# The Inverse Problem for Hamilton-Jacobi equations and Semiconcave Envelopes

Esteve C., Zuazua E.. The Inverse Problem for Hamilton-Jacobi equations and Semiconcave Envelopes SIAM J. Math. Anal. (2020)

Abstract. We study the inverse problem, or inverse design problem, for a time-evolution Hamilton-Jacobi equation. More precisely, given a target function u_T and a time horizon T > 0, we aim to construct all the initial conditions for which the viscosity solution coincides with u_T at time T. As it is common in this kind of nonlinear equations, the target might not be reachable. We first study the existence of at least one initial condition leading the system to the given target. The natural candidate, which indeed allows determining the reachability of u_T , is the one obtained by reversing the direction of time in the equation, considering u_T as terminal condition. In this case, we use the notion of backward viscosity solution, that provides existence and uniqueness for the terminal-value problem. We also give an equivalent reachability condition based on a differential inequality, that relates the reachability of the target with its semiconcavity properties. Then, for the case when u_T is reachable, we construct the set of all initial conditions for which the solution coincides with u_T at time T. Note that in general, such initial conditions are not unique. Finally, for the case when the target u_T is not necessarily reachable, we study the projection of u_T on the set of reachable targets, obtained by solving the problem backward and then forward in time. This projection is then identified with the solution of a fully nonlinear obstacle problem, and can be interpreted as the semiconcave envelope of u_T , i.e. the smallest reachable target bounded from below by u_T .