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In this paper, we review some basic results on linear quadratic differential games. We consider both the cooperative and non-cooperative case. For the non-cooperative game we consider the open-loop and (linear) feedback information structure. Furthermore, the effect of adding uncertainty is considered. The overview is based on [9]. Readers interested in detailed proofs and additional results are referred to this book.

International audience; In this paper we study a linear programming problem with a linear perturbation introduced through a parameter $\epsilon > 0$. We identify and analyze an unusual asymptotic phenomenon in such a linear program. Namely, discontinuous limiting behavior of the optimal objective function value of such a linear program may occur even when the rank of the coefficient matrix of the constraints is unchanged by the perturbation. We show that, under mild conditions, this phenomenon is a result of the classical Slater constraint qualification being violated at the limit and propose an iterative, constraint augmentation approach for resolving this problem.

This paper is focusing on finding smooth approximate solutio ns of the HJB inequality that corresponds to the infinite horizon optimal control problem with discount- ing. We establish that such approximate solutions exist (un der a simple controllability type condition) and that they can be used for construction of near optimal controls. We also show that these approximate solutions of the HJB inequa lity can be found by solving certain semi-infinite linear programming problems and we pr opose an algorithm for the solution of the latter. We discuss a numerical solution of a n on-trivial optimal control problem obtained with the help of a software implementation of the new algorithm. Refereed/Peer-reviewed

A dynamic Cournot duopoly game, characterized by firms with bounded rationality, is represented by a discrete-time dynamical system of the plane. Conditions ensuring the local stability of a Nash equilibrium, under a local (or myopic) adjustment process, are given, and the influence of marginal costs and speeds of adjustment of the two firms on stability is studied. The stability loss of the Nash equilibrium, as some parameter of the model is varied, gives rise to more complex (periodic or chaotic) attractors. The main result of this paper is given by the exact determination of the basin of attraction of the locally stable Nash equilibrium (or other more complex bounded attractors around it), and the study of the global bifurcations that change the structure of the basin from a simple to a very complex one, with consequent loss of predictability, as some parameters of the model are allowed to vary. These bifurcations are studied by the use of critical curves, a relatively new and powerful method for the study of noninvertible two-dimensional maps