Lipschitzian regularity of minimizers and Lavrentiev phenomenon in the calculus of variations and optimal control

Abstract

We study properties of minimizing trajectories for various problems of the calculus of variations and optimal control. One of the important issues is Lipschitzian regularity of the minimizers. It is interesting for its own sake and also because standard optimality conditions may fail for non-Lipschitzian minimizers. Another important issue we address is the occurrence of the Lavrentiev phenomenon (M.A. Lavrentiev, 1927), when the infimum of a problem in a class of (non-Lipschitzian) functions with unbounded derivative is strictly less then the one in the respective class of functions with bounded derivative. We provide some new conditions which guarantee Lipschitzian regularity of minimizing trajectories of Lagrange problems of optimal control. Concerning the Lavrentiev phenomenon, we establish its prevalence by demonstrating that in many situations its occurrence can be guaranteed by a finite number of relations (equalities and inequalities) imposed on the jets of the functions, which appear in the integrands of the problem.