Section: Research Program
Dynamic nonregular systems
mechanical systems, impacts, unilateral constraints, complementarity, modeling, analysis, simulation, control, convex analysis
Dynamical systems (we limit ourselves to finitedimensional ones) are said to be nonregular whenever some nonsmoothness of the state arises. This nonsmoothness may have various roots: for example some outer impulse, entailing socalled differential equations with measure. An important class of such systems can be described by the complementarity system
$\left\{\begin{array}{c}\dot{x}=f(x,u,\lambda )\phantom{\rule{0.166667em}{0ex}},\hfill \\ 0\le y\perp \lambda \ge 0\phantom{\rule{0.166667em}{0ex}},\hfill \\ g(y,\lambda ,x,u,t)=0\phantom{\rule{0.166667em}{0ex}},\hfill \\ \text{reinitialization}\phantom{\rule{4.pt}{0ex}}\text{law}\phantom{\rule{4.pt}{0ex}}\text{of}\phantom{\rule{4.pt}{0ex}}\text{the}\phantom{\rule{4.pt}{0ex}}\text{state}\phantom{\rule{4.pt}{0ex}}x(\xb7)\phantom{\rule{4.pt}{0ex}}\text{,}\hfill \end{array}\right.$  (1) 
where $\perp $ denotes orthogonality; $u$ is a control input. Now (1 ) can be viewed from different angles.

Hybrid systems: it is in fact natural to consider that (1 ) corresponds to different models, depending whether ${y}_{i}=0$ or ${y}_{i}>0$ (${y}_{i}$ being a component of the vector $y$). In some cases, passing from one mode to the other implies a jump in the state $x$; then the continuous dynamics in (1 ) may contain distributions.

Differential inclusions: $0\le y\perp \lambda \ge 0$ is equivalent to $\lambda \in {\mathrm{N}}_{K}\left(y\right)$, where $K$ is the nonnegative orthant and ${\mathrm{N}}_{K}\left(y\right)$ denotes the normal cone to $K$ at $y$. Then it is not difficult to reformulate (1 ) as a differential inclusion.

Dynamic variational inequalities: such a formalism reads as $\langle \dot{x}\left(t\right)+F(x\left(t\right),t),vx\left(t\right)\rangle \ge 0$ for all $v\in K$ and $x\left(t\right)\in K$, where $K$ is a nonempty closed convex set. When $K$ is a polyhedron, then this can also be written as a complementarity system as in (1 ).
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 tutorialsurvey paper has been published [5] , 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.