  We use the continuation and bifurcation package pde2path to numerically
analyze infinite time horizon optimal control problems for parabolic systems of
PDEs. The basic idea is a two step approach to the canonical systems, derived
from Pontryagin's maximum principle. First we find branches of steady or
time-periodic states of the canonical systems, i.e., canonical steady states
(CSS) respectively canonical periodic states (CPS), and then use these results
to compute time-dependent canonical paths connecting to a CSS or a CPS with the
so called saddle point property. This is a (high dimensional) boundary value
problem in time, which we solve by a continuation algorithm in the initial
states. We first explain the algorithms and then the implementation via some
example problems and associated pde2path demo directories. The first two
examples deal with the optimal management of a distributed shallow lake, and of
a vegetation system, both with (spatially, and temporally) distributed
controls. These examples show interesting bifurcations of so called patterned
CSS, including patterned optimal steady states. As a third example we discuss
optimal boundary control of a fishing problem with boundary catch. For the case
of CPS-targets we first focus on an ODE toy model to explain and validate the
method, and then discuss an optimal pollution mitigation PDE model.
