Optimal Control Problems and Variational Methods

Abstract

In the previous discussions on controllability, we have been concerned with the possibility of bringing a state (vector) from an initial position to an assigned position, namely the target, in a finite amount of time. In practice, many factors must be brought into consideration. For instance, the state may not be allowed to travel outside a certain region and the control (function) has certain limited capacity. Another important consideration is that there are certain quantities that we wish to optimize. Usually the quantities to be minimized are time, fuel, energy, cost, etc. and those to be maximized include speed, efficiency, profit, etc. The problem under consideration is, therefore, to optimize a quantity, called a functional, which usually depends on the control function, the state vector, and the time parameter, and at the same time, to satisfy certain constraints, namely: the control equation of the state-space description, a region the state vector is confined to, and an admissible collection of functions to which the control function belongs.