Welcome to my website. I am a Professor of Mechanical Engineering at the University of California at Santa Barbara (UCSB). I also have a courtesy appointment in the department of Electrical and Computer Engineering, and I am a member of the Center for Control, Dynamical Systems and Computation (CCDC). This interdisciplinary center brings together faculty and graduate students from across the College of Engineering departments and Mathematics.

My core research area is Controls and Dynamical Systems (CDS), and I do quite a bit of cross-disciplinary work at the interface between CDS and other fields such as Network Science, Fluid Mechanics, Statistical Physics, Machine Learning and Mathematics. The “Research” links to the left contain more information about the research work of my group.

Note: This site is currently under construction.

• Under Construction

These papers represent my attempts at viewing well-known textbook results with a bit of a mental shift. The main theme is to connect concepts across different areas, as well as emphasizing the process of “discovery”, i.e. how would you invent those concepts yourself if you had never heard of them? The links below are to the arXiv versions.

• Discovering Transforms: A Tutorial on Circulant Matrices, Circular Convolution, and the Discrete Fourier Transform: How would you “discover” the Fourier (and other) transform if you had never heard of it? This is a different way to think about a large class of transform methods as a linear algebra (or functional analysis) problem of simultaneous diagonalization of a class of matrices or operators.

• A Tutorial on Solution Properties of State Space Models of Dynamical Systems: The matrix exponential, the Peano-Baker Series, the Variations of Constants (Cauchy) formula, and the Picard iteration are various manifestations of one concept, namely the Neumann Series. All these various formulas can be “discovered” (including the definition of the matrix exponential) through various applications of the Neumann series when interpreted with a slight bit of abstraction.

• A Tutorial on Matrix Perturbation Theory (using compact matrix notation): Analytic matrix perturbation theory is an immensely useful tool in many areas. It is very ironic (to me) that standard treatments of this theory do not use matrix notation! Here I present this theory using compact matrix notation which I believe simplifies the usually messy formulas and gives some additional insight. In particular, the matrix Sylvester Equation plays a prominent role in finding the eigenvectors perturbations.

• A Short Introduction to the Koopman Representation of Dynamical Systems: The title describes what this is. I try to develop the basic concept with the minimum of unnecessary distractions. In particular, special attention is paid to the duality between initial conditions and output maps in both the original system and its Koopman representation (i.e. initial conditions of the original system become output operators in the Koopman representation and vice versa).