Logarithmic Norms. An introduction, with applications to stability theory

January 13, 2023, ESB 2001

Gustaf Soderlind


The logarithmic norm was introduced in 1958 for matrices, and for the purpose of estimating growth rates in initial value problems. Since then, the concept has been extended to nonlinear maps, differential operators and function spaces. There are applications in operator equations in general, including evolution equations as well as boundary value problems. In Part 1 of this talk, we give an introduction to the logarithmic norm, starting with modal analysis, i.e., if stability is investigated one eigenvalue at a time, the norm of a complex number equals its absolute value, while the logarithmic norm equals its real part. Generalizing to a matrix $A$, the logarithmic norm is the extremal value of a quadratic form associated with $A$, but may also be viewed as the right-hand Gateaux differential of the operator norm at the identity operator, in the direction of the matrix $A$. This pattern extends to all Lipschitz continuous nonlinear maps, and further to unbounded operators in Hilbert space. The framework rests on the algebraic properties of the class of operators under consideration. In Part 2, we focus on applications in stability theory. Starting from the Uniform Monotonicity Theorem (Browder & Minty, 1963), we show that the logarithmic Lipschitz constant offers an improved "signed" version of the classical Neumann Lemma, which is commonly used in fixed-point and contractivity theory. We then apply the same approach to the classical Small Gain Theorem in control theory, transforming it into a Large Gain Theorem. As the logarithmic Lipschitz constant accounts for negative feedback, one can in effect apply arbitrarily large control gains without losing stability.

Speaker's Bio

Gustaf Soderlind received his PhD degree from the Royal Institute of Technology, in Stockholm, Sweden, in 1982. In 1990 he was appointed Professor of Numerical Mathematics at Lund University, where he has been working in scientific computing, in particular in initial value problems, stability theory, and in the control theoretic design of adaptive time-stepping algorithms, including digital filter theory. He is Professor emeritus at Lund University since 2018, and is currently working on a monograph on logarithmic norms, with applications in stability theory, ranging from ordinary differential equations to initial-boundary value PDEs. postdoctoral researcher with Dr. Warren E. Dixon, he was appointed as the 2015-16 MAE postdoctoral teaching fellow. In 2016 he joined the School of Mechanical and Aerospace Engineering at the Oklahoma State University as an Assistant professor. His primary research interests lie on the intersection of machine learning and systems theory.

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