  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 $u_T$.
