Hybrid dynamical systems vs. ordinary differential equations: Examples of a "pathological" behavior

E. Litsyn, The Weizmann Institute of Science, Rehovot, Israel
Yu. Nepomnyaschchikh, Perm State University,Perm,Russia
A. Ponosov, Institutt for matematiske fag,NLH, As, Norway

E. J. Qualitative Theory of Diff. Equ., No. 9. (2000), pp. 1-10.

Abstract: We investigate the controlled harmonic oscillator
\begin{equation}\label{eq3.1}
\ddot{\xi}+\xi=u,
\end{equation}
where an external force (the control function) $u$ depends on the coordinate $\xi$, only. It can be shown that no ordinary (even nonlinear) feedback controls of the form $u=f(\xi(t))$ can asymptotically stabilize the solutions of the system (\ref{eq3.1}). However, one is able to make the system (\ref{eq3.1}) asymptotically stable if one designs a special feedback control $u$ depending on $\xi(\cdot)$ which is called {\it a hybrid feedback control}. We demonstrate in the paper that the dynamics of a typical linear system of ordinary differential equations equipped with a linear hybrid feedback control possesses some irregular properties that dynamical systems without delay do not have. For example, solutions with different initial conditions may cross or even partly coincide. This proves that the hybrid dynamics cannot, in general, be described by a system of ordinary differential equations, neither linear, nor nonlinear, so that time-delays have to be incorporated into the system.

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