Differential Games and Nonconvexities: A Conjugate Points Approach

Emilio Barucci and Pier Luigi Zezza
University of Florence
Barucci@stat.ds.unifi.it

Abstract

We study differential games relaxing the classical assumptions allowing the optimality of the candidate solution obtained by means of the Pontryagin Maximum Principle, i.e. convexity in the control and in the state. We are able to prove the existence of a Nash Equilibrium strategy under nonconvexities in the state, provided that the Legendre condition is satisfied. The analysis is developed by means of conjugate points. We provide a complete analysis for the existence of a saddle point solution in zero-sum linear-quadratic differential games. The existence of the solution is verified by means of the Riccati equation. Economic applications to dynamic duopoly pricing and to capital accumulation games are provided.