A General Minimum Principle for End-point Control Problems

Abstract

ABSTRACT A very brood class of end-point control problems is formulated in general terms, and correspondingly general procedures in the calculus of variations are developed for their solution The generality of the treatment lies in a systematic application of the point of view of functional analysis. The state of the system being controlled is described by an abstract vector in a suitable linear space ; the evolution of the system state with time is described in terms of an operator on the state vector ; the measure of system performance to be maximized or minimized is taken as a functional of the final state. The linearized system operator and its adjoint with respect to an inner product in the state space play central roles in the variational arguments The variational methods are in the spirit of those leading to the familiar Pontrjagin maximum principle, and indeed this principle appears as the finite-dimensional case of the present results. These results however apply equally to infinite-dimensional cases such as systems with distributed parameters whose mathematical description is given in terms of partial rather than ordinary differential equations. The methods are illustrated here for a problem in heat exchanger control. General numerical procedures are discussed, but for distributed parameter systems the calculations required to solve practical problems will be/extrcmely_formidable.