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 of 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.

提出了仅依赖于状态在有限维控制空间上的正交投影的稳定反馈算子。考虑了一类将有限维控制空间映射到自身中的单调反馈算子。包括可伸缩身份运算符的特殊情况。在执行器组和单调性的大小上给出了条件，这些条件保证了一类半线性抛物线方程的反馈的半全局稳定特性。随后，利用反馈仅依赖于状态的有限维分量这一事实，由深度神经网络计算出使二次能量成本最小的最优反馈控制。数值模拟显示了明确缩放的正交投影反馈的稳定性能，