Constrained Nonsmooth Problems of the Calculus of Variations and Nonsmooth Noether Equations

TLDR: In this article, the optimality conditions for nonsmooth variational problems of the calculus of variations with various types of constraints, such as additional constraints at the boundary, isoperimetric constraints, and nonholonomic inequality constraints, are derived.

Abstract: The paper is devoted to an analysis of optimality conditions for nonsmooth multidimensional problems of the calculus of variations with various types of constraints, such as additional constraints at the boundary, isoperimetric constraints, and nonholonomic inequality constraints. To derive optimality conditions, we study generalised concepts of differentiability of nonsmooth functions called codifferentiability and quasidifferentiability. Under some natural and easily verifiable assumptions we prove that a nonsmooth integral functional defined on the Sobolev space is continuously codifferentiable and compute its codifferential and quasidifferential. Then we apply general optimality conditions for nonsmooth optimisation problems in Banach spaces to obtain optimality conditions for nonsmooth problems of the calculus of variations. Through a series of simple examples we demonstrate that our optimality conditions are sometimes better than existing ones in terms of various subdifferentials, in the sense that our optimality conditions can detect the non-optimality of a given point when subdifferential-based optimality conditions fail to disqualify this point as non-optimal. Apart from standard optimality conditions, we also study so-called inner variations of nonsmooth integral functionals and utilise them to extend the Noether equations for variational problems to a nonsmooth setting. With the use of these equations we obtain a nonsmooth law of conservation of energy for autonomous nonsmooth variational problems. Finally, we discuss some difficulties one faces when trying to extend famous Noether's theorem on symmetries and conservation laws to a nonsmooth case.