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Uniqueness of the asymptotic behaviour of autonomous and non‐autonomous, switched and non‐switched, linear and non‐linear systems of dimension 2
Author(s) -
Azzouz Abdennasser,
Hasler Martin
Publication year - 1988
Publication title -
international journal of circuit theory and applications
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.364
H-Index - 52
eISSN - 1097-007X
pISSN - 0098-9886
DOI - 10.1002/cta.4490160206
Subject(s) - mathematics , linear system , exponential stability , uniqueness , dimension (graph theory) , differentiable function , classification of discontinuities , generalization , piecewise linear function , connection (principal bundle) , linear differential equation , piecewise , stability (learning theory) , differential equation , mathematical analysis , control theory (sociology) , nonlinear system , pure mathematics , computer science , physics , geometry , control (management) , quantum mechanics , machine learning , artificial intelligence
An explicit criterion is given which guarantees that the solutions of a system of two first‐order differential equations have a unique asymptotic behaviour as t'→ + ∞. the equations are allowed to be time‐dependent, with isolate discontinuities (switching), and non‐linear. In particular, continuously differentiable and piecewise linear systems, as far as the dependence on the state variables is concerned, are considered. the particular case of a linear system switched back and forth between two 2 × 2 matrices is treated in detail, with an outline of an algorithm to decide on the converence to zero of the solutions. The case of switching between N matrices is a straightforward generalization that is, however, not covered in this paper. It is shown that the problem of unique asymptotic behaviour of piecewise linear systems can always be treated by considering such switched linear systems. Furthermore, an example shows that even some continuously differentiable systems may be reduced to this case. Finally, the connection with the work of Brayton and Tong on the stability of differential equations, also using sets of matrices, is given.