Section: Application Domains

Introduction

Many systems (either actual or abstract) can be represented by (1 ). Some typical examples are:

• Mechanical systems with unilateral constraints and dry friction (the biped robot is a typical example, hair and fiber dynamics is another example).

• Electrical circuits with ideal diodes and/or transistors Mos.

• Optimal control with constraints on the state, closed loop of a system controlled by an mpc algorithm (model predictive control), discontinuous feedback controllers like sliding-mode control, etc.

This class of models is not too large (to allow thorough studies), yet rich enough to include many applications. This goes in contrast to a study of general hybrid systems. Note for example that (1 ) is a “continuous” hybrid system, in that the continuous variables $x$ and $u$ prevail in the evolution (there is no discrete control to commute from a mode to the other: only the input $u$ can be used). The main tools for the analysis and simulation of such dynamical systems come from Convex Analysis, Non-smooth Analysis, Complementarity Theory (we make a strong use of complementarity problems solvers for numerical simulation), Variational Inequalities. Let us cite some specific applications.