Deterministic Learning-Based Tracking Control for Parabolic PDE Systems With Infinite-Dimensional Nonlinear Uncertain Dynamics

In this article, we investigate the joint problem of dynamics learning and tracking control for a class of parabolic partial differential equation (PDE) systems with infinite-dimensional uncertain nonlinear dynamics. A new learning control scheme is proposed based on the deterministic learning (DL) theory. One key feature of the proposed scheme is its capability of accurately learning the system's nonlinear uncertain dynamics during real-time tracking control with provable stability and convergence of the overall PDE closed-loop system. Specifically, the Galerkin method is first employed to deal with the infinite dimensionality of the PDE system; a novel DL-based adaptive learning control scheme is then proposed using dual radial basis function neural networks (RBF NNs), in which a pair of RBF NNs are employed to address, respectively, the matched and unmatched components of uncertain nonlinear system dynamics. This control scheme is finally examined on the original PDE system, and it is rigorously proved that: first the PDE system's state tracks the prescribed reference trajectory with guaranteed closed-loop stability and tracking accuracy; and second locally accurate identification of the PDE system's dominant nonlinear uncertain dynamics can be achieved with provable convergence of associated NN weights to their optimal values, thereby the learned knowledge can be ultimately stored and represented by the convergent constant RBF NN models. Based on this, an experience-based control scheme is further proposed, which is capable of recalling the associated learned knowledge in real-time to further improve control performance and reduce computational complexity with maintained provable stabilization. It is worth stressing that although this work is focused particularly on parabolic PDE systems, it is groundbreaking with important technical breakthroughs that would facilitate a more complete extension of the DL theory from traditional ordinary differential equation systems to PDE systems in the future. Extensive simulation studies have been conducted to demonstrate effectiveness and advantage of the proposed results.