Some important PDE models in Continuum Physics, such as **hyperbolic conservation laws**, represent a major challenge from a control viewpoint for two (closely related) reasons:

**solutions lack regularity properties**and develop shock discontinuities in finite time, making linearization methods inapplicable- the property of
**backward uniqueness**is lost in the absence of viscosity effects and the most elementary control problem (but relevant in applications), that of**inverse design**, aimed at identifying the initial source leading to the available measurements at the final time, is severely**ill-posed**.

Similar issues arise in the context of Hamilton-Jacobi equations.

The existing methods, both the classical ones based on adjoint methodology or the more recent ones relying on sparsity and l1-minimization, provide numerical approximations of one of the possible initial sources. But these tools are not yet capable to provide all the feasible realizations for strongly time-irreversible systems as hyperbolic conservation laws. This issue is one of our main objectives, because of its implications in a number of key areas, such as management of natural resources.

We aim at developing a theory allowing the inverse design in the absence of **backward uniqueness** to be addressed both for linear and nonlinear problems. We shall take advantage of the fact that initial data recovered by backward **weak but not entropic solutions** can lead to the desired target by the forward entropic flow. This leads to the interesting and non-standard question of building numerical schemes to approximate non-entropic weak solutions. This program will also be developed in the context of **Hamilton-Jacobi** equations where, among the wide class of weak solutions, the physical ones are characterized by the viscosity criterion.