doi: 10.3934/mcrf.2021045
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A nonlinear version of Halanay's inequality for the uniform convergence to the origin

Center of Excellence for Research DEWS, Department of Information Engineering, Computer Science, and Mathematics, University of L'Aquila, Via Vetoio Coppito I, 67100 L'Aquila, Italy

* Corresponding author: e-mail pierdomenico.pepe@univaq.it

This article was recruited by Fabian Wirth

Received  September 2019 Revised  February 2021 Early access September 2021

Fund Project: The work has been supported in part by grant MIUR-FFABR 2017

A nonlinear version of Halanay's inequality is studied in this paper as a sufficient condition for the convergence of functions to the origin, uniformly with respect to bounded sets of initial values. The same result is provided in the case of forcing terms, for the uniform convergence to suitable neighborhoods of the origin. Related Lyapunov methods for the global uniform asymptotic stability and the input-to-state stability of systems described by retarded functional differential equations, with possibly nonconstant time delays, are provided. The relationship with the Razumikhin methodology is shown.

Citation: Pierdomenico Pepe. A nonlinear version of Halanay's inequality for the uniform convergence to the origin. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2021045
References:
[1]

M. Adivar and E. A. Bohner, Halanay type inequalities on time scales with applications, Nonlinear Anal., 74 (2011), 7519-7531.  doi: 10.1016/j.na.2011.08.007.  Google Scholar

[2]

R. P. AgarwalY.-H. Kim and S. K. Sen, New discrete Halanay inequalities: Stability of difference equations, Commun. Appl. Anal., 12 (2008), 83-90.   Google Scholar

[3]

C. T. H. Baker, Development and application of Halanay-type theory: Evolutionary differential and difference equations with time lag, J. Comput. Appl. Math., 234 (2010), 2663-2682.  doi: 10.1016/j.cam.2010.01.027.  Google Scholar

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C. T. H. Baker and E. Buckwar, Exponential stability in $p-th$ mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations, J. Comput. Appl. Math., 184 (2005), 404-427.  doi: 10.1016/j.cam.2005.01.018.  Google Scholar

[5]

D. Bresh-Pietri, J. Chauvin and N. Petit, Invoking Halanay inequality to conclude on closed-loop stability of a process with input-varying delay, Proceedings of the $10^{th}$ IFAC Workshop on Time-Delay Systems, Boston, USA, (2012), 266–271. Google Scholar

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E. Fridman, Introduction to Time-Delay Systems: Analysis and Control, Birkhäuser, 2014. doi: 10.1007/978-3-319-09393-2.  Google Scholar

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M. T. Grifa and P. Pepe, On stability analysis of discrete-time systems with constrained time-delays via nonlinear halanay-type inequality, IEEE Control Syst. Lett., 5 (2021), 869-874.  doi: 10.1109/LCSYS.2020.3007096.  Google Scholar

[8] A. Halanay, Differential Equations, Stability, Oscillations, Time Lags, Academic Press, New York, 1966.   Google Scholar
[9]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer Verlag, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[10]

J. K. Hale and S. M. Verduyn Lunel, Strong stabilization of neutral functional differential equations, IMA J. Math. Control Inform., 19 (2002), 5-23.  doi: 10.1093/imamci/19.1_and_2.5.  Google Scholar

[11]

L. V. HienV. N. Phat and H. Trinh, New generalized Halanay inequalities with applications to stability of nonlinear non-autonomous time-delay systems, Nonlinear Dynam., 82 (2015), 563-575.  doi: 10.1007/s11071-015-2176-0.  Google Scholar

[12]

M. Jankovic, Control Lyapunov-Razumikhin functions and robust stabilization of time delay systems, IEEE Trans. Automat. Control, 46 (2001), 1048-1060.  doi: 10.1109/9.935057.  Google Scholar

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I. Karafyllis and Z.-P. Jiang, Stability and Stabilization of Nonlinear Systems, Springer, London, 2011. doi: 10.1007/978-0-85729-513-2.  Google Scholar

[14]

I. Karafyllis, M. Malisoff, F. Mazenc and P. Pepe, Stabilization of nonlinear delay systems: A tutorial on recent results, I. Karafyllis, M. Malisoff, F. Mazenc, P. Pepe (EDs), Recent Results on Nonlinear Delay Control Systems, In Honor of Miroslav Krstic, Series Advances in Delays and Dynamics, Springer, 4 (2015), 1–41. doi: 10.1007/978-3-319-18072-4_1.  Google Scholar

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I. KarafyllisP. Pepe and Z.-P. Jiang, Global output stability for systems described by retarded functional differential equations: Lyapunov characterizations, Eur. J. Control, 14 (2008), 516-536.  doi: 10.3166/ejc.14.516-536.  Google Scholar

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H. K. Khalil, Nonlinear Systems, Prentice Hall, International Edition, Third Edition, Upper Saddle River, New Jersey, 2000. Google Scholar

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A. V. Kim, Functional Differential Equations, Application of I-Smooth Calculus, Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-94-017-1630-7.  Google Scholar

[18] N. N. Krasovskii, Stability of Motion, Stanford University Press, 1963.   Google Scholar
[19]

Y. LinE. D. Sontag and Y. Wang, A smooth converse Lyapunov theorem for robust stability, SIAM J. Control Optim., 34 (1996), 124-160.  doi: 10.1137/S0363012993259981.  Google Scholar

[20]

E. LizA. Ivanov and J. B. Ferreiro, Discrete Halanay-type inequalities and applications, Nonlinear Anal., 55 (2003), 669-678.  doi: 10.1016/j.na.2003.07.013.  Google Scholar

[21]

F. Mazenc and M. Malisoff, Extensions of Razumikhin's theorem and Lyapunov-Krasovskii functional constructions for time-varying systems with delay, Automatica J. IFAC, 78 (2017), 1-13.  doi: 10.1016/j.automatica.2016.12.005.  Google Scholar

[22]

A. Mironchenko and F. Wirth, Characterizations of input-to-state stability for infinite-dimensional systems, IEEE Trans. Automat. Control, 63 (2018), 1602-1617.  doi: 10.1109/tac.2017.2756341.  Google Scholar

[23]

S. Mohamad and K. Gopalsamy, Continuous and discrete Halanay-type Inequalities, Bull. Austral. Math. Soc., 61 (2000), 371-385.  doi: 10.1017/S0004972700022413.  Google Scholar

[24]

A. D. Myshkis, Razumikhin's method in the qualitative theory of processes with delay, J. Appl. Math. Stochastic Anal., 8 (1995), 233-247.  doi: 10.1155/S1048953395000219.  Google Scholar

[25]

P. Pepe, On Liapunov-Krasovskii functionals under Caratheodory conditions, Automatica J. IFAC, 43 (2007), 701-706.  doi: 10.1016/j.automatica.2006.10.024.  Google Scholar

[26]

P. Pepe, On control Lyapunov-Razumikhin functions, nonconstant delays, nonsmooth feedbacks, and nonlinear sampled-data stabilization, IEEE Trans. Automat. Control, 62 (2017), 5604-5619.  doi: 10.1109/TAC.2017.2689500.  Google Scholar

[27]

P. Pepe and E. Fridman, On global exponential stability preservation under sampling for globally Lipschitz time-delay systems, Automatica J. IFAC, 82 (2017), 295-300.  doi: 10.1016/j.automatica.2017.04.055.  Google Scholar

[28]

P. Pepe and Z.-P. Jiang, A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems, Systems Control Lett., 55 (2006), 1006-1014.  doi: 10.1016/j.sysconle.2006.06.013.  Google Scholar

[29]

P. Pepe and I. Karafyllis, Converse Lyapunov-Krasovskii theorems for systems described by neutral functional differential equations in Hale's form, Internat. J. Control, 86 (2013), 232-243.  doi: 10.1080/00207179.2012.723137.  Google Scholar

[30]

P. PepeI. Karafyllis and Z.-P. Jiang, Lyapunov-Krasovskii characterization of the input-to-state stability for neutral systems in Hale's form, Systems Control Lett., 102 (2017), 48-56.  doi: 10.1016/j.sysconle.2017.01.008.  Google Scholar

[31]

B. S. Razumikhin, On stability of systems with delay, Prykl. Mat. Mekh., 20 (1956), 500–512, (in Russian).  Google Scholar

[32]

E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34 (1989), 435-443.  doi: 10.1109/9.28018.  Google Scholar

[33]

A. R. Teel, Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem, IEEE Trans. Automat. Control, 43 (1998), 960-964.  doi: 10.1109/9.701099.  Google Scholar

[34]

S. Udpin and P. Niamsup, New discrete type inequalities and global stability of nonlinear difference equations, Appl. Math. Lett., 22 (2009), 856-859.  doi: 10.1016/j.aml.2008.07.011.  Google Scholar

[35]

W. Wang, A generalized Halanay inequality for stability of nonlinear neutral functional differential equations, J. Inequal. Appl., (2010), Article ID 475019, 16 pp. doi: 10.1155/2010/475019.  Google Scholar

[36]

L. WenY. Yu and W. Wang, Generalized Halanay inequalities for dissipativity of Volterra functional differential equations, J. Math. Anal. Appl., 347 (2008), 169-178.  doi: 10.1016/j.jmaa.2008.05.007.  Google Scholar

[37]

L. Xu and D. He, A nonlinear nonautonomous delay differential inequality for dissipativity of Lotka-Volterra functional differential equations, Tbilisi Math. J., 7 (2014), 37-43.  doi: 10.2478/tmj-2014-0004.  Google Scholar

[38]

C. ZhangF. DengH. Mo and H. Ren, Dissipativity and stability of a class of nonlinear multiple time delay integro-differential equations, J. Nonlinear Sci. Appl., 12 (2019), 363-375.  doi: 10.22436/jnsa.012.06.03.  Google Scholar

show all references

References:
[1]

M. Adivar and E. A. Bohner, Halanay type inequalities on time scales with applications, Nonlinear Anal., 74 (2011), 7519-7531.  doi: 10.1016/j.na.2011.08.007.  Google Scholar

[2]

R. P. AgarwalY.-H. Kim and S. K. Sen, New discrete Halanay inequalities: Stability of difference equations, Commun. Appl. Anal., 12 (2008), 83-90.   Google Scholar

[3]

C. T. H. Baker, Development and application of Halanay-type theory: Evolutionary differential and difference equations with time lag, J. Comput. Appl. Math., 234 (2010), 2663-2682.  doi: 10.1016/j.cam.2010.01.027.  Google Scholar

[4]

C. T. H. Baker and E. Buckwar, Exponential stability in $p-th$ mean of solutions, and of convergent Euler-type solutions, of stochastic delay differential equations, J. Comput. Appl. Math., 184 (2005), 404-427.  doi: 10.1016/j.cam.2005.01.018.  Google Scholar

[5]

D. Bresh-Pietri, J. Chauvin and N. Petit, Invoking Halanay inequality to conclude on closed-loop stability of a process with input-varying delay, Proceedings of the $10^{th}$ IFAC Workshop on Time-Delay Systems, Boston, USA, (2012), 266–271. Google Scholar

[6]

E. Fridman, Introduction to Time-Delay Systems: Analysis and Control, Birkhäuser, 2014. doi: 10.1007/978-3-319-09393-2.  Google Scholar

[7]

M. T. Grifa and P. Pepe, On stability analysis of discrete-time systems with constrained time-delays via nonlinear halanay-type inequality, IEEE Control Syst. Lett., 5 (2021), 869-874.  doi: 10.1109/LCSYS.2020.3007096.  Google Scholar

[8] A. Halanay, Differential Equations, Stability, Oscillations, Time Lags, Academic Press, New York, 1966.   Google Scholar
[9]

J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer Verlag, 1993. doi: 10.1007/978-1-4612-4342-7.  Google Scholar

[10]

J. K. Hale and S. M. Verduyn Lunel, Strong stabilization of neutral functional differential equations, IMA J. Math. Control Inform., 19 (2002), 5-23.  doi: 10.1093/imamci/19.1_and_2.5.  Google Scholar

[11]

L. V. HienV. N. Phat and H. Trinh, New generalized Halanay inequalities with applications to stability of nonlinear non-autonomous time-delay systems, Nonlinear Dynam., 82 (2015), 563-575.  doi: 10.1007/s11071-015-2176-0.  Google Scholar

[12]

M. Jankovic, Control Lyapunov-Razumikhin functions and robust stabilization of time delay systems, IEEE Trans. Automat. Control, 46 (2001), 1048-1060.  doi: 10.1109/9.935057.  Google Scholar

[13]

I. Karafyllis and Z.-P. Jiang, Stability and Stabilization of Nonlinear Systems, Springer, London, 2011. doi: 10.1007/978-0-85729-513-2.  Google Scholar

[14]

I. Karafyllis, M. Malisoff, F. Mazenc and P. Pepe, Stabilization of nonlinear delay systems: A tutorial on recent results, I. Karafyllis, M. Malisoff, F. Mazenc, P. Pepe (EDs), Recent Results on Nonlinear Delay Control Systems, In Honor of Miroslav Krstic, Series Advances in Delays and Dynamics, Springer, 4 (2015), 1–41. doi: 10.1007/978-3-319-18072-4_1.  Google Scholar

[15]

I. KarafyllisP. Pepe and Z.-P. Jiang, Global output stability for systems described by retarded functional differential equations: Lyapunov characterizations, Eur. J. Control, 14 (2008), 516-536.  doi: 10.3166/ejc.14.516-536.  Google Scholar

[16]

H. K. Khalil, Nonlinear Systems, Prentice Hall, International Edition, Third Edition, Upper Saddle River, New Jersey, 2000. Google Scholar

[17]

A. V. Kim, Functional Differential Equations, Application of I-Smooth Calculus, Kluwer Academic Publishers, Dordrecht, 1999. doi: 10.1007/978-94-017-1630-7.  Google Scholar

[18] N. N. Krasovskii, Stability of Motion, Stanford University Press, 1963.   Google Scholar
[19]

Y. LinE. D. Sontag and Y. Wang, A smooth converse Lyapunov theorem for robust stability, SIAM J. Control Optim., 34 (1996), 124-160.  doi: 10.1137/S0363012993259981.  Google Scholar

[20]

E. LizA. Ivanov and J. B. Ferreiro, Discrete Halanay-type inequalities and applications, Nonlinear Anal., 55 (2003), 669-678.  doi: 10.1016/j.na.2003.07.013.  Google Scholar

[21]

F. Mazenc and M. Malisoff, Extensions of Razumikhin's theorem and Lyapunov-Krasovskii functional constructions for time-varying systems with delay, Automatica J. IFAC, 78 (2017), 1-13.  doi: 10.1016/j.automatica.2016.12.005.  Google Scholar

[22]

A. Mironchenko and F. Wirth, Characterizations of input-to-state stability for infinite-dimensional systems, IEEE Trans. Automat. Control, 63 (2018), 1602-1617.  doi: 10.1109/tac.2017.2756341.  Google Scholar

[23]

S. Mohamad and K. Gopalsamy, Continuous and discrete Halanay-type Inequalities, Bull. Austral. Math. Soc., 61 (2000), 371-385.  doi: 10.1017/S0004972700022413.  Google Scholar

[24]

A. D. Myshkis, Razumikhin's method in the qualitative theory of processes with delay, J. Appl. Math. Stochastic Anal., 8 (1995), 233-247.  doi: 10.1155/S1048953395000219.  Google Scholar

[25]

P. Pepe, On Liapunov-Krasovskii functionals under Caratheodory conditions, Automatica J. IFAC, 43 (2007), 701-706.  doi: 10.1016/j.automatica.2006.10.024.  Google Scholar

[26]

P. Pepe, On control Lyapunov-Razumikhin functions, nonconstant delays, nonsmooth feedbacks, and nonlinear sampled-data stabilization, IEEE Trans. Automat. Control, 62 (2017), 5604-5619.  doi: 10.1109/TAC.2017.2689500.  Google Scholar

[27]

P. Pepe and E. Fridman, On global exponential stability preservation under sampling for globally Lipschitz time-delay systems, Automatica J. IFAC, 82 (2017), 295-300.  doi: 10.1016/j.automatica.2017.04.055.  Google Scholar

[28]

P. Pepe and Z.-P. Jiang, A Lyapunov-Krasovskii methodology for ISS and iISS of time-delay systems, Systems Control Lett., 55 (2006), 1006-1014.  doi: 10.1016/j.sysconle.2006.06.013.  Google Scholar

[29]

P. Pepe and I. Karafyllis, Converse Lyapunov-Krasovskii theorems for systems described by neutral functional differential equations in Hale's form, Internat. J. Control, 86 (2013), 232-243.  doi: 10.1080/00207179.2012.723137.  Google Scholar

[30]

P. PepeI. Karafyllis and Z.-P. Jiang, Lyapunov-Krasovskii characterization of the input-to-state stability for neutral systems in Hale's form, Systems Control Lett., 102 (2017), 48-56.  doi: 10.1016/j.sysconle.2017.01.008.  Google Scholar

[31]

B. S. Razumikhin, On stability of systems with delay, Prykl. Mat. Mekh., 20 (1956), 500–512, (in Russian).  Google Scholar

[32]

E. D. Sontag, Smooth stabilization implies coprime factorization, IEEE Trans. Automat. Control, 34 (1989), 435-443.  doi: 10.1109/9.28018.  Google Scholar

[33]

A. R. Teel, Connections between Razumikhin-type theorems and the ISS nonlinear small gain theorem, IEEE Trans. Automat. Control, 43 (1998), 960-964.  doi: 10.1109/9.701099.  Google Scholar

[34]

S. Udpin and P. Niamsup, New discrete type inequalities and global stability of nonlinear difference equations, Appl. Math. Lett., 22 (2009), 856-859.  doi: 10.1016/j.aml.2008.07.011.  Google Scholar

[35]

W. Wang, A generalized Halanay inequality for stability of nonlinear neutral functional differential equations, J. Inequal. Appl., (2010), Article ID 475019, 16 pp. doi: 10.1155/2010/475019.  Google Scholar

[36]

L. WenY. Yu and W. Wang, Generalized Halanay inequalities for dissipativity of Volterra functional differential equations, J. Math. Anal. Appl., 347 (2008), 169-178.  doi: 10.1016/j.jmaa.2008.05.007.  Google Scholar

[37]

L. Xu and D. He, A nonlinear nonautonomous delay differential inequality for dissipativity of Lotka-Volterra functional differential equations, Tbilisi Math. J., 7 (2014), 37-43.  doi: 10.2478/tmj-2014-0004.  Google Scholar

[38]

C. ZhangF. DengH. Mo and H. Ren, Dissipativity and stability of a class of nonlinear multiple time delay integro-differential equations, J. Nonlinear Sci. Appl., 12 (2019), 363-375.  doi: 10.22436/jnsa.012.06.03.  Google Scholar

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