  We develop an optimization framework for the design of acoustic cloaks, with
the aim of overcoming the limitations of usual transformation-based cloaks in
terms of microstructure complexity and shape arbitrarity of the obstacle. This
is achieved by recasting the acoustic cloaking design as a nonlinear optimal
control problem constrained by a linear elliptic partial differential equation.
In this setting, isotropic material properties' distributions realizing the
cloak take the form of control functions and a system of first-order optimality
conditions is derived accordingly. Such isotropic media can then be obtained in
practice with simple hexagonal lattices of inclusions in water.
  For this reason, the optimization problem is directly formulated to take into
account suitable partitions of the control domain Two types of inclusions are
considered, and long-wavelength homogenization is used to define the feasible
set of material properties that is employed as a constraint in the optimization
problem. In this manner, we link the stage of material properties optimization
with that of microstructure design, aiming at finding the optimal implementable
solution. As a test benchmark, cloaking of the silhouette of a ship is
considered, for various frequencies and directions of incidence. The resulting
cloak is numerically tested via coupled structural/acoustic simulations.
