Doctor of Philosophy (PhD)



Document Type



The goal of this dissertation is to expand Berhard Koopman's operator theoretic global linearization approach to the study of nonautonomous flows. Given a system with states x in a set \Omega (the state space), a map t\to \gamma(t,s,x) (t\geq s \geq 0) is called a global flow if it describes the time evolution of a system with the initial state x \in \Omega at time t \geq s \geq 0. Koopman's approach to the study of flows is to look at the dynamics of the observables of the states instead of studying the dynamics of the states directly. To do so, one considers a vector space Z containing observables (measurements) and a vector space \mathcal{M}:=\mathcal{F}([0,\infty)\times \Omega,Z) of functions containing observations g: [0,\infty) \times \Omega \to Z. Then every global flow \gamma induces a family T(t)(t\geq 0) of linear maps on \mathcal{M}, where \begin{equation}\label{abstract} T(t)g: (s,x) \mapsto g(t+s,\gamma(t+s,s,x)). \end{equation} Since every global flow \gamma satisfies \gamma(s,s,x) = x and \gamma(t,r,\gamma(r,s,x)) = \gamma(t,s,x) for t\geq r \geq s \geq 0 and x \in \Omega, the linear maps T(t)(t\geq 0) define an operator semigroup on \mathcal{M}; that is, T(0) = I \text and T(t+s) = T(t)T(s) for t,s \geq 0. Following Koopman's approach, in pursuit of understanding the flow \gamma, we investigate the linear flow semigroup T(t)(t\geq 0) on \M given by \eqref{abstract}, and if \gamma(t,s,x) = U(t,s)x for some linear evolution family U(t,s), an associated special evolution semigroups on a subspace of \M given by S(t)f: s\mapsto f(t+s)U(t+s,s). Of primary concern are continuity properties of the associated linear evolution semigroups on different function spaces (Chapters 1-3). The Lie generator of the flow and a collection of open problems concerning general flow semigroups \eqref{abstract}, asymptotics and/or finite time blow-up, and Lie-Totter type approximations are described in Chapter 4.



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Committee Chair

Neubrander, Frank