Optimal arcs and the minimum value function in problems of Lagrange

Existence theorems are proved for basic problems of Lagrange in the calculus of variations and optimal control theory, in particular problems for arcs with both endpoints fixed. Emphasis is placed on deriving continuity and growth properties of the minimum value of the integral as a function of the endpoints of the arc and the interval of integration. Control regions are not required to be bounded. Some results are also obtained for problems of Bolza. Conjugate convex functions and duality are used extensively in the development, but the problems themselves are not assumed to be especially "convex". Constraints are incorporated by the device of allowing the Lagrangian function to be extended-real-valued. This necessitates a new approach to the question of what technical conditions of regularity should be imposed that will not only work, but will also be flexible and general enough to meet thediverse applications. One of the underlying purposes of the paper is to present an answer to this question. 1. Statement of main results. Let [a, b] be a real interval, and let L be a function on [a, b] x R' x R' with values in (no, + no]. For each subinterval [t0, t1l C [a, b] and endpoint pair (c0, c1) E R' x R', we consider the problem of Lagrange in which the integral