Analysis of min-max systems

The paper will study extensions of known results of max-plus (or min-plus) systems to systems where the underlying algebra consists of the three operations max, min and addition simultaneously (such systems are called min-max systems). Such systems are nonlinear in both the max-plus and the min-plus algebra. The notion of a ‘pulsative’ circuit is introduced, which characterizes the speed of a stationary solution of the system. In general, the cycle mean of a pulsative circuit is neither maximal (as is the case for max-plus systems) nor minimal (as is the case for min-plus systems). Subject to certain conditions, the solutions of min-max systems show a periodic behaviour, which is directly related to the pulsative circuits. Solutions starting from arbitrary initial conditions converge in a finite number of steps to such a periodic behaviour. Game-theoretic notions are helpful for the construction of pulsative circuits. Both analytic and graph-theoretic reasoning is used in the deviation of the various results.