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 in adjoint methodology, or the more recent ones relying in sparsity and l^{1} minimization, provide numerical approximations of one of the possible initial sources. But these tools were not yet developed to the point where they can 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.

**This working package will lead, in particular, to innovative computational tools for inverse design in the absence of backward uniqueness and for processes with extreme backward sensitivity such as hyperbolic conservation laws**