Fundamental matrix solutions of piecewise smooth differential systems

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Abstract

We consider the fundamental matrix solution associated to piecewise smooth differential systems of Filippov type, in which the vector field varies discontinuously as solution trajectories reach one or more surfaces. We review the cases of transversal intersection and of sliding motion on one surface. We also consider the case when sliding motion takes place on the intersection of two or more surfaces. Numerical results are also given.

Section snippets

Introduction and background

Our purpose in this paper is to survey definitions and properties of the fundamental matrix solution associated to piecewise smooth differential equations. Many of the results we give are available in the literature, but are not all readily available. Moreover, some of the extensions we consider herein, such as when there is sliding motion on intersection of surfaces, appear new.

We study differential equations with discontinuous right-hand side, and more precisely equations in which the

Fundamental matrix solution: cross and/or slide on one surface

The fundamental matrix solution associated to the linearized system is a very useful tool in performing stability and bifurcation study of a smooth dynamical systems. It is natural to suspect that it should be a useful tool also for nonsmooth dynamical systems. In this and the next sections, we consider the fundamental matrix solutions for piecewise smooth systems. In this section, we look at the case in which we cross and/or slide on one surface. In the next section we also consider the case

Fundamental matrix solution: cross and/or slide on two surfaces

Now, suppose that the state space is split into four regions R1, R2, R3 and R4 by two intersecting hypersurfaces Σ1 and Σ2 which are defined by the scalar functions h1:RnR and h2:RnR, that is:R1={xRn|h1(x)<0,h2(x)<0},R2={xRn|h1(x)<0,h2(x)>0},R3={xRn|h1(x)>0,h2(x)>0},R4={xRn|h1(x)>0,h2(x)<0},(see Fig. 5). Consider the system with discontinuous right-hand side:x(t)=f(x(t))=f1(x),xR1,f2(x),xR2,f3(x),xR3,f4(x),xR4,with initial point x(0)=x0Rn. The functions h1(x) and h2(x) are assumed

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