Variational problems with inequality constraints

In an earlier paper [I] we showed how the local characterization of optimality of dynamic programming led to simple derivations of many of the results of the calculus of variations. Except for one brief section, we restricted our attention to the Problem of Lagrange. In that section, we considered a minimum time problem of the Mayer type and derived the multiplier rule for the problem. In this paper we shall reconsider the Problem of Mayer. We shall state the problem in a more general way than in [l]. After deducing the appropriate form of the multiplier rule we shall consider variational problems restricted by inequality constraints. We first show, as we did in [I], that no further theory is required if the constraint explicitly involves what we shall call the decision variable. We shall then derive the considerably more complex conditions necessary for optimality for curves constrained to lie in a region of the state variable space. In a forthcoming third paper we shall consider the computational solution of variational problems. We shall present a practical function-space gradient technique. While this technique is known to a few practioners of the art [2,3], we feel that the simplicity of our derivation is of interest. In that paper we shall illustrate both the contents of this paper and the computational technique by discussing the numerical solution of a problem with state variable inequality constraints. Much of the content of this paper is the result of research conducted in applied mathematics at Harvard University and will appear as part of the author’s forthcoming dissertation,