In heavy-duty diesel exhaust systems, selective catalytic reduction (SCR) is used to reduce NOx to nitrogen to meet environmental regulations. Diesel exhaust after-treatment involves a set of components that are best characterized as distributed parameter systems. Thus, the optimal ammonia dosage in the SCR is an important and challenging problem in diesel exhaust treatment. In this work, we propose a method to synthesize an optimal controller for the SCR section of the diesel exhaust after-treatment system, which is based on a system model consisting of coupled hyperbolic and parabolic partial differential equations (PDEs). This results in a boundary control problem; where the control objectives are to reduce the amount of NOx emissions and ammonia slip to the fullest extent possible using the inlet concentration of ammonia as the manipulated variable and assuming that the concentrations of nitric oxide and nitrogen dioxide and ammonia, are measured at the SCR inlet and outlet. The proposed method combines the method of characteristics, spectral decomposition and the model predictive control (MPC) approach. For performance comparison purposes, the open-loop dynamic optimization problem is solved via Direct transcription (DT) to compute the upper performance limit for the optimal SCR problem. The results show that the proposed approach is able to achieve a very high level of control performance in terms of NO and ammonia slip reduction. (C) 2016 Elsevier Ltd. All rights reserved.