  It is shown that an explicit oblique projection nonlinear feedback controller
is able to stabilize semilinear parabolic equations, with time-dependent
dynamics and with a polynomial nonlinearity. The actuators are typically
modeled by a finite number of indicator functions of small subdomains. No
constraint is imposed on the sign of the polynomial nonlinearity. The norm of
the initial condition can be arbitrarily large, and the total volume covered by
the actuators can be arbitrarily small. The number of actuators depend on the
operator norm of the oblique projection, on the polynomial degree of the
nonlinearity, on the norm of the initial condition, and on the total volume
covered by the actuators. The range of the feedback controller coincides with
the range of the oblique projection, which is the linear span of the actuators.
The oblique projection is performed along the orthogonal complement of a
subspace spanned by a suitable finite number of eigenfunctions of the diffusion
operator. For rectangular domains, it is possible to explicitly construct/place
the actuators so that the stability of the closed-loop system is guaranteed.
Simulations are presented, which show the semiglobal stabilizing performance of
the nonlinear feedback.
