  In this paper we describe a numerical framework for achieving passive thermal
cloaking of arbitrary shapes in both static and transient regimes. The design
strategy is cast as the solution of an optimal control problem (OCP) for the
heat equation where the coefficients of the thermal diffusivity matrix take the
role of control functions and the distance between the uncloaked and the
cloaked field is minimized in a suitable observation domain. The control
actions enter bilinearly in the heat equation, thus making the resulting OCP
nonlinear, and its analysis nontrivial. We show that optimal diffusivity
coefficients exist both for the static and the transient case; we derive a
system of first-order necessary optimality conditions; finally, we carry out
their numerical approximation using the Finite Element Method. A series of
numerical test cases assess the capability of our strategy to tackle passive
thermal cloaking of arbitrarily complex two-dimensional objects.
