The Generalized Problem of Bolza

Abstract

We consider the problem of minimizing a functional of the type \[l(x(0),x(1)) + \int_0^1 {L(t,x,\dot x)dt,} \] where l and L are permitted to attain the value $ + \infty $. We show that many standard variational and optimal control problems may be expressed in this form. In terms of certain generalized gradients, we obtain necessary conditions satisfied by solutions to the problem, in the form of a generalized Euler–Lagrange equation. We also extend the necessary condition of Weierstrass to this setting. The results obtained allow one to treat not only the standard problems but others as well, bringing under one roof the classical (differentiable) situation, the cases where convexity assumptions replace differentiability, and new problems where neither intervene. We apply the results in the final section to derive a new version of the maximum principle of optimal control theory.