December  2012, 32(12): 4409-4427. doi: 10.3934/dcds.2012.32.4409

A metric proof of the converse Lyapunov theorem for semicontinuous multivalued dynamics

1. 

Dipartimento di Matematica, Università degli Studi di Roma La Sapienza, P.le Aldo Moro 2, 00185 Roma, Italy

2. 

Instituto Superior Técnico, Universidade Técnica de Lisboa, Departamento de Matemática, Av. Rovisco Pais, 1049-001 Lisboa

Received  January 2011 Revised  June 2012 Published  August 2012

We extend the metric proof of the converse Lyapunov Theorem, given in [13] for continuous multivalued dynamics, by means of tools issued from weak KAM theory, to the case where the set-valued vector field is just upper semicontinuous. This generality is justified especially in view of application to discontinuous ordinary differential equations. The more relevant new point is that we introduce, to compensate the lack of continuity, a family of perturbed dynamics, obtained through internal approximation of the original one, and perform some stability analysis of it.
Citation: Antonio Siconolfi, Gabriele Terrone. A metric proof of the converse Lyapunov theorem for semicontinuous multivalued dynamics. Discrete & Continuous Dynamical Systems - A, 2012, 32 (12) : 4409-4427. doi: 10.3934/dcds.2012.32.4409
References:
[1]

J. P. Aubin and A. Cellina, "Differential Inclusions,", Springer-Verlag, (1984). doi: 10.1007/978-3-642-69512-4. Google Scholar

[2]

M. Bardi and I. Capuzzo Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellmann Equations,", Birkhäuser, (1997). Google Scholar

[3]

E. N. Barron and R. Jensen, Lyapunov stability using minimum distance control,, Nonlinear Analysis, 43 (2001), 923. Google Scholar

[4]

A. Briani and A. Davini, Monge solution for discontinuous Hamiltonians,, ESAIM Control, 11 (2005), 229. doi: 10.1051/cocv:2005004. Google Scholar

[5]

G. Buttazzo, M. Giaquinta and S. Hildebrandt, "One-dimensional Variational Problems,", Oxford Lecture Series in Mathematics and its Applications, (1998). Google Scholar

[6]

F. Camilli and A. Siconolfi, Time-dependent measurable Hamilton-Jacobi equations,, Comm. Partial Differential Equations, 30 (2005), 813. Google Scholar

[7]

F. Clarke, "Optimization and Nonsmooth Analysis,", Wiley, (1983). Google Scholar

[8]

F. H. Clarke, Yu. S. Ledyaev and R. J. Stern, Asymptotic stability and smooth Lyapunov functions,, J. Differential Equations, 149 (1998), 69. Google Scholar

[9]

A. Fathi, Partitions of unity for countable covers,, Amer. Math. Monthly, 104 (1997), 720. Google Scholar

[10]

A. Fathi and A. Siconolfi, PDE aspects of Aubry-Mather theory for semiconvex Hamiltonians,, Calc. Var. Partial Differential Equations, 22 (2005), 185. Google Scholar

[11]

Y. Lin, E. D. Sontag and Y. Wang, A Smooth Converse Lyapunov Theorem for robust stability,, SIAM J. Control Optim., 34 (1996), 124. Google Scholar

[12]

A. Siconolfi, Metric Character of Hamilton-Jacobi Equations,, Trans. Am. Math. Soc., 355 (2003), 1987. doi: 10.1090/S0002-9947-03-03237-9. Google Scholar

[13]

A. Siconolfi and G. Terrone, A metric approach to the converse Lyapunov Theorem for continuous multivalued dynamics,, Nonlinearity, 20 (2007), 1077. Google Scholar

[14]

G. Terrone, "Stabilizzazione di Sistemi Controllati e Insieme di Aubry,", Master Thesis, (2004). Google Scholar

show all references

References:
[1]

J. P. Aubin and A. Cellina, "Differential Inclusions,", Springer-Verlag, (1984). doi: 10.1007/978-3-642-69512-4. Google Scholar

[2]

M. Bardi and I. Capuzzo Dolcetta, "Optimal Control and Viscosity Solutions of Hamilton-Jacobi-Bellmann Equations,", Birkhäuser, (1997). Google Scholar

[3]

E. N. Barron and R. Jensen, Lyapunov stability using minimum distance control,, Nonlinear Analysis, 43 (2001), 923. Google Scholar

[4]

A. Briani and A. Davini, Monge solution for discontinuous Hamiltonians,, ESAIM Control, 11 (2005), 229. doi: 10.1051/cocv:2005004. Google Scholar

[5]

G. Buttazzo, M. Giaquinta and S. Hildebrandt, "One-dimensional Variational Problems,", Oxford Lecture Series in Mathematics and its Applications, (1998). Google Scholar

[6]

F. Camilli and A. Siconolfi, Time-dependent measurable Hamilton-Jacobi equations,, Comm. Partial Differential Equations, 30 (2005), 813. Google Scholar

[7]

F. Clarke, "Optimization and Nonsmooth Analysis,", Wiley, (1983). Google Scholar

[8]

F. H. Clarke, Yu. S. Ledyaev and R. J. Stern, Asymptotic stability and smooth Lyapunov functions,, J. Differential Equations, 149 (1998), 69. Google Scholar

[9]

A. Fathi, Partitions of unity for countable covers,, Amer. Math. Monthly, 104 (1997), 720. Google Scholar

[10]

A. Fathi and A. Siconolfi, PDE aspects of Aubry-Mather theory for semiconvex Hamiltonians,, Calc. Var. Partial Differential Equations, 22 (2005), 185. Google Scholar

[11]

Y. Lin, E. D. Sontag and Y. Wang, A Smooth Converse Lyapunov Theorem for robust stability,, SIAM J. Control Optim., 34 (1996), 124. Google Scholar

[12]

A. Siconolfi, Metric Character of Hamilton-Jacobi Equations,, Trans. Am. Math. Soc., 355 (2003), 1987. doi: 10.1090/S0002-9947-03-03237-9. Google Scholar

[13]

A. Siconolfi and G. Terrone, A metric approach to the converse Lyapunov Theorem for continuous multivalued dynamics,, Nonlinearity, 20 (2007), 1077. Google Scholar

[14]

G. Terrone, "Stabilizzazione di Sistemi Controllati e Insieme di Aubry,", Master Thesis, (2004). Google Scholar

[1]

Michael Schönlein. Asymptotic stability and smooth Lyapunov functions for a class of abstract dynamical systems. Discrete & Continuous Dynamical Systems - A, 2017, 37 (7) : 4053-4069. doi: 10.3934/dcds.2017172

[2]

Andrea Davini, Maxime Zavidovique. Weak KAM theory for nonregular commuting Hamiltonians. Discrete & Continuous Dynamical Systems - B, 2013, 18 (1) : 57-94. doi: 10.3934/dcdsb.2013.18.57

[3]

Diogo Gomes, Levon Nurbekyan. An infinite-dimensional weak KAM theory via random variables. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6167-6185. doi: 10.3934/dcds.2016069

[4]

Xifeng Su, Lin Wang, Jun Yan. Weak KAM theory for HAMILTON-JACOBI equations depending on unknown functions. Discrete & Continuous Dynamical Systems - A, 2016, 36 (11) : 6487-6522. doi: 10.3934/dcds.2016080

[5]

Michele V. Bartuccelli, G. Gentile, Kyriakos V. Georgiou. Kam theory, Lindstedt series and the stability of the upside-down pendulum. Discrete & Continuous Dynamical Systems - A, 2003, 9 (2) : 413-426. doi: 10.3934/dcds.2003.9.413

[6]

Maxime Zavidovique. Existence of $C^{1,1}$ critical subsolutions in discrete weak KAM theory. Journal of Modern Dynamics, 2010, 4 (4) : 693-714. doi: 10.3934/jmd.2010.4.693

[7]

Volodymyr Pichkur. On practical stability of differential inclusions using Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2017, 22 (5) : 1977-1986. doi: 10.3934/dcdsb.2017116

[8]

Honglv Ma, Jin Zhang, Chengkui Zhong. Global existence and asymptotic behavior of global smooth solutions to the Kirchhoff equations with strong nonlinear damping. Discrete & Continuous Dynamical Systems - B, 2019, 24 (9) : 4721-4737. doi: 10.3934/dcdsb.2019027

[9]

Olga Bernardi, Matteo Dalla Riva. Analytic dependence on parameters for Evans' approximated Weak KAM solutions. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4625-4636. doi: 10.3934/dcds.2017199

[10]

Regina S. Burachik, Xiaoqi Yang. Asymptotic strong duality. Numerical Algebra, Control & Optimization, 2011, 1 (3) : 539-548. doi: 10.3934/naco.2011.1.539

[11]

Anatoli F. Ivanov, Musa A. Mammadov. Global asymptotic stability in a class of nonlinear differential delay equations. Conference Publications, 2011, 2011 (Special) : 727-736. doi: 10.3934/proc.2011.2011.727

[12]

Rinaldo M. Colombo, Graziano Guerra. Differential equations in metric spaces with applications. Discrete & Continuous Dynamical Systems - A, 2009, 23 (3) : 733-753. doi: 10.3934/dcds.2009.23.733

[13]

Michael Hochman. Lectures on dynamics, fractal geometry, and metric number theory. Journal of Modern Dynamics, 2014, 8 (3&4) : 437-497. doi: 10.3934/jmd.2014.8.437

[14]

Masaki Hibino. Gevrey asymptotic theory for singular first order linear partial differential equations of nilpotent type — Part I —. Communications on Pure & Applied Analysis, 2003, 2 (2) : 211-231. doi: 10.3934/cpaa.2003.2.211

[15]

Sergio Grillo, Jerrold E. Marsden, Sujit Nair. Lyapunov constraints and global asymptotic stabilization. Journal of Geometric Mechanics, 2011, 3 (2) : 145-196. doi: 10.3934/jgm.2011.3.145

[16]

Roland Gunesch, Anatole Katok. Construction of weakly mixing diffeomorphisms preserving measurable Riemannian metric and smooth measure. Discrete & Continuous Dynamical Systems - A, 2000, 6 (1) : 61-88. doi: 10.3934/dcds.2000.6.61

[17]

Thanh-Anh Nguyen, Dinh-Ke Tran, Nhu-Quan Nguyen. Weak stability for integro-differential inclusions of diffusion-wave type involving infinite delays. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3637-3654. doi: 10.3934/dcdsb.2016114

[18]

Thomas Lorenz. Mutational inclusions: Differential inclusions in metric spaces. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 629-654. doi: 10.3934/dcdsb.2010.14.629

[19]

Luis Barreira, César Silva. Lyapunov exponents for continuous transformations and dimension theory. Discrete & Continuous Dynamical Systems - A, 2005, 13 (2) : 469-490. doi: 10.3934/dcds.2005.13.469

[20]

Carlos Arnoldo Morales, M. J. Pacifico. Lyapunov stability of $\omega$-limit sets. Discrete & Continuous Dynamical Systems - A, 2002, 8 (3) : 671-674. doi: 10.3934/dcds.2002.8.671

2018 Impact Factor: 1.143

Metrics

  • PDF downloads (6)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]