Adaptive neural boundary control design for a class of nonlinear spatially distributed systems

In this paper, an adaptive neural network (NN) control design is proposed for a class of parabolic partial differential equation (PDE) systems with boundary control actuation and unknown nonlinearities. Initially, applying the Galerkin´s method, the PDE system is represented by a finite-dimensional slow subsystem and a coupled infinite-dimensional fast residual subsystem. Subsequently, a modal-feedback controller is designed according to dissipative theory and small gain theorem, such that the closed-loop PDE system is practically stable. In the proposed boundary control scheme, a radial basis function (RBF) NN is employed to approximate the unknown nonlinearities of the slow subsystem. The outcome of the control problem is formulated as a linear matrix inequality (LMI) problem. Finally, the proposed design method is applied to the control of the temperature profile of a catalytic rod to illustrate its effectiveness.