This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination. IEEE TRANSACTIONS ON NEURAL NETWORKS AND LEARNING SYSTEMS 1 Adaptive Optimal Control of Highly Dissipative Non

Abstract — Highly dissipative nonlinear partial differential equations (PDEs) are widely employed to describe the system dynamics of industrial spatially distributed processes (SDPs). In this paper, we consider the optimal control problem of the general highly dissipative SDPs, and propose an adaptive optimal control approach based on neuro-dynamic programming (NDP). Initially, Karhunen-Loève decomposition is employed to compute empirical eigenfunctions (EEFs) of the SDP based on the method of snapshots. These EEFs together with singular perturbation technique are then used to obtain a finite-dimensional slow subsystem of ordinary differential equations that accurately describes the dominant dynamics of the PDE system. Subse-quently, the optimal control problem is reformulated on the basis of the slow subsystem, which is further converted to solve a Hamilton–Jacobi–Bellman (HJB) equation. HJB equa-tion is a nonlinear PDE that has proven to be impossible to solve analytically. Thus, an adaptive optimal control method is developed via NDP that solves the HJB equation online using neural network (NN) for approximating the value function; and an online NN weight tuning law is proposed without requiring an initial stabilizing control policy. Moreover, by involving the NN estimation error, we prove that the original closed-loop PDE system with the adaptive optimal control policy is semiglob-ally uniformly ultimately bounded. Finally, the developed method is tested on a nonlinear diffusion-convection-reaction process and applied to a temperature cooling fin of high-speed aerospace vehicle, and the achieved results show its effectiveness. Index Terms — Adaptive optimal control, empirical eigenfunction (EEF), highly dissipative partial differential equations (PDEs), neuro-dynamic programming (NDP), spatially distributed processes (SDPs)