This new textbook on ordinary differential equations offers a surprisingly fresh approach to the subject by incorporating an introduction to control theory into a rigorous treatment of initial value problems. The work is written at an intermediate level and aimed at students of mathematics and mathematically-oriented engineering. There are natural connections between control theory and ordinary differential equations, but they are rarely combined at this level. When they are it is usually in a more specialized engineering class.
James Clerk Maxwell — he whose name is attached to the fundamental equations of electromagnetism — wrote a paper called “On Governors” in 1868. He looked at differential equations and their solutions not as unchangeable things but as objects that could be altered, using external inputs, to cause solutions to exhibit certain prescribed properties. Those external inputs could be generated by a feedback process that creates them based on observations of the system. The feedback synthesis that this implies is the basis of control theory.
The authors begin with examples that provide a preview of the interplay between differential equations and control. These include applications in electric circuits, mechanics of nonlinear and inverted controlled pendulums, satellite dynamics and the like. From the beginning students get a glimpse of the power of the theory to come. In the circuit application with capacitor, inductor and twin-tunnel diode, for example, they learn that the soon-to-be-seen Poincaré-Bendixson theorem guarantees the existence of at least one periodic solution.
The book then moves to a more formal treatment of initial value problems, linear systems and the associated existence and uniqueness theorems. There is also a long section on systems with periodic coefficients and the associated Floquet theory.
The basics of linear control theory follow. The authors carefully introduce the critical concepts of controllability and observability — what they are and how they fit naturally into the representation of a linear system. The authors then immediately tie this back to the satellite control example presented in the first chapter. They go on to establish the basic criteria for observability, then briefly discuss the impulse response and transfer functions and investigate when a given transfer function can be realized in a control system.
The authors shift back to discuss nonlinear equations where they again treat existence and uniqueness questions in a couple of different ways (first, using the Peano approach and then the Contraction Mapping theorem). A section in this chapter focuses on planar systems and the Poincaré-Bendixson theorem. The book concludes with two chapters on stability. The first of these is a more or less standard treatment, but the second is about stability and stabilization of feedback systems. The authors have a special interest in stabilizing potentially or inherently unstable processes, so they have chosen to include more recent developments in feedback stabilization and input-to-state stability.
This is an elegant treatment at a level that would be accessible to graduate students or strong advanced undergraduates. Analysis and linear algebra are prerequisites, and both are used in relatively sophisticated ways. The interplay between linear algebra and control theory is important, and it is developed very nicely here.
The book is characterized throughout by strong writing, clear and complete proofs, good examples and plenty of exercises. Solutions of selected exercises are provided in an appendix.
Bill Satzer (firstname.lastname@example.org) is a senior intellectual property scientist at 3M Company, having previously been a lab manager at 3M for composites and electromagnetic materials. His training is in dynamical systems and particularly celestial mechanics; his current interests are broadly in applied mathematics and the teaching of mathematics.