# Enrique Zuazua

## Friedrich-Alexander-Universität Erlangen-Nürnberg, Fundación Deusto, Universidad Autónoma de Madrid

### Inverse design for conservation laws and Hamilton–Jacobi equations

We shall discuss the inverse problem, or inverse design problem, for a time-evolution Hamilton–Jacobi equation and the 1-d analog in the context of scalar conservation laws.

More precisely, given a target function and a time horizon, we aim to construct all the initial conditions for which the viscosity solution coincides the given target at the given time. As is common in this kind of nonlinear equation, 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 in fact allows determining the reachability of the target, is the one obtained by reversing the direction of time in the equation. When is reachable, we construct the set of all the initial conditions for which the viscosity solutions lead to the final target.

We shall also discuss a number of open problems arising in this area and the possible link with other control problems through value functions, in particular in the context of reinforcement learning.