Lyapunov functions for infinite-dimensional systems. (English) Zbl 1040.93062

Summary: We study Lyapunov functions for infinite-dimensional dynamical systems governed by general maximal monotone operators. We obtain a characterization of Lyapunov pairs by means of the associated Hamilton-Jacobi partial differential equations, whose solutions are meant in the viscosity sense, as evolved in works of Tataru and Crandall-Lions. Our approach also leads to a new sufficient condition for Lyapunov pairs, generalizing a classical result of Pazy.


93D30 Lyapunov and storage functions
93C25 Control/observation systems in abstract spaces
93B28 Operator-theoretic methods
49L20 Dynamic programming in optimal control and differential games
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[1] Attouch, H.; Azé, D., Approximation and regularization of arbitrary functions in Hilbert spaces by the lasry – lions method, Ann. inst. H. Poincaré anal. non linéaire, 10, 289-312, (1993) · Zbl 0780.41021
[2] Ph. Bénilan, M. G. Crandall, and, A. Pazy, book in preparation.
[3] Brézis, H., Operateurs maximaux monotones et semi-groupes de contractions dans LES espaces de Hilbert, (1973), North-Holland Amsterdam · Zbl 0252.47055
[4] Crandall, M.G.; Lions, P.L., Hamilton – jacobi equations in infinite dimensions, part VI. nonlinear A and Tataru’s method refined, (), 51-89 · Zbl 0793.35118
[5] Deimling, K., Multivalued differential equations, (1992), Walter de Gruyter Berlin · Zbl 0760.34002
[6] Henry, D., Geometric theory of semilinear parabolic equations, (1981), Springer-Verlag Berlin/New York · Zbl 0456.35001
[7] Ishii, H., Viscosity solutions for a class of hamilton – jacobi equations in Hilbert spaces, J. funct. anal., 105, 301-341, (1992) · Zbl 0753.70014
[8] Kocan, M.; Soravia, P., A viscosity approach to infinite-dimensional hamilton – jacobi equations arising in optimal control with state constraints, SIAM J. control optim., 36, 1348-1375, (1998) · Zbl 0918.49025
[9] Kocan, M.; Soravia, P., Nonlinear, dissipative, infinite dimensional systems, (), 75-93 · Zbl 0946.49021
[10] Kocan, M.; Soravia, P., Differential games and nonlinear H∞ control in infinite dimensions, SIAM J. control optim., 39, 1296-1322, (2000) · Zbl 0986.93023
[11] Kocan, M.; Soravia, P.; Swiech, A., On differential games for infinite dimensional systems with nonlinear, unbounded operators, J. math. anal. appl., 211, 395-423, (1997) · Zbl 0884.90148
[12] Kocan, M.; Swiech, A., Perturbed optimization on product spaces, Nonlin. anal. theory methods appl., 26, 81-90, (1996) · Zbl 1018.49020
[13] Kocan, M.; Swiech, A., Second order unbounded parabolic equations in separated form, Studia math., 115, 291-310, (1995) · Zbl 0832.49017
[14] Pazy, A., The Lyapunov method for semigroups of nonlinear contractions in Banach spaces, J. anal. math., 40, 239-262, (1981) · Zbl 0507.47042
[15] Pazy, A., On the Lyapunov method for evolution equations governed by accretive operators, Research notes in mathematics, 68, (1982), Pitman London, p. 166-189 · Zbl 0491.34051
[16] Soravia, P., Stability of dynamical systems with competitive controls: the degenerate case, J. math. anal. appl., 191, 428-449, (1995) · Zbl 0821.93059
[17] Soravia, P., H∞ control of nonlinear systems: differential games and viscosity solutions, SIAM J. control optim., 34, 1071-1097, (1996) · Zbl 0926.93019
[18] Soravia, P., Optimality principles and representation formulas for viscosity solutions of hamilton – jacobi equations. I. equations of unbounded and degenerate control problems without uniqueness, Adv. differential equations, 4, 275-296, (1999) · Zbl 0955.49016
[19] Tataru, D., Viscosity solutions for hamilton – jacobi equations with unbounded nonlinear terms, J. math. anal. appl., 163, 345-392, (1992) · Zbl 0757.35034
[20] Tataru, D., Viscosity solutions for hamilton – jacobi equations with unbounded nonlinear term: A simplified approach, J. differential equations, 111, 123-146, (1994) · Zbl 0810.34060
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