Control and inverse problems for the wave equation on metric graphs
Chair
Avdonin, SergeiCommittee
Rhodes, JohnRybkin, Alexei
Avdonina, Nina
Keyword
Inverse problemsDifferential equations
Partial differential equations
Wave equation
Boundary value problems
Dirichlet problem
Neumann problem
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Show full item recordAbstract
This thesis focuses on control and inverse problems for the wave equation on finite metric graphs. The first part deals with the control problem for the wave equation on tree graphs. We propose new constructive algorithms for solving initial boundary value problems on general graphs and boundary control problems on tree graphs. We demonstrate that the wave equation on a tree is exactly controllable if and only if controls are applied at all or all but one of the boundary vertices. We find the minimal controllability time and prove that our result is optimal in the general case. The second part deals with the inverse problem for the wave equation on tree graphs. We describe the dynamical Leaf Peeling (LP) method. The main step of the method is recalculating the response operator from the original tree to a peeled tree. The LP method allows us to recover the connectivity, potential function on a tree graph and the lengths of its edges from the response operator given on a finite time interval. In the third part we consider the control problem for the wave equation on graphs with cycles. Among all vertices and edges we choose certain active vertices and edges, and give a constructive proof that the wave equation on the graph is exactly controllable if Neumann controllers are placed at the active vertices and Dirichlet controllers are placed at the active edges. The control time for this construction is determined by the chosen orientation and path decomposition of the graph. We indicate the optimal time that guarantees the exact controllability for all systems of a described class on a given graph. While the choice of the active vertices and edges is not unique, we find the minimum number of controllers to guarantee the exact controllability as a graph invariant.Description
Dissertation (Ph.D.) University of Alaska Fairbanks, 2022Table of Contents
Chapter 1: General Introduction. Chapter 2: Control problems for the wave equation on metric tree graphs -- 2.1. Introduction -- 2.2. Preliminaries -- 2.3. The forward and control problems for the wave equation on a finite length interval -- 2.4. The forward and control problems in a star-shaped neighborhood graph of an internal vertex -- 2.5. Solving the forward problem for wave equations on general graphs -- 2.6. Controllability on a tree graph. Chapter 3: Inverse problem for the wave equation on graphs -- 3.1 Introduction --3.2 Preliminaries -- 3.3 The forward problem and the Duhamel's principle -- 3.4 The response function and the inverse problem -- 3.5 Leaf peeling method on a rooted tree. Chapter 4: Control problems for the wave equations on graphs with cycles -- 4.1. Introduction -- 4.2. Preliminaries -- 4.2.1. Metric graphs and Hilbert spaces on graphs -- 4.2.2. Observation and control problems of the wave equation -- 4.2.3. Directed acyclic graphs and linear ordering of vertices -- 4.2.4. The forward problem on an interval -- 4.2.5. Solution to the forward problem on a general graph -- 4.3. The forward and control problems on a DAG with controllers placed on a single-track active set -- V4.3.1. The tangle-free path union and single-track active set of a DAG -- 4.3.2. The forward problem when {I∗, j∗} is a ST active set -- 4.3.3. Shape and velocity controllability on an interval -- 4.3.4. Shape and velocity controllability on graphs -- 4.3.5. Exact controllability on graphs -- 4.3.6. Connectivity of the graph -- 4.3.7. The number of controllers -- 4.4. Appendix. Chapter 5: Conclusions -- References.Date
2022-12Type
DissertationCollections
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