  Stabilizing feedback operators are presented which depend only on the
orthogonal projection of the state onto the finite-dimensional control space. A
class of monotone feedback operators mapping the finite-dimensional control
space into itself is considered. The special case of the scaled identity
operator is included. Conditions are given on the set of actuators and on the
magnitude of the monotonicity, which guarantee the semiglobal stabilizing
property of the feedback for a class semilinear parabolic-like equations.
Subsequently an optimal feedback control minimizing the quadratic energy cost
is computed by a deep neural network, exploiting the fact that the feedback
depends only on a finite dimensional component of the state. Numerical
simulations demonstrate the stabilizing performance of explicitly scaled
orthogonal projection feedbacks, and of deep neural network feedbacks.
