Delay Switching across the Stability Boundary
DOI:
https://doi.org/10.51094/jxiv.44Keywords:
Delay, Feedback, Stochasticity, Asymptotic Stability, Stability Control, Switched SystemsAbstract
We propose a simple delay differential equation with a delay switching. In this model, the delay is a time-dependent variable taking two values across the stability boundary. With both stochastic and regular periodic switching of the delay, there are cases where the region of asymptotic stability is enhanced. We also show that this is in contrast to the analogous case of switching coefficient parameters in the equation. Also, the direction of switching across the stability boundary affects the stability.
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References
N. D. Hayes. Roots of the transcendental equation associated with a certain difference–differential equation. J. Lond. Math. Soc., 25:226–232, 1950.
R. Bellman & K. Cooke. Differential–Difference Equations. Academic Press, New York, 1963.
M. C. Mackey & L. Glass. Oscillation and chaos in physiological control systems. Science, 197:287–289, 1977.
G. Stepan. Retarded dynamical systems: Stability and characteristic functions. Wiley & Sons, New York, 1989.
A. Longtin & J. G. Milton. Insight into the transfer function, gain and oscillation onset for the pupil light reflex using delay-differential equations. Biol. Cybern., 61:51–58, 1989.
J. J. Collins & C. J. de Luca. Random walking during quiet standing. Phys. Rev. Lett., 73:764–767, 1994.
J. L. Cabrera & J. G. Milton. On-off intermittency in a human balancing task. Phys. Rev. Lett., 89:158702, 2002.
S. A. Campbell, R. Edwards, & P. van den Driessche. Delayed Coupling Between Two Neural Network Loops. SIAM J. Appl. Math., 65: 316–335, 2004.
T. Insperger, G. Stepan. Semi-Discretization for Time Delay Systems. Springer, 2011.
U. Kuchler & B. Mensch. Langevin’s stochastic differential equation extended by a time-delayed term. Stoch. Stoch. Rep., 40:23–42, 1992.
T. Ohira & J. G. Milton. Delayed random walks. Phys. Rev. E, 52:3277–3280, 1995.
T. Ohira &Y. Sato. Resonance with noise & delay. Phys. Rev. Lett., 82:2811–2815, 1999.
T. Ohira &T. Yamane. Delayed stochastic systems. Phys. Rev. E, 61:1247–1257, 2000.
T. D. Frank & P.J. Beek. Stationary solutions of linear stochastic delay differential equations: Applications to biological systems. Phys. Rev. E, 64: 021917, 2001.
L. Giuggioli & Z. Neu. Fokker-Planck representations of non-Markov Langevin equations: application
to delayed systems. Philosophical Transactions of the Royal Society A: Mathematical, Physical & Engineering Sciences, 377: 2018131, 2019.
T. Ohira. Stability Enhancement with Stochastic Delay Switching. Nagoya Repository, 0002001219, 2021.
T. Ohira. Enhancement of Stability with Delay Switching. In the proceedings of 7th International Symposium on BioComplexity, January 25-27, Beppu, Japan, 2022.
W. Michiels, V. Van Assche & S. -I. Niculescu. Stabilization of time-delay systems with a Controlled time-varying delay & applications, IEEE Transactions on Automatic Control, 50:493-504, 2005.
M. Sadeghpour, D. Breda & G. Orosz. Stability of Linear Continuous-Time Systems with Stochastically Switching Delays. IEEE Transactions on Automatic Contro, DOI 10.1109/TAC.2019.2904491 2019.
M. Wicks, P. Peleties ＆ R. DeCarlo. Switched Controller Synthesis for the Quadratic Stabilisation of a Pair of Unstable Linear Systems. European Journal of Control, 4:140-147, 1998.
M. S. Branicky. Multiple Lyapunov Functions & Other Analysis Tools for Switched & Hybrid Systems. IEEE Transactions on Automatic Control, 43:475-482, 1998.
G. Zhai, B. Hu, K. Yasuda & A. N. Michel. Stability analysis of switched systems with stable & unstable subsystems: an average dull time approach. International Journal of Systems Science, 32:1055-1061, 2001.
W. Xiang & J.Xiao. Stabilization of switched continuous-time systems with all modes unstable via dwell time switching. Automatica, 50:940-945, 2014
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Submitted: 2022-04-06 10:08:11 UTC
Published: 2022-04-12 01:59:36 UTC
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Toru Ohira
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