Mathematical Methods in Systems, Optimization, and Control
When J.W. Helton started his career in the 1970s, systems theory was mainly dominated by the mean square criterion, i.e. $H_2$ approximation. One of Helton's main contributions is that with his work, mixing operator theory and systems engineering, he has caused a shift towards $H_\infty$, i.e. worst case analysis, which is obviously essential in (optimal) control. You want to keep a plane in the air under all circumstances, and not just under average conditions. Systems theory in general has grown out of an engineering approach, often based on expertise built up from practical experience, and on the other hand it has a strong mathematical tradition as well, which gave rise to new research fields in mathematics. Think for example of Wiener's work. Certainly control theory has been inspired by the engineering approach, but Helton has given a strong jerk towards the more pure mathematics approach. By a combination of events and circumstances, this has given a spur to start a flourishing recrudescence of an area mixing pure mathematics and engineering systems theory in the 1970s. The biennial symposium on Mathematical Theory of Networks and Systems (MTNS) started in 1973 and has been a platform to discuss these ideas. The International Workshop on Operator Theory and Applications (IWOTA) started in 1983 and is now taking place every year. It serves the same purpose as MTNS but somewhat more oriented towards the pure mathematicians.
Helton's early work was related to electrical engineering, by hooking it up with Adamjan-Arov-Krein theory (also known as AAK, or Hankel-norm approximation), the commutant lifting theory of Foias-Nagy-Sarason, the Nehari theory, moment problems and complex interpolation. In this way, the field has matured to a quite broad field as it is now The original objectives do still hold: join the interests of mathematicians and engineers which involves, as the MTNS and IWOTA objectives state, "a broad range of fields of pure and applied mathematics, including ordinary and partial differential equations, real and complex analysis, numerical analysis, probability theory and stochastic analysis, operator theory, linear and commutative algebra as well as algebraic and differential geometry. There are a wide range of applications ranging from problems in biology, communications and mathematical finance to problems in chemical engineering, aerospace engineering and robotics."
Many of the main topics in this area are covered by the 19 contributions, all written by the main players in the field. An incomplete selection of topics: Carathéodory-Julia theorem, Nevanlinna-Pick interpolation, Nehari's problem, Szegő's limit theorem, semiseparable matrices, spectral inclusion, positive completion problems, matrix inequalities, Riccati equations, control problems, etc. This is the list of technical papers.
- J. Agler: The Carathéodory-Julia Theorem and the Network Realization Formula in Two Variables
- J.A. Ball and Q. Fang: Nevanlinna-Pick Interpolation via Graph Spaces and Kreĭn-space Geometry: A Survey
- E. Basor: A Brief History of the Strong Szegő Limit Theorem
- P. Dewilde: Riccati or Square Root Equation? The Semi-separable Case
- R.G. Douglas and J. Eschmeier: Spectral Inclusion Theorems
- H. Dym: Tutorial on a Nehari Problem and Related Reproducing Kernel Spaces
- A.E. Frazho, S. ter Horst an M.A. Kaashoek: Optimal Solutions to Matrix-valued Nehari Problems and Related Limit Theorems
- P.A. Fuhrmann and U. Helmke: On Theorems of Halmos and Roth
- M. Harrison: Pfisterfs Theorem Fails in the Free Case
- J.W. Helton, I. Klep and S. McCullough: Free Analysis, Convexity and LMI Domains
- R. Howe: Traces of Commutators of Integral Operators . the Aftermath
- M.R. James: Information States in Control Theory: From Classical to Quantum
- J. Nie: Convex Hulls of Quadratically Parameterized Sets With Quadratic Constraints
- D. Plaumann, B. Sturmfels and C. Vinzant: Computing Linear Matrix Representations of Helton-Vinnikov Curves
- L. Rodman and H.J. Woerdeman: Positive Completion Problems Over C*-algebras
- M.S. Takyar and T.T. Georgiou: Fractional-order Systems and the Internal Model Principle
- E. Tannenbaum, T. Georgiou and A. Tannenbaum: Optimal Mass Transport for Problems in Control, Statistical Estimation, and Image Analysis
- V. Vinnikov: LMI Representations of Convex Semialgebraic Sets and Determinantal Representations of Algebraic Hypersurfaces: Past, Present, and Future
- N.J. Young: Some Analysable Instances of μ-synthesis
We think this is not the place to discuss each paper individually. As you can see, many of these topics are quite classical. What the papers bring is then a recent generalisation, an extension from one to two or more variables, from stationary to time-varying systems, from the scalar to the matrix case, or just placing it in a broader, more general setting. That can be brought in the form of a tutorial, a survey, or even a shorter note, but all very readable and up-to-date. Therefore, it can be safely recommended to specialists and to students at an advanced mathematics or engineering level as well.
The late Israel Gohberg who founded this Birkhäuser book series on Operator Theory: Advances and Applications in 1979 (this is volume 222), was one of the main players in this area of MTNS topics and was IWOTA president for a long time. In fact the whole series initiative fits exactly into this framework.