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  August 2019 Early access  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 and 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.

[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. 

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[11]

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

[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.

[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.

[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.

[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.

[16]

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

[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.

[18]

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

[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.

[20]

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

[21]

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

[22]

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

[23]

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

[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.

[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.

[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.

[27]

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

[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.

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.

[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. 

[3]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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. 

[11]

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

[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.

[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.

[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.

[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.

[16]

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

[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.

[18]

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

[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.

[20]

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

[21]

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

[22]

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

[23]

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

[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.

[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.

[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.

[27]

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

[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.

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