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Section: Scientific Foundations

Dynamic non-regular systems

Dynamical systems (we limit ourselves to finite-dimensional ones) are said to be non-regular whenever some nonsmoothness of the state arises. This nonsmoothness may have various roots: for example some outer impulse, entailing so-called differential equations with measure . An important class of such systems can be described by the complementarity system

Im1 $\mfenced o={  \mtable{...}$(1)

where $ \bottom$ denotes orthogonality; u is a control input. Now (1 ) can be viewed from different angles.

Thus, the 2nd and 3rd lines in (1 ) define the modes of the hybrid systems, as well as the conditions under which transitions occur from one mode to another. The 4th line defines how transitions are performed by the state x . There are several other formalisms which are quite related to complementarity. A tutorial-survey paper has been published [3] , whose aim is to introduce the dynamics of complementarity systems and the main available results in the fields of mathematical analysis, analysis for control (controllability, observability, stability), and feedback control.


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