The hybrid matching of Hurwitz systems

In this paper we study planar hybrid systems composed by two stable linear systems, defined by Hurwitz matrices, in addition with a jump that can be a piecewise linear, a polynomial or an analytic function. We provide an explicit analytic necessary and sufficient condition for this class of hybrid systems to be asymptotically stable. We also prove the existence of limit cycles in this class of hybrid systems. Our results can be seen as generalizations of results already obtained in the literature. This was possible due to an embedding of piecewise smooth vector fields in a hybrid structure.

💡 Research Summary

The paper investigates a class of planar hybrid dynamical systems that consist of two stable linear subsystems—each defined by a Hurwitz matrix—together with a discontinuous “jump” map that may be piecewise‑linear, polynomial, or analytic. The authors first motivate the problem by recalling the Markus‑Yamabe conjecture, which provides a sufficient (but not necessary) condition for global asymptotic stability (GAS) in two dimensions, and they point out that many piecewise‑smooth systems fall outside the scope of that condition.

A hybrid system is formally defined as a quadruple (X=(X^{+},X^{-};\Sigma_{\rho},\varphi_{\rho})). The switching manifold (\Sigma_{\rho}) is the “broken line” composed of the negative‑x axis ((x\le0, y=0)) and the ray (y=\rho x) for (x>0). The crossing condition (\langle X^{\pm}(q),\nabla h_{\rho}(q)\rangle>0) for all (q\in\Sigma_{\rho}\setminus{0}) guarantees that trajectories always cross (\Sigma_{\rho}) in the same direction, eliminating sliding or sticking phenomena. The reset map (\varphi_{\rho}) is given by
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