Generalized Kalman-type rank conditions for oligopolies

Abstract

A generalized final state control model is introduced, in which the state transition relation depends on the current state and input as usual, and in addition it contains the input in the previous time period in the discrete case, or the derivative of the input function in the continuous case. The discrete case is discussed in detail. Sufficient and necessary conditions are given for the complete controllability of the final state and a system of linear algebraic equations is derived, the solutions of which provide the controlling input sequence. These results are straightforward generalizations of the corresponding Kalman-type rank conditions. The general methodology is illustrated in the case of the dynamic oligopoly game.