This appendix provides some background material on those aspects of optimization, game theory, and optimal and robust control theory which are frequently used in the text. It also serves to introduce the reader to our notation and terminology. For more detailed expositions on the topics covered here, standard references are for game theory, for optimal control, and for robust (H∞) control.

Introduction to optimization

We discuss in this section elements of and some key results from optimization in finite-dimensional spaces, including nonlinear, convex and linear programming, and distributed computation. Before we do this, however, it will be useful to introduce the notions of sets, spaces, norms, and functionals, which are building blocks of a theory of optimization.

Sets, spaces, and norms

A set S is a collection of elements. If s is a member (element) of S, we write s ∈ S; if s does not belong to S, we write s ∈ S. If S contains a finite number of elements, it is called a finite set; otherwise it is called an infinite set. If the number of elements of an infinite set is countable (i.e. if there is a one-to-one correspondence between its elements and positive integers), we say that it is a denumerable (countable) set, otherwise it is a nondenumerable (uncountable) set.

We are a lucky generation for witnessing the microprocessor and Internet revolutions, the type of technological marvels that mark the start of a new era: the information age. Just like electricity, railroads, and automobiles, the information technologies have a profound effect on our way of life and will stay with us for decades and centuries to come. Thanks to these advances, we have been building complex communication and computing networks on a global scale. However, it is still difficult today to predict how this information age will progress in the future or to fully grasp its consequences. We can hope for a complete understanding perhaps in decades to come, as past history tells us.

Although we have engineered and built the Internet, the prime example of the information revolution, our (mathematical) understanding of its underlying systems is cursory at best, since their complexity is orders of magnitude greater than that of their predecessors, e.g. the plain telephone network. Each disruptive technology brings its own set of problems along with enormous opportunities. Just as we are still trying to solve various issues associated with automobiles, the challenges put forward by the information and communication networks will be there not only for us but also for the next generations to address.

An important challenge today is security of complex computing and communication networks. Our limited understanding of these systems has a very unexpected side-effect: partial loss of “observability” and “control” of the very systems we build.