Graduation Year

2023

Document Type

Dissertation

Degree

Ph.D.

Degree Name

Doctor of Philosophy (Ph.D.)

Degree Granting Department

Mathematics and Statistics

Major Professor

Razvan Teodorescu, Ph.D.

Co-Major Professor

Joel A. Rosendfeld, Ph.D.

Committee Member

Catherine A. Beneteau, Ph.D.

Committee Member

Dmitry Khavinson, Ph.D.

Committee Member

Benjamin P. Russo, Ph.D.

Keywords

System Identification, Dynamical Systems, Integral Transform, Kernelized Reconstruction

Abstract

Consider a nonautonomous nonlinear evolution $\dot{x}=f(x,t,\mu)$, where the vector $x(t) \in \mathbb{R}^n$ represents the state of the dynamical system at time $t$, $\mu$ contains system parameters, and $f(\cdot)$ represents a dynamic constraint. In most practical applications, the nonlinear dynamic constraint $f$ is unknown analytically. The problem of approximating $f$ directly from data measurements generated by the system is a main goal of this manuscript. In the postulates of the Nonlinear Autoregressive (NAR) framework, we show that the problem of approximating $f$ can be studied through symbols of densely defined multiplication operators over a Reproducing Kernel Hilbert Spaces (RKHS). In this formulation, data is mapped into a RKHS by virtue of occupation kernels which are special functions that reside in a RKHS owing to an integration functional. The resulting scheme is a parameter identification algorithm where system parameters are approximated according to some induced structure on the symbols of the operator. The action of the adjoint multiplication on an occupation kernel induces a kernelized transform which is the subject of study in the second part of the dissertation.

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