An approximate dynamic programming (ADP)-based suboptimal neurocontroller to obtain desired temperature for a high-speed aerospace vehicle is synthesized in this paper. A 1-D distributed parameter model of a fin is developed from basic thermal physics principles. ldquoSnapshotrdquo solutions of the dynamics are generated with a simple dynamic inversion-based feedback controller. Empirical basis functions are designed using the ldquoproper orthogonal decompositionrdquo (POD) technique and the snapshot solutions. A low-order nonlinear lumped parameter system to characterize the infinite dimensional system is obtained by carrying out a Galerkin projection. An ADP-based neurocontroller with a dual heuristic programming (DHP) formulation is obtained with a single-network-adaptive-critic (SNAC) controller for this approximate nonlinear model. Actual control in the original domain is calculated with the same POD basis functions through a reverse mapping. Further contribution of this paper includes development of an online robust neurocontroller to account for unmodeled dynamics and parametric uncertainties inherent in such a complex dynamic system. A neural network (NN) weight update rule that guarantees boundedness of the weights and relaxes the need for persistence of excitation (PE) condition is presented. Simulation studies show that in a fairly extensive but compact domain, any desired temperature profile can be achieved starting from any initial temperature profile. Therefore, the ADP and NN-based controllers appear to have the potential to become controller synthesis tools for nonlinear distributed parameter systems.