Mathematical Control Theory

Now online version available (click on link for pdf file, 544 pages)

(Please note: book is copyrighted by Springer-Verlag. Springer has kindly allowed me to place a copy on the web, as a reference and for ease of web searches. Please consider buying your own hardcopy.)

Precise reference:
     Eduardo D. Sontag, Mathematical Control Theory: Deterministic Finite Dimensional Systems.
     Second Edition
, Springer, New York, 1998.
(531+xvi pages, ISBN 0-387-984895)
     Series: Textbooks in Applied Mathematics, Number 6. Hardcover, approx $55.00
     Order in USA from 1-800-SPRINGER or from

Errata for 2nd edition

First Edition's web page.
     (Errata, revisions, and some comments, all regarding the first edition, are included there. No errata posted since mid 1997.)

Math Reviews

Publicity Blurb:

This textbook introduces the core concepts and results of Control and System Theory. Unique in its emphasis on foundational aspects, it takes a "hybrid" approach in which basic results are derived for discrete and continuous time scales, and discrete and continuous state variables. Primarily geared towards mathematically advanced undergraduate or graduate students, it may also be suitable for a second engineering course in control which goes beyond the classical frequency domain and state-space material. The choice of topics, together with detailed end-of-chapter links to the bibliography, makes it an excellent research reference as well.

The Second Edition constitutes a substantial revision and extension of the First Edition, mainly adding or expanding upon advanced material, including: Lie-algebraic accessibility theory, feedback linearization, controllability of neural networks, reachability under input constraints, topics in nonlinear feedback design (such as backstepping, damping, control-Lyapunov functions, and topological obstructions to stabilization), and introductions to the calculus of variations, the maximum principle, numerical optimal control, and linear time-optimal control.

Also covered, as in the First Edition, are notions of systems and automata theory, and the algebraic theory of linear systems, including controllability, observability, feedback equivalence, and minimality; stability via Lyapunov, as well as input/output methods; linear-quadratic optimal control; observers and dynamic feedback; Kalman filtering via deterministic optimal observation; parametrization of stabilizing controllers, and facts about frequency domain such as the Nyquist criterion.

From the reviews of the first edition:

"This book will be very useful for mathematics and engineering students interested in a modern and rigorous systems course, as well as for experts in control theory and applications"
Mathematical Reviews
"An excellent book... gives a thorough and mathematically rigorous treatment of control and system theory"
Zentralblatt fur Mathematik
"The style is mathematically precise... fills an important niche... serves as an excellent bridge (to topics treated in traditional engineering courses). The book succeeds in conveying the important basic ideas of mathematical control theory, with appropriate level and style"
IEEE Transactions on Automatic Control

Chapter and Section Headings:

  What Is Mathematical Control Theory?
  Proportional-Derivative Control
  Digital Control
  Feedback Versus Precomputed Control
  State-Space and Spectrum Assignment
  Outputs and Dynamic Feedback
  Dealing with Nonlinearity
  A Brief Historical Background
  Some Topics Not Covered Systems
Basic Definitions
  I/O Behaviors
  Linear Discrete-Time Systems
  Smooth Discrete-Time Systems
  Linear Continuous-Time Systems
  Linearizations Compute Differentials
  More on Differentiability
  Volterra Expansions
  Notes and Comments
Reachability and Controllability
  Basic Reachability Notions
  Time-Invariant Systems
  Controllable Pairs of Matrices
  Controllability Under Sampling
  More on Linear Controllability
  Bounded Controls
  First-Order Local Controllability
  Controllability of Recurrent Nets
  Piecewise Constant Controls
  Notes and Comments
Nonlinear Controllability
  Lie Brackets
  Lie Algebras and Flows
  Accessibility Rank Condition
  Ad, Distributions, and Frobenius' Theorem
  Necessity of Accessibility Rank Condition
  Additional Problems
  Notes and Comments
Feedback and Stabilization
  Constant Linear Feedback
  Feedback Equivalence
  Feedback Linearization
  Disturbance Rejection and Invariance
  Stability and Other Asymptotic Notions
  Unstable and Stable Modes
  Lyapunov and Control-Lyapunov Functions
  Linearization Principle for Stability
  Introduction to Nonlinear Stabilization
  Notes and Comments
  Basic Observability Notions
  Time-Invariant Systems
  Continuous-Time Linear Systems
  Linearization Principle for Observability
  Realization Theory for Linear Systems
  Recursion and Partial Realization
  Rationality and Realizability
  Abstract Realization Theory
  Notes and Comments
Observers and Dynamic Feedback
  Observers and Detectability
  Dynamic Feedback
  External Stability for Linear Systems
  Frequency-Domain Considerations
  Parametrization of Stabilizers
  Notes and Comments
Optimality: Value Function
  Dynamic Programming
  Linear Systems with Quadratic Cost
  Tracking and Kalman Filtering
  Infinite-Time (Steady-State) Problem
  Nonlinear Stabilizing Optimal Controls
  Notes and Comments
Optimality: Multipliers
  Review of Smooth Dependence
  Unconstrained Controls
  Excursion into the Calculus of Variations
  Gradient-Based Numerical Methods
  Constrained Controls: Minimum Principle
  Notes and Comments
Optimality: Minimum-Time for Linear Systems
  Existence Results
  Maximum Principle for Time-Optimality
  Applications of the Maximum Principle
  Remarks on the Maximum Principle
  Additional Exercises
  Notes and Comments
Appendix: Linear Algebra
  Operator Norms
  Singular Values
  Jordan Forms and Matrix Functions
  Continuity of Eigenvalues
Appendix: Differentials
  Finite Dimensional Mappings
  Maps Between Normed Spaces
Appendix: Ordinary Differential Equations
  Review of Lebesgue Measure Theory
  Initial-Value Problems
  Existence and Uniqueness Theorem Linear Differential Equations
  Stability of Linear Equations
List of Symbols

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