Structural problems in adaptive neuro control

We study the relation between neural network based control and nonlinear parametric adaptive control. We consider the static problem and give results which show that, for smooth systems, continuous stabilising controls imply the existence of neural network controls for compact sets of initial conditions. We discuss the possibilities for global results and show that this is possible in a very restrictive sense. Controllability and observability are briefly considered. We illustrate a control problem (speed field tracking) which cannot even be approximately solved by neural networks. These static results serve to illustrate the fundamental limitations of approximate parameterisations which will impose limitations on the adaptive schema. Finally, we illustrate the adaptive situation by a feedback linearisation scheme.