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Spectral Properties of the Koopman Operator in the Analysis of Nonstationary Dynamical Systems

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Spectral Properties of the Koopman Operator in the Analysis of Nonstationary Dynamical Systems UNIVERSITY of CALIFORNIA Santa Barbara Spectral Properties of the Koopman Operator in the Analysis of Nonstationary Dynamical Systems A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mechanical Engineering by Ryan M. Mohr Committee in charge: Professor Igor Mezi´c,Chair Professor Bassam Bamieh Professor Jeffrey Moehlis Professor Mihai Putinar June 2014 The dissertation of Ryan M. Mohr is approved: Bassam Bamieh Jeffrey Moehlis Mihai Putinar Igor Mezi´c,Committee Chair May 2014 Spectral Properties of the Koopman Operator in the Analysis of Nonstationary Dynamical Systems Copyright c 2014 by Ryan M. Mohr iii To my family and dear friends iv Acknowledgements Above all, I would like to thank my family, my parents Beth and Jim, and brothers Brennan and Gralan. Without their support, encouragement, confidence and love, I would not be the person I am today. I am also grateful to my advisor, Professor Igor Mezi´c,who introduced and guided me through the field of dynamical systems and encouraged my mathematical pursuits, even when I fell down the (many) proverbial (mathematical) rabbit holes. I learned many fascinating things from these wandering forays on a wide range of topics, both contributing to my dissertation and not. Our many interesting discussions on math- ematics, research, and philosophy are among the highlights of my graduate school career. His confidence in my abilities and research has been a constant source of encouragement, especially when I felt uncertain in them myself. His enthusiasm for science and deep, meaningful work has set the tone for the rest of my career and taught me the level of research problems I should consider and the quality I should strive for. My introduction to high quality research and top researchers in my field and others, both here in the UC Santa Barbara community and in the wider scientific community, has enriched me immensely. Aside from my advisor, I must foremost acknowledge and thank my good friend and colleague Dr. Marko Budiˇsi´cfor all our conversations, the interesting and interestingly crazy ideas we bounced off each other, and his support as we traveled this path together. I would also like to thank Professor v Jeff Moehlis both as a teacher and a colleague, for what he has taught me, the advice he has given me, and the discussions we have had. Professor Mihai Putinar's enthusiasm and interest in my research has both encouraged and reassured me that I was pursuing meaningful and interesting work. Finally, I'd like to acknowledge and thank all my friends who have had, and continue to have, a meaningful and positive impact on my life. Of special note are Mai, Marko, Lina, Veronica, Mike, Chris, and Gabri whose friendship, comraderie, and humor (and willingness to nod politely when I went off on my many and varied math tangents) I wouldn't trade for anything. Marko, Lina, Gabri, our Saturday barbecues are a source of many good memories. Mai, your continuous and unfailing friendship, support, and humor these past 12 years has been a constant source of strength and happiness in my life. If I had failed to know even one of you, my life would have been the worse for it. Thank you. vi Curriculum Vitae Education 2006 { PhD Candidate in Mechanical Engineering, University of California, Santa Barbara, USA 2001 { 2006 Bachelor of Science, Mechanical Engineering & Physics (Majors), Math- ematics (Minor), Summa Cum Laude, Virginia Commonwealth University, Vir- ginia, USA Relevant Experience Research 2007 { Graduate Student Researcher, UC Santa Barbara, Research group of Dr. Igor Mezi´c 2005 { 2006 Undergraduate Researcher, Virginia Commonwealth University, Control and Mechatronic Systems Laboratory, Research group of Dr. Kam Leang 2004 { 2005 Undergraduate Researcher, Medical College of Virginia, Smooth Muscles Laboratory, Research group of Dr. Paul Ratz (Dept. Biochemistry, Med. Col. of Virginia), Co-PI: Dr. John Speich (Dept. Mech Eng., Virginia Commonwealth Univ. ) vii Teaching 2006 { 2012 Teaching Assistant, UC Santa Barbara, 6 academic quarters; courses: \Mechanical Vibrations (Lead TA)", \Nonlinear Phenomena" (twice), \System Identification and Control", \ME Design II", \Sensors" Fall 2005 Laboratory Assistant, Virginia Commonwealth University; course: \Mecha- tronics" Professional Jun 2004 { Aug 2005 Advanced Fiber Systems, BSME Cooperative Edu- cation Engineer, Kevlar SBU, Spruance Plant, DuPont, Richmond, Virginia, USA; Created and maintained Kevlar Production database and analysis pro- grams for a Six Sigma project; analysis of historical meter pump performance and reliability Fall 2004 Advanced Fiber Systems, BSME Cooperative Education Engi- neer, Nomex SBU, Spruance Plant, DuPont, Richmond, Virginia, USA; Prod- uct portfolio optimization study, created analysis program identifying opportune areas for improvement viii Academic Activities Speaker May 2013 SIAM Conference on Applications of Dynamical Systems DS13, Snow- bird, UT, USA; Koopman Mode Analysis of Networks, Minisymposium Talk (MS60) May 2011 SIAM Conference on Applications of Dynamical Systems DS11, Snow- bird, UT, USA; Low-dimensional models and anomaly detection for TCP-like networks using the Koopman operator; Contributed Talk (CP15) Dec 2010 49th IEEE Conference on Decision and Control, Atlanta , GA, USA, The Use of Ergodic Theory in Designing Dynamics for Search Problems; Presenta- tion and a Proceedings Paper [see MM10] May 2009 SIAM Conference on Applications of Dynamical Systems DS09, Snow- bird, UT, USA; Designing Search Dynamics Robust Under Sensor Uncertainty; Minisymposium Talk (MS33) Attendee Jan 2008 Dynamics Days, Knoxville, TN, USA; Dec 2007 Workshop on Chaos and Ergodicity of Realistic Hamiltonian Systems, Centre de recherches math´ematiques(CRM), Universit´ede Montr´eal,Montr´eal, Qu´ebec, Canada; ix May 2007 SIAM Conference on Applications of Dynamical Systems DS07, Snow- bird, UT, USA; Member of Professional Societies 2013 { present American Mathematical Society (AMS), Student Member 2006 { present Society for Industrial and Applied Mathematics (SIAM), Student Member Awards June 2013 Outstanding Teaching Assistant for the Academic Year 2012/13 ; De- partment of Mechanical Engineering, UC Santa Barbara May 2006 Invited Speaker; School of Engineering Graduation Ceremony, Virginia Commonwealth University Fall 2005 Best Physics Student; Dept. of Physics, Virginia Commonwealth Univer- sity 2004 National Dean's List; 2003{2004 Edition 2003 National Dean's List; 2002{2003 Edition 2001 { 2006 Henrico County Merit Scholarship for full tuition and fees for duration of undergraduate engineering education x Fields of Study Major Field: Dynamical Systems Studies in Operator and Spectral Theory with Professor Igor Mezi´c Studies in Ergodic Theory with Professor Igor Mezi´c xi Abstract Spectral Properties of the Koopman Operator in the Analysis of Nonstationary Dynamical Systems by Ryan M. Mohr The dominating methodology used in the study of dynamical systems is the ge- ometric picture introduced by Poincar´e. The focus is on the structure of the state space and the asymptotic behavior of trajectories. Special solutions such as fixed points and limit cycles, along with their stable and unstable manifolds, are of interest due to their ability to organize the trajectories in the surrounding state space. Another viewpoint that is becoming more prevalent is the operator-theoretic / functional-analytic one which describes the system in terms of the evolution of func- tions or measures defined on the state space. Part I of this doctoral dissertation focuses on the Koopman, or composition, operator that determines how a function on the state space evolves as the state trajectories evolve. Most current studies involv- ing the Koopman operator have dealt with its spectral properties that are induced by dynamical systems that are, in some sense, stationary (in the probabilistic sense). The dynamical systems studied are either measure-preserving or initial conditions for trajectories are restricted to an attractor for the system. In these situations, only the xii point spectrum on the unit circle is considered; this part of the spectrum is called the unimodular spectrum. This work investigates relaxations of these situations in two different directions. The first is an extension of the spectral analysis of the Koopman operator to dynamical systems possessing either dissipation or expansion in regions of their state space. The second is to consider switched, stochastically-driven dynamical systems and the associated collection of semigroups of Koopman operators. In the first direction, we develop the Generalized Laplace Analysis (GLA) for both spectral operators of scalar type (in the sense of Dunford) and non spectral operators. The GLA is a method of constructing eigenfunctions of the Koopman operator corresponding to non-unimodular eigenvalues. It represents an extension of the ergodic theorems proven for ergodic, measure-preserving, on-attractor dynamics to the case where we have off-attractor dynamics. We also give a general procedure for constructing an appropriate Banach space of functions on which the Koopman operator is spectral. We explicitly construct these spaces for attracting fixed points and limit cycles. The spaces that we introduce and construct are generalizations of the familiar Hilbert Hardy spaces in the complex unit disc. In the second direction, we develop the theory of switched semigroups of Koopman operators. Each semigroup is assumed to be spectral of scalar-type
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