Finer analysis of characteristic curves and its application to shock profile, exact and optimal controllability of a scalar conservation law with strict convex flux

Abstract

Here we consider scalar conservation law in one space dimension with strictly convex flux. Goal of this paper is to study two problems. First problem is to know the profile of the entropy solution. In spite of the fact that, this was studied extensively in last several decades, the complete profile of the entropy solution is not well understood. Second problem is the exact controllability. This was studied for Burgers equation and some partial results are obtained for large time. It was a challenging problem to know the controllability for all time and also for general convex flux. In a seminal paper, Dafermos introduces the characteristic curves and obtain some qualitative properties of a solution of a convex conservation law. In this paper, we further study the finer properties of these characteristic curves. As a bi-product we solve these two problems in complete generality. In view of the explicit formulas of Lax - Oleinik, Joseph - Gowda, target functions must satisfy some necessary conditions. In this paper we prove that it is also sufficient. Method of the proof depends highly on the characteristic methods and explicit formula given by Lax - Oleinik and the proof is constructive. This method allows to solve the optimal controllability problem in a trackable way.