Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities

Free access article

Issue		ESAIM: COCV Volume 5, 2000


Page(s)		369 - 393
DOI		10.1051/cocv:2000114

DOI: 10.1051/cocv:2000114

ESAIM: COCV, July 2000, Vol. 5, 369-393

Value functions for Bolza problems with discontinuous Lagrangians and Hamilton-Jacobi inequalities

Gianni Dal Maso
SISSA, via Beirut 2, 34014 Trieste, Italy.

Hélène Frankowska
CNRS, ERS2064, Centre de Recherche Viabilité, Jeux, Contrôle, Université de Paris-Dauphine, 75775 Paris Cedex 16, France; (frankows@viab.dauphine.fr)

Received July 26, 1999. Revised April 28, 2000

Abstract: We investigate the value function of the Bolza problem of the Calculus of Variations $\begin{equation*}% V (t,x)=\inf \left\{ \int_{0}^{t} L (y (s),y' (s))ds + \varphi (y(t)) : y \in W^{1,1} (0,t;\mathbb{R}^n),\; y(0)=x \right\}, \end{equation*}$
with a lower semicontinuous Lagrangian L and a final cost $\varphi$ , and show that it is locally Lipschitz for t>0whenever L is locally bounded. It also satisfies Hamilton-Jacobi inequalities in a generalized sense. When the Lagrangian is continuous, then the value function is the unique lower semicontinuous solution to the corresponding Hamilton-Jacobi equation, while for discontinuous Lagrangian we characterize the value function by using the so called contingent inequalities.

Keywords and phrases: Discontinuous Lagrangians, Hamilton-Jacobi equations, viability theory, viscosity solutions.

AMS Subject Classification: 49L20, 49L25

Article without figures

What is OpenURL?

The OpenURL standard is a protocol for transmission of metadata describing the resource that you wish to access. An OpenURL link contains article metadata and directs it to the OpenURL server of your choice. The OpenURL server can provide access to the resource and also offer complementary services (specific search engine, export of references...). The OpenURL link can be generated by different means.

If your librarian has set up your subscription with an OpenURL resolver, OpenURL links appear automatically on the abstract pages.
You can define your own OpenURL resolver with your EDPS Account. In this case your choice will be given priority over that of your library.
You can use an add-on for your browser (Firefox or I.E.) to display OpenURL links on a page (see http://www.openly.com/openurlref/). You should disable this module if you wish to use the OpenURL server that you or your library have defined.