June  2022, 27(6): 3345-3374. doi: 10.3934/dcdsb.2021188

Matrix measures, stability and contraction theory for dynamical systems on time scales

1. 

University of Salerno, Department of Information and Electrical Engineering and Applied Mathematics, Salerno, Italy

2. 

University of Passau, Faculty of Computer Science and Mathematics, Passau, Germany

* Corresponding author: Giovanni Russo

Received  August 2020 Revised  May 2021 Published  June 2022 Early access  July 2021

This paper is concerned with the study of the stability of dynamical systems evolving on time scales. We first formalize the notion of matrix measures on time scales, prove some of their key properties and make use of this notion to study both linear and nonlinear dynamical systems on time scales. Specifically, we start with considering linear time-varying systems and, for these, we prove a time scale analogous of an upper bound due to Coppel. We make use of this upper bound to give stability and input-to-state stability conditions for linear time-varying systems. Then, we consider nonlinear time-varying dynamical systems on time scales and establish a sufficient condition for the convergence of the solutions. Finally, after linking our results to the existence of a Lyapunov function, we make use of our approach to study certain epidemic dynamics and complex networks. For the former, we give a sufficient condition on the parameters of a SIQR model on time scales ensuring that its solutions converge to the disease-free solution. For the latter, we first give a sufficient condition for pinning synchronization of complex time scale networks and then use this condition to study certain collective opinion dynamics. The theoretical results are complemented with simulations.

Citation: Giovanni Russo, Fabian Wirth. Matrix measures, stability and contraction theory for dynamical systems on time scales. Discrete and Continuous Dynamical Systems - B, 2022, 27 (6) : 3345-3374. doi: 10.3934/dcdsb.2021188
References:
[1]

Z. Aminzare and E. D. Sontag, Contraction methods for nonlinear systems: A brief introduction and some open problems, in 53rd IEEE Conference on Decision and Control, (2014), 3835–3847.

[2]

S. BabenkoM. DefoortM. Djemai and S. Nicaise, On the consensus tracking investigation for multi-agent systems on time scale via matrix-valued Lyapunov functions, Automatica J. IFAC, 97 (2018), 316-326.  doi: 10.1016/j.automatica.2018.08.003.

[3]

Z. Bartosiewicz and E. Pawłuszewicz, Realizations of linear control systems on time scales, Control Cybernet., 35 (2006), 769-786. 

[4]

Z. Bartosiewicz and E. Piotrowska, Lyapunov functions in stability of nonlinear systems on time scales, J. Difference Equ. Appl., 17 (2011), 309-325.  doi: 10.1080/10236190902932734.

[5]

M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, 2001. doi: 10.1007/978-1-4612-0201-1.

[6]

M. BotnerY. ZaraiM. Margaliot and L. Grüne, On approximating contractive systems, IEEE Trans. Automat. Control, 62 (2017), 6451-6457.  doi: 10.1109/TAC.2017.2651649.

[7]

L. Bourdin and E. Trélat, Pontryagin maximum principle for finite dimensional nonlinear optimal control problems on time scales, SIAM J. Control Optim., 51 (2013), 3781-3813.  doi: 10.1137/130912219.

[8]

Z. Cao, W. Feng, X. Wen, L. Zu and M. Cheng, Dynamics of a stochastic SIQR epidemic model with standard incidence, Phys. A, 527 (2019), 121180, 12 pp. doi: 10.1016/j.physa.2019.121180.

[9]

Q. Cheng and J. Cao, Synchronization of complex dynamical networks with discrete time delays on time scales, Neurocomputing, 151 (2015), 729-736.  doi: 10.1016/j.neucom.2014.10.033.

[10]

S. K. Choi, Y. Cui and N. Koo, Variationally stable dynamic systems on time scales, Adv. Difference Equ., 2012 (2012), 17pp. doi: 10.1186/1687-1847-2012-129.

[11]

S. Coogan and M. Arcak, A note on norm-based Lyapunov functions via contraction analysis, 2013, Available at https://arXiv.org/abs/1308.0586.

[12]

W. A. Coppel, Dichotomies in Stability Theory, vol. 629, Springer, Berlin, 1978.

[13]

J. J. DaCunha, Stability for time varying linear dynamic systems on time scales, J. Comput. Appl. Math., 176 (2005), 381-410.  doi: 10.1016/j.cam.2004.07.026.

[14]

J. J. DaCunha, Transition matrix and generalized matrix exponential via the Peano-Baker series, J. Difference Equ. Appl., 11 (2005), 1245-1264.  doi: 10.1080/10236190500272798.

[15]

G. Dahlquist, Stability and Error Bounds in the Numerical Integration of Ordinary Differential Equations, Inaugural dissertation, University of Stockholm, Almqvist & Wiksells Boktryckeri AB, Uppsala, 1958.

[16]

F. Della Rossa, D. Salzano, A. Di Meglio, F. De Lellis, M. Coraggio, C. Calabrese, A. Guarino, R. Cardona-Rivera, P. De Lellis, D. Liuzza, F. Lo Iudice, G. Russo and M. di Bernardo, A network model of italy shows that intermittent regional strategies can alleviate the covid-19 epidemic, Nature Communications, 11 (2020), 5106.

[17]

C. A. Desoer and H. Haneda, The measure of a matrix as a tool to analyze computer algorithms for circuit analysis, IEEE Trans. Circuit Theory CT, 19 (1972), 480-486. 

[18]

C. A. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties, Society for Industrial and Applied Mathematics, 2009. doi: 10.1137/1.9780898719055.ch1.

[19]

O. DiekmannJ. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.

[20]

J. Eckhardt and G. Teschl, On the connection between the Hilger and Radon-Nikodym derivatives, J. Math. Anal. Appl., 385 (2012), 1184-1189.  doi: 10.1016/j.jmaa.2011.07.041.

[21]

G. FanG. Russo and P. C. Bressloff, Node-to-node and node-to-medium synchronization in quorum sensing networks affected by state-dependent noise, SIAM J. Appl. Dyn. Syst., 18 (2019), 1934-1953.  doi: 10.1137/19M1249515.

[22]

R. A. C. Ferreira and C. M. Silva, A nonautonomous epidemic model on time scales, J. Difference Equ. Appl., 24 (2018), 1295-1317.  doi: 10.1080/10236198.2018.1479400.

[23]

B. R. Flay, Catastrophe theory in social psychology: Some applications to attitudes and social behavior, Behavioral Science, 23 (1978), 335-350.  doi: 10.1002/bs.3830230404.

[24]

N. E. Friedkin and F. Bullo, How truth wins in opinion dynamics along issue sequences, Proceedings of the National Academy of Sciences, 114 (2017), 11380-11385.  doi: 10.1073/pnas.1710603114.

[25]

T. Gard and J. Hoffacker, Asymptotic behavior of natural growth on time scales, Dynam. Systems Appl., 12 (2003), 131-147. 

[26]

M. GattoE. BertuzzoL. MariS. MiccoliL. CarraroR. Casagrandi and A. Rinaldo, Spread and dynamics of the COVID-19 epidemic in italy: Effects of emergency containment measures, Proceedings of the National Academy of Sciences, 117 (2020), 10484-10491.  doi: 10.1073/pnas.2004978117.

[27]

J. Ghaderi and R. Srikant, Opinion dynamics in social networks with stubborn agents: Equilibrium and convergence rate, Automatica J. IFAC, 50 (2014), 3209-3215.  doi: 10.1016/j.automatica.2014.10.034.

[28]

G. S. Guseinov, Integration on time scales, J. Math. Anal. Appl., 285 (2003), 107-127.  doi: 10.1016/S0022-247X(03)00361-5.

[29]

J. A. P. Heesterbeek, A brief history of $\mathcal{R}_0$ and a recipe for its calculation, Acta Biotheoretica, 50 (2002), 189-204. 

[30]

J. M. HeffernanR. J. Smith and L. M. Wahl, Perspectives on the basic reproductive ratio, Journal of The Royal Society Interface, 2 (2005), 281-293.  doi: 10.1098/rsif.2005.0042.

[31]

S. Hilger, Analysis on measure chains–a unified approach to continuous and discrete calculus, Results Math., 18 (1990), 18-56.  doi: 10.1007/BF03323153.

[32]

D. Hinrichsen and A. J. Pritchard, Mathematical Systems Theory I, Springer-Verlag, Berlin, 2005. doi: 10.1007/b137541.

[33]

S. Jafarpour, P. Cisneros-Velarde and F. Bullo, Weak and semi-contraction for network systems and diffusively-coupled oscillators, IEEE Transactions on Automatic Control, (2021), 1–1.

[34]

P. JiaA. MirTabatabaeiN. E. Friedkin and F. Bullo, Opinion dynamics and the evolution of social power in influence networks, SIAM Rev., 57 (2015), 367-397.  doi: 10.1137/130913250.

[35]

B. Kaymakçalan, Lyapunov stability theory for dynamic systems on time scales, J. Appl. Math. Stochastic Anal., 5 (1992), 275-282.  doi: 10.1155/S1048953392000224.

[36]

B. Kaymakçalan, Existence and comparison results for dynamic systems on a time scale, J. Math. Anal. Appl., 172 (1993), 243-255.  doi: 10.1006/jmaa.1993.1021.

[37]

V. Lakshmikantham, S. Sivasundaram and B. Kaymakcaln, Dynamic Systems on Measure Chains, Kluwer Academic, 1996. doi: 10.1007/978-1-4757-2449-3.

[38]

X. Liu and K. Zhang, Impulsive Systems on Hybrid Time Domains, Springer-Verlag, 2019. doi: 10.1007/978-3-030-06212-5.

[39]

X. Liu and K. Zhang, Synchronization of linear dynamical networks on time scales: Pinning control via delayed impulses, Automatica J. IFAC, 72 (2016), 147-152.  doi: 10.1016/j.automatica.2016.06.001.

[40]

W. Lohmiller and J.-J. E. Slotine, On contraction analysis for non-linear systems, Automatica J. IFAC, 34 (1998), 683-696.  doi: 10.1016/S0005-1098(98)00019-3.

[41]

S. M. Lozinskii, Error estimate for numerical integration of ordinary differential equations. Ⅰ, Izv. Vtss. Uchebn. Zaved Matematika, 5 (1959), 222-222. 

[42]

X. LuX. Zhang and Q. Liu, Finite-time synchronization of nonlinear complex dynamical networks on time scales via pinning impulsive control, Neurocomputing, 275 (2018), 2104-2110. 

[43]

A. A. Martynyuk, Stability Theory for Dynamic Equations on Time Scales, Birkhäuser, Basel, 2016. doi: 10.1007/978-3-319-42213-8.

[44]

M. Mobilia, Does a single zealot affect an infinite group of voters?, Phys. Rev. Lett., 91 (2003), 028701. doi: 10.1103/PhysRevLett.91.028701.

[45]

J. Monteil and G. Russo, On the design of nonlinear distributed control protocols for platooning systems, IEEE Control Syst. Lett., 1 (2017), 140-145.  doi: 10.1109/LCSYS.2017.2710907.

[46]

J. Monteil and G. Russo, On the coexistence of human-driven and automated vehicles within platoon systems, in 2019 18th European Control Conference (ECC), (2019), 3173–3178. doi: 10.23919/ECC.2019.8796304.

[47]

J. MonteilG. Russo and R. Shorten, On $\mathcal{L}_\infty$ string stability of nonlinear bidirectional asymmetric heterogeneous platoon systems, Automatica J. IFAC, 105 (2019), 198-205.  doi: 10.1016/j.automatica.2019.03.025.

[48]

A. Ogulenko, Asymptotical properties of social network dynamics on time scales, J. Comput. Appl. Math., 319 (2017), 413-422.  doi: 10.1016/j.cam.2017.01.031.

[49]

M. Pedersen and M. Meneghini, Quantifying undetected covid19 cases and effects of containment measures in Italy, available at: https://www.researchgate.net/publication/339915690_Quantifying_undetected_COVID-19_cases_and_effects_of_containment_measures_in_Italy.

[50]

M. Porfiri and M. di Bernardo, Criteria for global pinning-controllability of complex networks, Automatica J. IFAC, 44 (2008), 3100-3106.  doi: 10.1016/j.automatica.2008.05.006.

[51]

T. Potston and I. Stewart, Catastrophe Theory and its Applications, Dover Publications, Inc., Mineola, NY, 1996.

[52]

C. PötzscheS. Siegmund and F. Wirth, A spectral characterization of exponential stability for linear time-invariant systems on time scales, Discrete Contin. Dyn. Syst., 9 (2003), 1223-1241.  doi: 10.3934/dcds.2003.9.1223.

[53]

D. R. PoulsenJ. M. Davis and I. A. Gravagne, Optimal control on stochastic time scales, IFAC-PapersOnLine, 50 (2017), 14861-14866.  doi: 10.1016/j.ifacol.2017.08.2518.

[54]

K. Prem, Y. Liu, T. Russell, A. Kucharski, R. Eggo, N. Davies, M. Jit and P. Klepac, The effect of control strategies to reduce social mixing on outcomes of the COVID-19 epidemic in Wuhan, China: A modelling study, The Lancet.

[55]

W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, New York, NY, 1976.

[56]

G. Russo and F. Wirth, Coppel's inequality for linear systems on time scales, in 2019 IEEE 58th Conference on Decision and Control (CDC), (2019), 5187–5192.

[57]

G. Russo, M. di Bernardo and E. D. Sontag, Global entrainment of transcriptional systems to periodic inputs, PLoS Comput. Biol., 6 (2010), e1000739, 26 pp. doi: 10.1371/journal.pcbi.1000739.

[58]

F. Z. TaousserM. Defoort and M. Djemai, Stability analysis of a class of switched linear systems on non-uniform time domains, Systems Control Lett., 74 (2014), 24-31.  doi: 10.1016/j.sysconle.2014.09.012.

[59]

Y. Tian and L. Wang, Opinion dynamics in social networks with stubborn agents: An issue-based perspective, Automatica J. IFAC, 96 (2018), 213-223.  doi: 10.1016/j.automatica.2018.06.041.

[60]

M. Vidyasagar, Nonlinear systems analysis (2nd Ed.), Prentice-Hall, Englewood Cliffs, NJ, 1993.

[61]

X. Wang and H. Su, Pinning control of complex networked systems: A decade after and beyond, Annual Reviews in Control, 38 (2014), 103-111. 

[62]

D. Watts and S. Strogatz, Collective dynamics of small-world networks, Nature, 393 (1998), 440-442. 

[63]

S. Xie, G. Russo and R. Middleton, Scalability in nonlinear network systems affected by delays and disturbances, IEEE Transactions on Control of Network Systems, (2021), 1–1.

[64]

Y. D. Zhong and N. E. Leonard, A continuous threshold model of cascade dynamics, in 2019 IEEE 58th Conference on Decision and Control (CDC), (2019), 1704–1709.

show all references

References:
[1]

Z. Aminzare and E. D. Sontag, Contraction methods for nonlinear systems: A brief introduction and some open problems, in 53rd IEEE Conference on Decision and Control, (2014), 3835–3847.

[2]

S. BabenkoM. DefoortM. Djemai and S. Nicaise, On the consensus tracking investigation for multi-agent systems on time scale via matrix-valued Lyapunov functions, Automatica J. IFAC, 97 (2018), 316-326.  doi: 10.1016/j.automatica.2018.08.003.

[3]

Z. Bartosiewicz and E. Pawłuszewicz, Realizations of linear control systems on time scales, Control Cybernet., 35 (2006), 769-786. 

[4]

Z. Bartosiewicz and E. Piotrowska, Lyapunov functions in stability of nonlinear systems on time scales, J. Difference Equ. Appl., 17 (2011), 309-325.  doi: 10.1080/10236190902932734.

[5]

M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birkhäuser, 2001. doi: 10.1007/978-1-4612-0201-1.

[6]

M. BotnerY. ZaraiM. Margaliot and L. Grüne, On approximating contractive systems, IEEE Trans. Automat. Control, 62 (2017), 6451-6457.  doi: 10.1109/TAC.2017.2651649.

[7]

L. Bourdin and E. Trélat, Pontryagin maximum principle for finite dimensional nonlinear optimal control problems on time scales, SIAM J. Control Optim., 51 (2013), 3781-3813.  doi: 10.1137/130912219.

[8]

Z. Cao, W. Feng, X. Wen, L. Zu and M. Cheng, Dynamics of a stochastic SIQR epidemic model with standard incidence, Phys. A, 527 (2019), 121180, 12 pp. doi: 10.1016/j.physa.2019.121180.

[9]

Q. Cheng and J. Cao, Synchronization of complex dynamical networks with discrete time delays on time scales, Neurocomputing, 151 (2015), 729-736.  doi: 10.1016/j.neucom.2014.10.033.

[10]

S. K. Choi, Y. Cui and N. Koo, Variationally stable dynamic systems on time scales, Adv. Difference Equ., 2012 (2012), 17pp. doi: 10.1186/1687-1847-2012-129.

[11]

S. Coogan and M. Arcak, A note on norm-based Lyapunov functions via contraction analysis, 2013, Available at https://arXiv.org/abs/1308.0586.

[12]

W. A. Coppel, Dichotomies in Stability Theory, vol. 629, Springer, Berlin, 1978.

[13]

J. J. DaCunha, Stability for time varying linear dynamic systems on time scales, J. Comput. Appl. Math., 176 (2005), 381-410.  doi: 10.1016/j.cam.2004.07.026.

[14]

J. J. DaCunha, Transition matrix and generalized matrix exponential via the Peano-Baker series, J. Difference Equ. Appl., 11 (2005), 1245-1264.  doi: 10.1080/10236190500272798.

[15]

G. Dahlquist, Stability and Error Bounds in the Numerical Integration of Ordinary Differential Equations, Inaugural dissertation, University of Stockholm, Almqvist & Wiksells Boktryckeri AB, Uppsala, 1958.

[16]

F. Della Rossa, D. Salzano, A. Di Meglio, F. De Lellis, M. Coraggio, C. Calabrese, A. Guarino, R. Cardona-Rivera, P. De Lellis, D. Liuzza, F. Lo Iudice, G. Russo and M. di Bernardo, A network model of italy shows that intermittent regional strategies can alleviate the covid-19 epidemic, Nature Communications, 11 (2020), 5106.

[17]

C. A. Desoer and H. Haneda, The measure of a matrix as a tool to analyze computer algorithms for circuit analysis, IEEE Trans. Circuit Theory CT, 19 (1972), 480-486. 

[18]

C. A. Desoer and M. Vidyasagar, Feedback Systems: Input-Output Properties, Society for Industrial and Applied Mathematics, 2009. doi: 10.1137/1.9780898719055.ch1.

[19]

O. DiekmannJ. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.  doi: 10.1007/BF00178324.

[20]

J. Eckhardt and G. Teschl, On the connection between the Hilger and Radon-Nikodym derivatives, J. Math. Anal. Appl., 385 (2012), 1184-1189.  doi: 10.1016/j.jmaa.2011.07.041.

[21]

G. FanG. Russo and P. C. Bressloff, Node-to-node and node-to-medium synchronization in quorum sensing networks affected by state-dependent noise, SIAM J. Appl. Dyn. Syst., 18 (2019), 1934-1953.  doi: 10.1137/19M1249515.

[22]

R. A. C. Ferreira and C. M. Silva, A nonautonomous epidemic model on time scales, J. Difference Equ. Appl., 24 (2018), 1295-1317.  doi: 10.1080/10236198.2018.1479400.

[23]

B. R. Flay, Catastrophe theory in social psychology: Some applications to attitudes and social behavior, Behavioral Science, 23 (1978), 335-350.  doi: 10.1002/bs.3830230404.

[24]

N. E. Friedkin and F. Bullo, How truth wins in opinion dynamics along issue sequences, Proceedings of the National Academy of Sciences, 114 (2017), 11380-11385.  doi: 10.1073/pnas.1710603114.

[25]

T. Gard and J. Hoffacker, Asymptotic behavior of natural growth on time scales, Dynam. Systems Appl., 12 (2003), 131-147. 

[26]

M. GattoE. BertuzzoL. MariS. MiccoliL. CarraroR. Casagrandi and A. Rinaldo, Spread and dynamics of the COVID-19 epidemic in italy: Effects of emergency containment measures, Proceedings of the National Academy of Sciences, 117 (2020), 10484-10491.  doi: 10.1073/pnas.2004978117.

[27]

J. Ghaderi and R. Srikant, Opinion dynamics in social networks with stubborn agents: Equilibrium and convergence rate, Automatica J. IFAC, 50 (2014), 3209-3215.  doi: 10.1016/j.automatica.2014.10.034.

[28]

G. S. Guseinov, Integration on time scales, J. Math. Anal. Appl., 285 (2003), 107-127.  doi: 10.1016/S0022-247X(03)00361-5.

[29]

J. A. P. Heesterbeek, A brief history of $\mathcal{R}_0$ and a recipe for its calculation, Acta Biotheoretica, 50 (2002), 189-204. 

[30]

J. M. HeffernanR. J. Smith and L. M. Wahl, Perspectives on the basic reproductive ratio, Journal of The Royal Society Interface, 2 (2005), 281-293.  doi: 10.1098/rsif.2005.0042.

[31]

S. Hilger, Analysis on measure chains–a unified approach to continuous and discrete calculus, Results Math., 18 (1990), 18-56.  doi: 10.1007/BF03323153.

[32]

D. Hinrichsen and A. J. Pritchard, Mathematical Systems Theory I, Springer-Verlag, Berlin, 2005. doi: 10.1007/b137541.

[33]

S. Jafarpour, P. Cisneros-Velarde and F. Bullo, Weak and semi-contraction for network systems and diffusively-coupled oscillators, IEEE Transactions on Automatic Control, (2021), 1–1.

[34]

P. JiaA. MirTabatabaeiN. E. Friedkin and F. Bullo, Opinion dynamics and the evolution of social power in influence networks, SIAM Rev., 57 (2015), 367-397.  doi: 10.1137/130913250.

[35]

B. Kaymakçalan, Lyapunov stability theory for dynamic systems on time scales, J. Appl. Math. Stochastic Anal., 5 (1992), 275-282.  doi: 10.1155/S1048953392000224.

[36]

B. Kaymakçalan, Existence and comparison results for dynamic systems on a time scale, J. Math. Anal. Appl., 172 (1993), 243-255.  doi: 10.1006/jmaa.1993.1021.

[37]

V. Lakshmikantham, S. Sivasundaram and B. Kaymakcaln, Dynamic Systems on Measure Chains, Kluwer Academic, 1996. doi: 10.1007/978-1-4757-2449-3.

[38]

X. Liu and K. Zhang, Impulsive Systems on Hybrid Time Domains, Springer-Verlag, 2019. doi: 10.1007/978-3-030-06212-5.

[39]

X. Liu and K. Zhang, Synchronization of linear dynamical networks on time scales: Pinning control via delayed impulses, Automatica J. IFAC, 72 (2016), 147-152.  doi: 10.1016/j.automatica.2016.06.001.

[40]

W. Lohmiller and J.-J. E. Slotine, On contraction analysis for non-linear systems, Automatica J. IFAC, 34 (1998), 683-696.  doi: 10.1016/S0005-1098(98)00019-3.

[41]

S. M. Lozinskii, Error estimate for numerical integration of ordinary differential equations. Ⅰ, Izv. Vtss. Uchebn. Zaved Matematika, 5 (1959), 222-222. 

[42]

X. LuX. Zhang and Q. Liu, Finite-time synchronization of nonlinear complex dynamical networks on time scales via pinning impulsive control, Neurocomputing, 275 (2018), 2104-2110. 

[43]

A. A. Martynyuk, Stability Theory for Dynamic Equations on Time Scales, Birkhäuser, Basel, 2016. doi: 10.1007/978-3-319-42213-8.

[44]

M. Mobilia, Does a single zealot affect an infinite group of voters?, Phys. Rev. Lett., 91 (2003), 028701. doi: 10.1103/PhysRevLett.91.028701.

[45]

J. Monteil and G. Russo, On the design of nonlinear distributed control protocols for platooning systems, IEEE Control Syst. Lett., 1 (2017), 140-145.  doi: 10.1109/LCSYS.2017.2710907.

[46]

J. Monteil and G. Russo, On the coexistence of human-driven and automated vehicles within platoon systems, in 2019 18th European Control Conference (ECC), (2019), 3173–3178. doi: 10.23919/ECC.2019.8796304.

[47]

J. MonteilG. Russo and R. Shorten, On $\mathcal{L}_\infty$ string stability of nonlinear bidirectional asymmetric heterogeneous platoon systems, Automatica J. IFAC, 105 (2019), 198-205.  doi: 10.1016/j.automatica.2019.03.025.

[48]

A. Ogulenko, Asymptotical properties of social network dynamics on time scales, J. Comput. Appl. Math., 319 (2017), 413-422.  doi: 10.1016/j.cam.2017.01.031.

[49]

M. Pedersen and M. Meneghini, Quantifying undetected covid19 cases and effects of containment measures in Italy, available at: https://www.researchgate.net/publication/339915690_Quantifying_undetected_COVID-19_cases_and_effects_of_containment_measures_in_Italy.

[50]

M. Porfiri and M. di Bernardo, Criteria for global pinning-controllability of complex networks, Automatica J. IFAC, 44 (2008), 3100-3106.  doi: 10.1016/j.automatica.2008.05.006.

[51]

T. Potston and I. Stewart, Catastrophe Theory and its Applications, Dover Publications, Inc., Mineola, NY, 1996.

[52]

C. PötzscheS. Siegmund and F. Wirth, A spectral characterization of exponential stability for linear time-invariant systems on time scales, Discrete Contin. Dyn. Syst., 9 (2003), 1223-1241.  doi: 10.3934/dcds.2003.9.1223.

[53]

D. R. PoulsenJ. M. Davis and I. A. Gravagne, Optimal control on stochastic time scales, IFAC-PapersOnLine, 50 (2017), 14861-14866.  doi: 10.1016/j.ifacol.2017.08.2518.

[54]

K. Prem, Y. Liu, T. Russell, A. Kucharski, R. Eggo, N. Davies, M. Jit and P. Klepac, The effect of control strategies to reduce social mixing on outcomes of the COVID-19 epidemic in Wuhan, China: A modelling study, The Lancet.

[55]

W. Rudin, Principles of Mathematical Analysis, McGraw-Hill, New York, NY, 1976.

[56]

G. Russo and F. Wirth, Coppel's inequality for linear systems on time scales, in 2019 IEEE 58th Conference on Decision and Control (CDC), (2019), 5187–5192.

[57]

G. Russo, M. di Bernardo and E. D. Sontag, Global entrainment of transcriptional systems to periodic inputs, PLoS Comput. Biol., 6 (2010), e1000739, 26 pp. doi: 10.1371/journal.pcbi.1000739.

[58]

F. Z. TaousserM. Defoort and M. Djemai, Stability analysis of a class of switched linear systems on non-uniform time domains, Systems Control Lett., 74 (2014), 24-31.  doi: 10.1016/j.sysconle.2014.09.012.

[59]

Y. Tian and L. Wang, Opinion dynamics in social networks with stubborn agents: An issue-based perspective, Automatica J. IFAC, 96 (2018), 213-223.  doi: 10.1016/j.automatica.2018.06.041.

[60]

M. Vidyasagar, Nonlinear systems analysis (2nd Ed.), Prentice-Hall, Englewood Cliffs, NJ, 1993.

[61]

X. Wang and H. Su, Pinning control of complex networked systems: A decade after and beyond, Annual Reviews in Control, 38 (2014), 103-111. 

[62]

D. Watts and S. Strogatz, Collective dynamics of small-world networks, Nature, 393 (1998), 440-442. 

[63]

S. Xie, G. Russo and R. Middleton, Scalability in nonlinear network systems affected by delays and disturbances, IEEE Transactions on Control of Network Systems, (2021), 1–1.

[64]

Y. D. Zhong and N. E. Leonard, A continuous threshold model of cascade dynamics, in 2019 IEEE 58th Conference on Decision and Control (CDC), (2019), 1704–1709.

Figure 1.  Time evolution of (26) with the representative set of parameters of Section 8. Top panel: dynamics evolving on the time scale $ \mathbb{P}_{a,b} $, with $ a = 1 $ and $ b = 0.24 $. Both conditions (C1) and (C2) are satisfied. Bottom panel: dynamics evolving, with the same parameters, on the discrete time scale of Section 8 with $ c = 0.24 $ so that both (C1) and (C2) are satisfied. Code at: https://github.com/GIOVRUSSO/Control-Group-Code
Figure 2.  Time behavior of (26) when $ {\mathbb T}\equiv {\mathbb R} $. The parameters are taken from [49]. In the simulation, we used $ \beta = 0.0373/N $ (i.e. the population is in lock-down), $ k_d = 1 $, $ k_{\Lambda} = N $ and initial conditions $ [0.25 N, 0.25 N, 0.25N, 0.25N] $. The code is available at: https://github.com/GIOVRUSSO/Control-Group-Code
Figure 3.  Top panel: graph of the small world network considered in Section 11. The number of nodes is 100 and the nodes pinned by the stubborn agent are highlighted in red in the figure (colors online). In total, 49 nodes were pinned. The network was built following the Watts-Strogatz model [62] and by setting the mean node degree to $ 2 $ and the rewiring probability to $ 0.7 $. Bottom panel: time evolution for the network (the time evolution for $ x_r(t) $ is highlighted with a dashed black line). Code for the simulations at: https://github.com/GIOVRUSSO/Control-Group-Code
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