# On a new necessary condition in the Calculus of Variations for Lagrangians that are highly discontinuous in the state and velocity

created by mariconda on 11 Sep 2019

[BibTeX]

Published Paper

Inserted: 11 sep 2019
Last Updated: 11 sep 2019

Journal: Atti Accad. Naz. Lincei Rend. Lincei Mat. Appl.
Volume: 30
Number: 3
Pages: 649-663
Year: 2019
Doi: 10.4171/RLM/865
We consider a local minimizer, in the sense of the $W^{1,1}$ norm, of the classical problem of the calculus of variations $\begin{cases} \text{Minimize}\quad &\displaystyle I(x):=\int_a^b\Lambda(t,x(t), x'(t))\,dt+\Psi(x(a), x(b))\\ \text{subject to:} &x\in W^{1,1}([a,b];\mathbb R),\\ &x'(t)\in C\,\text{ a.e., } \,x(t)\in \Sigma\quad\forall t\in [a,b].\\ \end{cases}\tag{P}$ where $\Lambda:[a,b]\times\mathbb R^n\times\mathbb R^n\to\mathbb R\cup\{+\infty\}$ is just Borel measurable, $C$ is a cone, $\Sigma$ is a nonempty subset of $\mathbb R^n$ and $\Psi$ is an arbitrary extended valued function: this allows to cover any kind of endpoint constraints. In the case where $\Lambda$ is real valued we do not assume further assumptions than Borel measurability and a local Lipschitz condition on $\Lambda$ with respect to $t$, allowing $\Lambda(t,x,\xi)$ to be possibly discontinuous, nonconvex in $x$ or $\xi$. In the case of an extended valued Lagrangian, we further impose the l.s.c. and a condition on the effective domain of $\Lambda(t,x,\cdot)$. This article is a survey of two recent papers of the authors (accepted in J. Differential Equations and in J. Optim. Theory Appl.). Consider a local minimizer $x_*$, in the sense of the norm of the absolutely continuous functions. We illustrate a new necessary condition: there exists an absolutely continuous function $p$ such that, for almost every $t$ in $[a,b]$, $\Lambda\Big(t, x_*(t),\dfrac{x_*'(t)}{v}\Big)v-\Lambda(t, x_*(t), x_*'(t))\ge p(t)(v-1)\quad\forall v>0,\tag{W}$ and moreover, $p'$ belongs to a suitable generalized subdifferential of $t\mapsto\Lambda (t, x_*(t), x_*'(t))$. The proof of (W) takes full advantage of a classical reparametrization technique, and of the most recent versions of the Maximum Principle. The variational inequality turns out to be equivalent to a generalized Erdmann -- DuBois-Reymond (EDBR) necessary type condition, that we are able to express just in terms of the classical tools of convex analysis (e.g. convex subdifferentials): in the autonomous, real valued case it hold true for \emph{every} Borel Lagrangian. Instead, one needs more regularity to formulate a version of the (EDBR) in terms of the limiting subdifferential.