Groupe d’études et de recherche en analyse des décisions

G-81-08

On the Global Asymptotic Stability of Equilibrium Solutions for Open-Loop Differential Games

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This paper is concerned with the extension to infinite-horizon open-loop differential games of the sufficient conditions for global-asymptotic-stability (CAS) of optimal solutions already obtained by Cass-Shell, Rockafellar and Brock in the realm of optimal control theory.

First a definition of open-loop equilibria for infinite horizon DG is proposed by extending the concept of overtaking optimality. The necessary conditions for an equilibrium can be cast in a "Pseudo-Hamiltonian" form under some regularity assumptions. A sufficient condition for GAS of bounded equilibrium solutions is thus obtained by considering a vector Lyapunov function in the (x,p1,p2) space, where x is the state varialbe and pj is the co-state variable associated with player j. The consideration of a vector rather than scalar-Lyapunov function permits a substantial simplification of the sufficient conditions when compared with previous results.

The particular case of DG with separation between state and controls and with separated state equations for all players (i.e., where the interaction only occurs in the pay-off) are then considered. For DF which feature both kinds of separation, the sufficient conditions for GAS are particularly simple.

Two applications to duapoly models are considered.

, 29 pages