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

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