Advances in artificial neural networks in the past two decades have witnessed empirical demonstrations of their capabilities, followed by rigorous mathematical explanations of their validity. For example, in the 1980s, simulation studies revealed that neural networks could approximate very nearly all functions encountered in practical problems. In the following years it was conclusively shown by a number of authors that neural networks are universal approximators in a precise mathematical sense. Similarly, in 1990, it was suggested that neural networks could be used as identifiers and controllers in dynamical systems. Following this, extensive simulation studies were carried out, and empirical evidence began to accumulate demonstrating that neural networks could outperform conventional linear controllers in many applications. It once again became apparent that more formal methods grounded in mathematical systems theory would be needed to quantitatively assess the capabilities as well as the limitations of neurocontrol.

When neural networks are included as parts of nonlinear adaptive control systems, they raise stability questions which are, in general, not mathematically tractable. The difficulties encountered depend upon the class of plants considered, the prior information assumed of the nonlinearities encountered, the domain in the state space in which the trajectories are to lie,
and the conditions under which the neural networks are trained. During much of the 1990s, rigorous methods based on linearization were developed, which are valid in a neighborhood of the equilibrium state.

In recent years, numerous papers in prestigious journals have described solutions to very complex neurocontrol problems. The assumptions made in these papers, the methods proposed, the proofs presented, and the claims made of global asymptotic stability have raised concerns among many members of the research community. The presentations and panel discussion in the proposed special session will examine the above problems, assumptions, claims, and the fundamental questions involved.