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Lyapunov functions for positive linear evolution problems
Author(s) -
Stephan H.
Publication year - 2005
Publication title -
zamm ‐ journal of applied mathematics and mechanics / zeitschrift für angewandte mathematik und mechanik
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.449
H-Index - 51
eISSN - 1521-4001
pISSN - 0044-2267
DOI - 10.1002/zamm.200410229
Subject(s) - mathematics , lyapunov function , monotone polygon , monotonic function , lyapunov equation , degenerate energy levels , mathematical optimization , mathematical analysis , nonlinear system , physics , geometry , quantum mechanics
We rigorously investigate the time monotonicity of Lyapunov functions for general positive linear evolution problems, including degenerate problems. This can be done by considering the problem in the convex set of probability measures and finding a general inequality for such Radon measures and Markov operators. For linear evolution problems (with discrete or continuous time), the existence of time monotone Lyapunov functions is not a consequence of any physical properties, but of the positivity and norm conservation of the equation. In some special cases the structure of such equations is given. Moreover, we describe completely the case of time constant Lyapunov functions – a property of deterministic dynamical systems.