article

Title: Nonsmooth multi-time Hamilton-Jacobi systems

Authors: Monica Motta and Franco Rampazzo

Issue: Volume 55 (2006), Issue 5, 1573-1614

Abstract:

$We establish existence of a solution for systems of Hamilton-Jacobi equations of the form \begin{equation} \begin{cases} \frac{\partial u}{\partial t_{1}} + H_{1}(x, D_{x}u) = 0,\\ \cdots\\ \frac{\partial u}{\partial t_{N}} + H_{N}(x, D_{x}u) = 0, \end{cases} \end{equation}, $(t_{1}, \dots, t_{N},x) \in \left] 0,T \right[^{N} \times \mathbb{R}^{n}$. A previous result---see [Guy Barles, Agn\`es Tourin, \textit{Commutation properties of semigroups for first-order Hamilton-Jacobi equations and application to multi-time equations}, Indiana Univ. Math. J. \textbf{50} (2001), 1523--1544] --- valid for $C^{1}$ Hamiltonians is here extended to the case where Hamiltonians are locally Lipschitz continuous. The main tool for dealing with this kind of non-smoothness consists in the interpretation of the existence issue in terms of commutativity of the minimum problems originating the Hamiltonians involved in equations of the form shown above. In turn, a sufficient condition for such commutativity is based on a notion of Lie bracket for nonsmooth vector-fields introduced in [ Franco Rampazzo, H\'ector Sussmann, \textit{Set-valued differentials and a nonsmooth version of Chow's theorem} (The 40th IEEE Conference on Decision and Control; Orlando, Florida, December 4 to 7, 2001), IEEE Publications \textbf{3} (2001), 2613--2618 ]. Besides existence, we establish uniqueness---actually, a comparison result---, regularity, and four different representations of the solution. Moreover, we prove a front-propagation property in the vector-valued time $(t_{1}, \dots, t_{N})$. The paper also contains results concerning semigroup properties of the solution and the additivity of a suitable defined exponential map.$