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On parametric generalized equations

CIBULKA, R. On parametric generalized equations. Vídeň, Rakousko, 2015.
Jazyk publikace: eng
Anglický název: On parametric generalized equations
Rok vydání: 2015
Autoři: Ing. Radek Cibulka Ph.D. ,
Abstrakt EN: By a parametric generalized equation we mean a problem to find a Lipschitz continuous function $z: [a,b] \to \mathbb{R}^n$ such that p(t) \in f(z(t)) + F (z(t)) \quad \mbox{whenever} \quad t\in [a,b], $$ where a single-valued function $f:\mathbb{R}^n \to \mathbb{R}^n$ is continuously differentiable and a~set-valued mapping $F: \mathbb{R}^n \rightrightarrows \mathbb{R}^n$ has closed graph. Hence $z(\cdot)$ is a selection for the solution mapping $$ S: [a,b] \ni t \longmapsto S(t):= \{z \in \mathbb{R}^n: \ p(t) \in f(z) + F(z) \}, $$ that is, $z(t) \in S(t)$ whenever $t \in [a,b]$. First, we present sufficient conditions for the existence of such a selection. Second, we investigate a modification of the Euler-Newton continuation method for tracking solution trajectories developed by Dontchev, Krastanov, Rockafellar, and Veliov. This allows us to deal with a non-differentiable input signal $p(\cdot)$. Finally, implementing this method (in Matlab) we provide a simulation of the behavior of some basic non-regular electrical circuits, that is, the circuits where various types of diodes are present.
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