August  2019, 24(8): 3653-3666. doi: 10.3934/dcdsb.2018309

Invariance principle in the singular perturbations limit

Department of Mathematics, The Weizmann Institute of Science, Rehovot 7610001, Israel

 

The paper is dedicated to my friend Peter Kloeden

Received  March 2018 Revised  July 2018 Published  November 2018

We examine the invariance principle in the stability theory of differential equations, within a general singularly perturbed system. The limit dynamics of such a system is depicted by the evolution of a Young measure whose values are invariant measures of the fast equation. We establish an invariance principle for the limit dynamics, and examine the relations, at times subtle, with the singularly perturbed system itself.

Citation: Zvi Artstein. Invariance principle in the singular perturbations limit. Discrete & Continuous Dynamical Systems - B, 2019, 24 (8) : 3653-3666. doi: 10.3934/dcdsb.2018309
References:
[1]

S. M. AfonsoE. M. BonottoM. Federson and Š. Schwabik, Discontinuous local semiflows for Kurzweil equations leading to LaSalle's invariance principle for differential systems with impulses at variable times, J. Differential Equations, 250 (2011), 2969-3001.  doi: 10.1016/j.jde.2011.01.019.  Google Scholar

[2]

J. AlvarezI. Orlov and L. Acho, An invariance principle for discontinuous dynamic systems with applications to a Coulomb friction oscillator, J. Dynamic Systems, Measurements and Control, 122 (2000), 687-699.   Google Scholar

[3]

Z. Artstein, On singularly perturbed ordinary differential equations with measure-valued limits, Mathematica Bohemica, 127 (2002), 139-152.   Google Scholar

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Z. Artstein, Asymptotic stability of singularly perturbed differential equations, J. Differential Equations, 262 (2017), 1603-1616.  doi: 10.1016/j.jde.2016.10.023.  Google Scholar

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Z. ArtsteinI. G. KevrekidisM. Slemrod and E. S. Titi, Slow observables of singularly perturbed differential equations, Nonlinearity, 20 (2007), 2463-2481.  doi: 10.1088/0951-7715/20/11/001.  Google Scholar

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Z. Artstein and M. Slemrod, The singular perturbation limit of an elastic structure in a rapidly flowing nearly invicid fluid, Quarterly Applied Mathematics, 59 (2001), 543-555.  doi: 10.1090/qam/1848534.  Google Scholar

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Z. Artstein and A. Vigodner, Singularly perturbed ordinary differential equations with dynamic limits, Proceedings Royal Society Edinburgh, 126 (1996), 541-569.  doi: 10.1017/S0308210500022903.  Google Scholar

[8]

A. Bacciotti and L. Mazzi, An invariance principle for nonlinear switch systems, Systems & Control letters, 54 (2005), 1109-1119.  doi: 10.1016/j.sysconle.2005.04.003.  Google Scholar

[9]

E. J. Balder, Lectures on Young measure theory and its applications to economics, Rend. Istit. Mat. Univ. Trieste, 31 (2000), supplemento 1, 1–69.  Google Scholar

[10]

I. Barkana, Can stability analysis be really simplified? (From Lyapunov to the new theorem of stability - Revisiting Lyapunov, Barbalat, LaSalle and all that), Mathematics in Engineering, Science and Aerospace, 8 (2017), 171-199.   Google Scholar

[11]

P. Billingsley, Convergence of Probability Measures, 2nd Ed. Wiley, New York, 1999. doi: 10.1002/9780470316962.  Google Scholar

[12]

E. M. Bonotto, LaSalle's theorem in impulsive dynamical systems, Nonlinear Analysis, 71 (2009), 2291-2297.  doi: 10.1016/j.na.2009.01.062.  Google Scholar

[13]

C. I. Byrnes and C. F. Martin, An integral invariance principle for nonlinear systems, IEEE transaction on Automatic Control, 40 (1995), 983-994.  doi: 10.1109/9.388676.  Google Scholar

[14]

G. ChenJ. Zhou and S. Čelikovský, On LaSalle's invariance principle and its application to robust synchronization of general vector Lienard equation, IEEE Transactions on Automatic Control, 50 (2005), 869-874.  doi: 10.1109/TAC.2005.849250.  Google Scholar

[15]

J. P. Hespanha, Uniform stability of switched linear systems: Extension of LaSalle's invariance principle, IEEE Transactions on Automatic Control, 49 (2004), 470-482.  doi: 10.1109/TAC.2004.825641.  Google Scholar

[16]

F. Hoppensteadt, Asymptotic stability in singular perturbation problems, J. Differential Equations, 4 (1968), 350-358.  doi: 10.1016/0022-0396(68)90021-1.  Google Scholar

[17]

A. KalitineB. Iggidr and R. Outbib, Semidefinite Lyapunov functions stability and stabilization, Mathematics Control Signals and Systems, 9 (1996), 95-106.  doi: 10.1007/BF01211748.  Google Scholar

[18]

P. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proceedings American Mathematical Society, 144 (2015), 259-268.  doi: 10.1090/proc/12735.  Google Scholar

[19]

N. Kryloff and N. Bogoliuboff, La théorie générale de la mesure dans son application à l'etude des systèmes dynamiques de la mécanique non linéaire, Annals of Mathematics, 38 (1937), 65-113.  doi: 10.2307/1968511.  Google Scholar

[20]

J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics 25, SIAM Publications, Philadelphia, 1976.  Google Scholar

[21]

V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, 1960.  Google Scholar

[22]

R. E. O'Malley Jr., Historical Developments in Singular Perturbations, Springer, New York, 2014. doi: 10.1007/978-3-319-11924-3.  Google Scholar

[23]

P. Pedregal, Parameterized Measures and Variational Principles, Birkhäuser Verlag, Basel, 1997. doi: 10.1007/978-3-0348-8886-8.  Google Scholar

[24]

C. Pötzsche, Chain rule and invariance principle on measure chains, J. Computational and Applied Mathematics, 141 (2002), 249-254.  doi: 10.1016/S0377-0427(01)00450-2.  Google Scholar

[25]

M. TaoH. Owhadi and J. E. Marsden, Nonintrusive and structure preserving multiscale integration of stiff ODEs, SDEs, and Hamiltonian systems with hidden slow dynamics via flow averaging, Multiscale Modeling Simulations, 8 (2010), 1269-1324.  doi: 10.1137/090771648.  Google Scholar

[26]

A. N. Tikhonov, A. B. Vasiléva and A. G. Sveshnikov, Differential Equations, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-642-82175-2.  Google Scholar

[27]

M. Valadier, A course on Young measures, Rend. Istit. Mat. Univ. Trieste, 26 (1994), supp., 349–394.  Google Scholar

[28]

F. Verhulst, Methods and Applications of Singular Perturbations, Texts in Applied Mathematics 50, Springer, New York, 2005. doi: 10.1007/0-387-28313-7.  Google Scholar

show all references

References:
[1]

S. M. AfonsoE. M. BonottoM. Federson and Š. Schwabik, Discontinuous local semiflows for Kurzweil equations leading to LaSalle's invariance principle for differential systems with impulses at variable times, J. Differential Equations, 250 (2011), 2969-3001.  doi: 10.1016/j.jde.2011.01.019.  Google Scholar

[2]

J. AlvarezI. Orlov and L. Acho, An invariance principle for discontinuous dynamic systems with applications to a Coulomb friction oscillator, J. Dynamic Systems, Measurements and Control, 122 (2000), 687-699.   Google Scholar

[3]

Z. Artstein, On singularly perturbed ordinary differential equations with measure-valued limits, Mathematica Bohemica, 127 (2002), 139-152.   Google Scholar

[4]

Z. Artstein, Asymptotic stability of singularly perturbed differential equations, J. Differential Equations, 262 (2017), 1603-1616.  doi: 10.1016/j.jde.2016.10.023.  Google Scholar

[5]

Z. ArtsteinI. G. KevrekidisM. Slemrod and E. S. Titi, Slow observables of singularly perturbed differential equations, Nonlinearity, 20 (2007), 2463-2481.  doi: 10.1088/0951-7715/20/11/001.  Google Scholar

[6]

Z. Artstein and M. Slemrod, The singular perturbation limit of an elastic structure in a rapidly flowing nearly invicid fluid, Quarterly Applied Mathematics, 59 (2001), 543-555.  doi: 10.1090/qam/1848534.  Google Scholar

[7]

Z. Artstein and A. Vigodner, Singularly perturbed ordinary differential equations with dynamic limits, Proceedings Royal Society Edinburgh, 126 (1996), 541-569.  doi: 10.1017/S0308210500022903.  Google Scholar

[8]

A. Bacciotti and L. Mazzi, An invariance principle for nonlinear switch systems, Systems & Control letters, 54 (2005), 1109-1119.  doi: 10.1016/j.sysconle.2005.04.003.  Google Scholar

[9]

E. J. Balder, Lectures on Young measure theory and its applications to economics, Rend. Istit. Mat. Univ. Trieste, 31 (2000), supplemento 1, 1–69.  Google Scholar

[10]

I. Barkana, Can stability analysis be really simplified? (From Lyapunov to the new theorem of stability - Revisiting Lyapunov, Barbalat, LaSalle and all that), Mathematics in Engineering, Science and Aerospace, 8 (2017), 171-199.   Google Scholar

[11]

P. Billingsley, Convergence of Probability Measures, 2nd Ed. Wiley, New York, 1999. doi: 10.1002/9780470316962.  Google Scholar

[12]

E. M. Bonotto, LaSalle's theorem in impulsive dynamical systems, Nonlinear Analysis, 71 (2009), 2291-2297.  doi: 10.1016/j.na.2009.01.062.  Google Scholar

[13]

C. I. Byrnes and C. F. Martin, An integral invariance principle for nonlinear systems, IEEE transaction on Automatic Control, 40 (1995), 983-994.  doi: 10.1109/9.388676.  Google Scholar

[14]

G. ChenJ. Zhou and S. Čelikovský, On LaSalle's invariance principle and its application to robust synchronization of general vector Lienard equation, IEEE Transactions on Automatic Control, 50 (2005), 869-874.  doi: 10.1109/TAC.2005.849250.  Google Scholar

[15]

J. P. Hespanha, Uniform stability of switched linear systems: Extension of LaSalle's invariance principle, IEEE Transactions on Automatic Control, 49 (2004), 470-482.  doi: 10.1109/TAC.2004.825641.  Google Scholar

[16]

F. Hoppensteadt, Asymptotic stability in singular perturbation problems, J. Differential Equations, 4 (1968), 350-358.  doi: 10.1016/0022-0396(68)90021-1.  Google Scholar

[17]

A. KalitineB. Iggidr and R. Outbib, Semidefinite Lyapunov functions stability and stabilization, Mathematics Control Signals and Systems, 9 (1996), 95-106.  doi: 10.1007/BF01211748.  Google Scholar

[18]

P. Kloeden and T. Lorenz, Construction of nonautonomous forward attractors, Proceedings American Mathematical Society, 144 (2015), 259-268.  doi: 10.1090/proc/12735.  Google Scholar

[19]

N. Kryloff and N. Bogoliuboff, La théorie générale de la mesure dans son application à l'etude des systèmes dynamiques de la mécanique non linéaire, Annals of Mathematics, 38 (1937), 65-113.  doi: 10.2307/1968511.  Google Scholar

[20]

J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics 25, SIAM Publications, Philadelphia, 1976.  Google Scholar

[21]

V. V. Nemytskii and V. V. Stepanov, Qualitative Theory of Differential Equations, Princeton University Press, Princeton, 1960.  Google Scholar

[22]

R. E. O'Malley Jr., Historical Developments in Singular Perturbations, Springer, New York, 2014. doi: 10.1007/978-3-319-11924-3.  Google Scholar

[23]

P. Pedregal, Parameterized Measures and Variational Principles, Birkhäuser Verlag, Basel, 1997. doi: 10.1007/978-3-0348-8886-8.  Google Scholar

[24]

C. Pötzsche, Chain rule and invariance principle on measure chains, J. Computational and Applied Mathematics, 141 (2002), 249-254.  doi: 10.1016/S0377-0427(01)00450-2.  Google Scholar

[25]

M. TaoH. Owhadi and J. E. Marsden, Nonintrusive and structure preserving multiscale integration of stiff ODEs, SDEs, and Hamiltonian systems with hidden slow dynamics via flow averaging, Multiscale Modeling Simulations, 8 (2010), 1269-1324.  doi: 10.1137/090771648.  Google Scholar

[26]

A. N. Tikhonov, A. B. Vasiléva and A. G. Sveshnikov, Differential Equations, Springer-Verlag, Berlin, 1985. doi: 10.1007/978-3-642-82175-2.  Google Scholar

[27]

M. Valadier, A course on Young measures, Rend. Istit. Mat. Univ. Trieste, 26 (1994), supp., 349–394.  Google Scholar

[28]

F. Verhulst, Methods and Applications of Singular Perturbations, Texts in Applied Mathematics 50, Springer, New York, 2005. doi: 10.1007/0-387-28313-7.  Google Scholar

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