POD-based reduced order optimal control of parabolic PDE systems via diffusivity-interior-boundary actuation

We present a framework to solve an optimal control problem for parabolic partial differential equations (PDEs) with diffusivity-interior-boundary actuators. The proposed approach is based on reduced order modeling (ROM) and successive optimal control computation. First we simulate the parabolic PDE system with given inputs to generate data ensembles, from which we then extract the most energetic modes to obtain a reduced order model based on the proper orthogonal decomposition (POD) method and Galerkin projection. The obtained reduced order model corresponds to a bilinear system. By solving the optimal control problem of the bilinear system successively, we update the given initial optimal inputs iteratively until the convergence is obtained. The simulation results demonstrate the effectiveness of the proposed method.