On the Discrete-Time Minimum Principle in Multiple-Mode Systems

This paper presents a robust version of the discrete-time minimum principle for uncertain systems. The uncertainty enters the dynamics of the system by an unknown value that belongs to a finite set. Each element of such a set represents a possible dynamic realization of the discrete-time trajectory. Following a mathematical programming approach, we reformulate the problem to obtain general necessary conditions that allow finding the worst-case optimality in the form of a weighted sum. These conditions are then applied to the time-varying linear affine quadratic problem with multiple modes and a general cost, which includes tracking signals in both: the state and the control variables. We illustrate our result with an application to a production-inventory control problem with uncertain dynamics.