On the application of dynamic programming to a class of implicit variational problems
A problem of this nature occurs when we wish to have the system in some specified state x0(T) at time T, without caring how the system gets there. This is usually an idealization, in the sense that a more realistic problem will involve a combination of a criterion of the type appearing in (2) together with some measure of the value of the final state. As has been shown in some recent publications, see [1], where further references may be found, a variety of problems of this nature arising in economic and engineering control processes may be solved computationally by combining the theory of dynamic programming with modern digital computers. In recent years, problems of less explicit nature have become more frequent. Thus, for example, what is called the "bang-bang" control problem requires that y be chosen so that the system tend to a specified equilibrium state as rapidly as possible; see [2], The upper limit of integration is thus not predetermined, but is rather a function of the choice of the vector y. In place of a formulation in precise analytic terms of the type appearing in (2) or (4), we encounter an implicit criterion of the following type: "When x satisfies a set of conditions Ci , C2 , ••• , Cp for the first time, we want a given scalar function of x to be as small as possible."