Strong and weak convexity for linear differential games

Zero-sum linear differential games on a fixed time segment with geometrically constrained controls are considered. We study the efficiency of an algorithm to calculate the game value function, which is based on Krasovskii's concept of the stable bridge for differential games. The second order estimates of the error of the discrete-time algorithm for games with strongly convex penalty functions and for games with strongly convex restriction sets are obtained. Convex analysis and, in particular, the theory of convex conjugate functions are important means of the investigation.