Well-Posedness of the Complementarity Class of Hybrid Systems

Abstract

One of the most fundamental properties of any class of dynamical systems is the study of well-posedness, i.e. the existence and uniqueness of a particular type of solution trajectories given an initial state. In case of interaction between continuous dynamics and discrete transitions this issue becomes highly non-trivial. In this survey an overview is given of the well-posedness results for complementarity systems, which form a class of hybrid systems described by the interconnection of differential equations and a specific combination of inequalities and Boolean expressions as appearing in the linear complementarity problem of mathematical programming.

The Complementarity Class of Hybrid Dynamical Systems

“…For complementarity systems one may develop several solution concepts (see [47]), which may be similar to the notion of an execution for hybrid automata [54], or to the solution concept for differential inclusions as used for differential equations with discontinuous right-hand sides [38]. A solution concept of the first type can for instance be formulated as follows.…”

“…In [47] an overview is given of available well-posedness results for complementarity systems. Here we just recall a few interesting results just to give you the flavor of them.…”

“…For further work on well-posedness for complementarity systems, see [47] and the references therein.…”

“…, which models a mass subject to Coulomb friction, cannot be written as an EVI as in (47). Indeed this would imply that the set .…”

Well Posedness of New Optimization Problems with Variational Inequality Constraints

“…Starting with the work of Tykhonov [1] for unconstrained optimization problems, various types of well posedness for variational problems have been considered (see, for instance, Levitin-Polyak well posedness [2][3][4][5], extended well posedness [6][7][8][9][10][11][12][13][14]), L-well posedness [15], α-well posedness [16,17]). Moreover, the concept of well posedness can be useful to study some related problems, such as variational inequality and fixed point problems [18][19][20][21][22], hemivariational inequality problems [23], complementary problems [24], equilibrium problems [25,26], Nash equilibrium problems [27] and variational inclusion problems [28]. Recently, the study of well posedness for vector variational inequalities and the associated optimization problems was formulated by Jayswal and Shalini [29].…”