This book is devoted to some mathematical methods that arise in two domains of artificial intelligence: neural networks and qualitative physics (which here we shall call “qualitative analysis”). These two topics are treated independently. Rapid advances in these two areas have left unanswered many mathematical questions that should motivate and challenge a wide range of mathematicians. The mathematical techniques that I choose to present in this book are as follows:

1. control and viability theory in neural networks and cognitive systems, regarded as dynamical systems controlled by synaptic matrices.

2. set-valued analysis, which plays a natural and crucial role in qualitative analysis and simulation by emphasizing properties common to a class of problems, data, and solutions. Set-valued analysis also underlies mathematical morphology, which provides useful techniques for image recognition.

This allows us to present in a unified way many examples of neural networks and to use several results on the control of linear and nonlinear systems to obtain a learning algorithm of pattern-classification problems (including time series in forecasting), such as the back-propagation formula, in addition to learning algorithms concerning feedback-regulation laws for solutions to control systems subject to state constraints (inverse dynamics).