470-482.  doi: 10.1109/TAC.2004.825641.

[13]

J. P. HespanhaD. LiberzonD. Angeli and E. D. Songtag, Nonlinear norm-observability notions and stability of switched system, IEEE Trans. Automat. Control, 50 (2005), 154-168.  doi: 10.1109/TAC.2004.841937.

[14]

J. P. Hespanha and A. S. Morse, Stability of switched systems by average dwell time, Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, AZ, USA, 1999. doi: 10.1109/CDC.1999.831330.

[15]

I. Hiskens, Stability of hybrid system limit cycles: Application to the compass gait biped robot, Proceeding of the 40th IEEE Conference on Decision and Control, Orlando, Florida, USA: IEEE, 2001. doi: 10.1109/CDC.2001.980200.

[16]

J. J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons, Proceedings of the National Academy of Sciences, 81 (1984), 3088-3092.  doi: 10.1073/pnas.81.10.3088.

[17]

D. Jeon and M. Tomizuka, Learning hybrid force and position control of robot manipulators, Proceedings 1992 IEEE International Conference on Robotics and Automation, 9 (1993), 423-431.  doi: 10.1109/ROBOT.1992.220146.

[18]

M. Johansson and A. Rantzer, Computation of piecewise quadratic Lyapunov functions for hybrid systems, IEEE Trans. Automat. Control, 43 (1998), 555-559.  doi: 10.1109/9.664157.

[19]

S. H. LeeT. H. Kim and J. T. Lim, A new stability of switched systems, Automatica, 36 (2000), 917-922.  doi: 10.1016/S0005-1098(99)00208-3.

[20]

S. M. LenhartT. I. Seidman and J. Yong, Optimal control of a bioreactor with modal switching, Math. Models Methods Appl. Sci., 11 (2001), 993-949.  doi: 10.1142/S0218202501001185.

[21]

B. LennartsonM. Tittus and B. Egardt, Hybrid systems in process control, Proceedings of 1994 33rd IEEE Conference on Decision and Control, 16 (2002), 45-56.  doi: 10.1109/CDC.1994.411706.

[22]

H. LiY. Peng and K. Wu, Properties of solution for the nonlinear on-off switched differential equation with switching at variable times, Nonlinear Dynamics, 103 (2021), 2287-2298. 

[23]

X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhuser, 1995. doi: 10.1007/978-1-4612-4260-4.

[24]

D. Liberzon, Switching in Systems and Control, Systems & Control: Foundations & Applications. Birkháuser Boston, Inc., Boston, MA, 2003. doi: 10.1007/978-1-4612-0017-8.

[25]

D. Liberzon, Switched systems, Handbook of Networked and Embedded Control Systems, (2005), 559–574. doi: 10.1007/0-8176-4404-0_24.

[26]

C. LiuZ. YangD. Sun and X. Liu, Stability of variable-time switched systems, Arab. J. Sci. Eng., 42 (2017), 2971-2980.  doi: 10.1007/s13369-017-2476-4.

[27]

C. LiuZ. YangD. SunX. Liu and W. Liu, Stability of switched neural networks with time-varying delays, Neural Computing and Applications, 30 (2018), 2229-2244.  doi: 10.1007/s00521-016-2805-7.

[28]

S. Liu and D. Liberzon, Global stability and asymptotic gain imply input-to-state stability for state-dependent switched systems, 2018 IEEE Conference on Decision and Control, (2018) 17–19. doi: 10.1109/CDC.2018.8619364.

[29]

A. S. Morse, Supervisory control of families of linear set-point controllers-part 1. Exact matching, IEEE Trans. Automat. Control, 41 (1996), 1413-1431.  doi: 10.1109/9.539424.

[30]

U. Oberst, Stability and stabilization of multidimensional input/Output systems, SIAM J. Control Optim., 45 (2006), 1467-1507.  doi: 10.1137/050639004.

[31]

P. Peleties and R. DeCarlo, Asymptotic stability of m-switched systems using lyapunov-like functions, 1991 American Control Conference, 2 (1991), 1679-1684.  doi: 10.23919/ACC.1991.4791667.

[32]

S. R. Sanders and G. C. Verghese, Lyapunov-based control for switched power converters, IEEE Transactions on Power Electronics, 7 (1992), 17-24. 

[33]

A. N. Shiryayev, Optimal Stopping Rules, Spring-Verlay, 1978.

[34]

R. ShortenF. WirthO. MasonW. Kai and C. King, Stability criteria for switched and hybrid systems, SIAM Rev., 49 (2007), 545-592.  doi: 10.1137/05063516X.

[35]

S. Stojanovic and J. Yong, Optimal switching for partial differential equations Ⅰ, Ⅱ, J. Math. Anal. Appl., 138 (1989), 418-438.  doi: 10.1016/0022-247X(89)90301-6.

[36]

Z. Sun and S. S. Ge, Switched Linear Systems: Control and Design, Springer, 2005.

[37]

F. A. Wyczalek, Hybrid electric vehicles: Year 2000 status, IEEE Aerospace and Electronic Systems Magazine, 16 (2001), 15-25.  doi: 10.1109/62.911316.

[38]

G. ZhaiB. Hu and K. Yasuda, Stability analysis of switched with stable and unstable subsystems: An average dwell time approach, Internat. J. Systems Sci., 32 (2001), 1055-1061.  doi: 10.1080/00207720116692.

[39]

G. Zhang and Y. Shen, Novel conditions on exponential stability of a class of delayed neural networks with state-dependent switching, Neural Networks, 71 (2015), 55-61.  doi: 10.1016/j.neunet.2015.07.016.

[40]

X. ZhangC. Li and T. Huang, Hybrid impulsive and switching Hopfield neural networks with state-dependent impulses, Neural Networks, 93 (2017), 176-184.  doi: 10.1016/j.neunet.2017.04.009.

show all references

References:
[1]

J. L. Aguilar and R. A. Garcia, An extension of LaSalle's invariance principle for switched systems, Systems Control Lett., 55 (2006), 376-384.  doi: 10.1016/j.sysconle.2005.07.009.

[2]

L. I. Allerhand and U. Shaked, Stability of stochastic nonlinear systems with state-dependent switching, IEEE Transactions on Automatic Control, 58 (2013), 994-1001. doi: 10.1109/TAC.2013.2246094.

[3]

J. BaekC. Kang and W. Kim, Practical approach for developing lateral motion control of autonomous lane change system, Applied Sciences, 10 (2020), 31-43.  doi: 10.3390/app10093143.

[4]

M. S. Branicky, Multiple Lyapunov functions and other analysis tools for switched and hybrid systems, IEEE Trans. Automat. Control, 43 (1998), 475-482.  doi: 10.1109/9.664150.

[5]

R. W. Brockett, Hybrid models for motion control systems, in essays on control: Perspectives in the theory and its applications, Progr. Systems Control Theory, 14 (1993), 29-53. 

[6]

C. Cai and A. Teel, Robust input-to-state stability for hybrid systems, SIAM J. Control Optim., 51 (2013), 1651-1678.  doi: 10.1137/110824747.

[7]

I. Capuzzo Dolcetta and L. C. Evans, Optimal switching for ordinary differential equations, SIAM J. Control Optim., 22 (1984), 143-161.  doi: 10.1137/0322011.

[8]

J. H. B. Deane and D. C. Hamill, Instability, subharmonics, and chaos in power electronic systems, 20th Annual IEEE Power Electronics Specialists Conference, 5 (1990), 260-268.  doi: 10.1109/PESC.1989.48470.

[9]

S. EngellS. KowalewskiC. Schulz and O. Stursberg, Continuous-discrete interactions in chemical processing plants, Proceedings of IEEE, 88 (2000), 1050-1068.  doi: 10.1109/5.871308.

[10]

L. C. Evans and A. Friedman, Optimal stochastic switching and the Dirichlet problem for the Bellman equation, Trans. Amer. Math. Soc., 253 (1979), 365-389.  doi: 10.1090/S0002-9947-1979-0536953-4.

[11]

J. Ezzine and A. H. Haddad, On the controllability and observability of hybrid systems, Internat. J. Control, 49 (1989), 2045-2055.  doi: 10.1080/00207178908559761.

[12]

J. P. Hespanha, Uniform stability of switched linear systems: Extensions of Lasalle's invariance principle, IEEE Trans. Automat. Control, 49 (2004), 470-482.  doi: 10.1109/TAC.2004.825641.

[13]

J. P. HespanhaD. LiberzonD. Angeli and E. D. Songtag, Nonlinear norm-observability notions and stability of switched system, IEEE Trans. Automat. Control, 50 (2005), 154-168.  doi: 10.1109/TAC.2004.841937.

[14]

J. P. Hespanha and A. S. Morse, Stability of switched systems by average dwell time, Proceedings of the 38th IEEE Conference on Decision and Control, Phoenix, AZ, USA, 1999. doi: 10.1109/CDC.1999.831330.

[15]

I. Hiskens, Stability of hybrid system limit cycles: Application to the compass gait biped robot, Proceeding of the 40th IEEE Conference on Decision and Control, Orlando, Florida, USA: IEEE, 2001. doi: 10.1109/CDC.2001.980200.

[16]

J. J. Hopfield, Neurons with graded response have collective computational properties like those of two-state neurons, Proceedings of the National Academy of Sciences, 81 (1984), 3088-3092.  doi: 10.1073/pnas.81.10.3088.

[17]

D. Jeon and M. Tomizuka, Learning hybrid force and position control of robot manipulators, Proceedings 1992 IEEE International Conference on Robotics and Automation, 9 (1993), 423-431.  doi: 10.1109/ROBOT.1992.220146.

[18]

M. Johansson and A. Rantzer, Computation of piecewise quadratic Lyapunov functions for hybrid systems, IEEE Trans. Automat. Control, 43 (1998), 555-559.  doi: 10.1109/9.664157.

[19]

S. H. LeeT. H. Kim and J. T. Lim, A new stability of switched systems, Automatica, 36 (2000), 917-922.  doi: 10.1016/S0005-1098(99)00208-3.

[20]

S. M. LenhartT. I. Seidman and J. Yong, Optimal control of a bioreactor with modal switching, Math. Models Methods Appl. Sci., 11 (2001), 993-949.  doi: 10.1142/S0218202501001185.

[21]

B. LennartsonM. Tittus and B. Egardt, Hybrid systems in process control, Proceedings of 1994 33rd IEEE Conference on Decision and Control, 16 (2002), 45-56.  doi: 10.1109/CDC.1994.411706.

[22]

H. LiY. Peng and K. Wu, Properties of solution for the nonlinear on-off switched differential equation with switching at variable times, Nonlinear Dynamics, 103 (2021), 2287-2298. 

[23]

X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhuser, 1995. doi: 10.1007/978-1-4612-4260-4.

[24]

D. Liberzon, Switching in Systems and Control, Systems & Control: Foundations & Applications. Birkháuser Boston, Inc., Boston, MA, 2003. doi: 10.1007/978-1-4612-0017-8.

[25]

D. Liberzon, Switched systems, Handbook of Networked and Embedded Control Systems, (2005), 559–574. doi: 10.1007/0-8176-4404-0_24.

[26]

C. LiuZ. YangD. Sun and X. Liu, Stability of variable-time switched systems, Arab. J. Sci. Eng., 42 (2017), 2971-2980.  doi: 10.1007/s13369-017-2476-4.

[27]

C. LiuZ. YangD. SunX. Liu and W. Liu, Stability of switched neural networks with time-varying delays, Neural Computing and Applications, 30 (2018), 2229-2244.  doi: 10.1007/s00521-016-2805-7.

[28]

S. Liu and D. Liberzon, Global stability and asymptotic gain imply input-to-state stability for state-dependent switched systems, 2018 IEEE Conference on Decision and Control, (2018) 17–19. doi: 10.1109/CDC.2018.8619364.

[29]

A. S. Morse, Supervisory control of families of linear set-point controllers-part 1. Exact matching, IEEE Trans. Automat. Control, 41 (1996), 1413-1431.  doi: 10.1109/9.539424.

[30]

U. Oberst, Stability and stabilization of multidimensional input/Output systems, SIAM J. Control Optim., 45 (2006), 1467-1507.  doi: 10.1137/050639004.

[31]

P. Peleties and R. DeCarlo, Asymptotic stability of m-switched systems using lyapunov-like functions, 1991 American Control Conference, 2 (1991), 1679-1684.  doi: 10.23919/ACC.1991.4791667.

[32]

S. R. Sanders and G. C. Verghese, Lyapunov-based control for switched power converters, IEEE Transactions on Power Electronics, 7 (1992), 17-24. 

[33]

A. N. Shiryayev, Optimal Stopping Rules, Spring-Verlay, 1978.

[34]

R. ShortenF. WirthO. MasonW. Kai and C. King, Stability criteria for switched and hybrid systems, SIAM Rev., 49 (2007), 545-592.  doi: 10.1137/05063516X.

[35]

S. Stojanovic and J. Yong, Optimal switching for partial differential equations Ⅰ, Ⅱ, J. Math. Anal. Appl., 138 (1989), 418-438.  doi: 10.1016/0022-247X(89)90301-6.

[36]

Z. Sun and S. S. Ge, Switched Linear Systems: Control and Design, Springer, 2005.

[37]

F. A. Wyczalek, Hybrid electric vehicles: Year 2000 status, IEEE Aerospace and Electronic Systems Magazine, 16 (2001), 15-25.  doi: 10.1109/62.911316.

[38]

G. ZhaiB. Hu and K. Yasuda, Stability analysis of switched with stable and unstable subsystems: An average dwell time approach, Internat. J. Systems Sci., 32 (2001), 1055-1061.  doi: 10.1080/00207720116692.

[39]

G. Zhang and Y. Shen, Novel conditions on exponential stability of a class of delayed neural networks with state-dependent switching, Neural Networks, 71 (2015), 55-61.  doi: 10.1016/j.neunet.2015.07.016.

[40]

X. ZhangC. Li and T. Huang, Hybrid impulsive and switching Hopfield neural networks with state-dependent impulses, Neural Networks, 93 (2017), 176-184.  doi: 10.1016/j.neunet.2017.04.009.

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