In this paper, we extend recent results on adaptive extrernum seeking control to a class of nonlinear distributed parameter systems. We address the real-time optimization of a chemical reaction that occurs in a tubular reactor described by an hyperbolic set of partial differential equations. An estimation and a control algorithm that take into account temperature constraints are developed based on a Lyapunov functional. We apply the algorithm to the on-line optimization of the Williams-Otto reaction where the kinetics are assumed a priori unknown. The result of this algorithm is a feedback profile control that steers the system to its optimum. (c) 2004 Elsevier Ltd. All rights reserved.