Systems Theory and PDEs: Open Problems, Recent Results, and New Directions

Preface of Systems Theory and PDEs—Open Problems, Recent Results, and New Directions
Contents
On some Impedance Boundary Conditions for a Thermo-Piezo-Electromagnetic System
	1 Introduction
	2 Preliminaries
		2.1 Evolutionary Equations
		2.2 Differential Operators and Classical Trace Spaces
		2.3 Abstract Boundary Data Spaces
	3 Boundary Conditions and Model System
		3.1 The Underlying System Equations
		3.2 Formulating New Boundary Conditions
		3.3 The Model for Thermo-Piezo-Electromagnetism with Boundary Dynamics
		3.4 Evolutionary Well-Posedness of the Model
	References
A Note on Some Non-Local Boundary Conditions and Their Use in Connection with Beltrami Fields
	1 Introduction
	2 Preliminaries
	3 Abstract Periodicity
	4 Conditions for a Closed Range of A#
	5 Applications
		5.1 The Operators grad# and div#
		5.2 The Operator curl#
	References
Spectral Theory for Schrödinger Operators on Compact Metric Graphs with δ and δ\' Couplings: A Survey
	1 Introduction
	2 Schrödinger Operators on Metric Graphs
		2.1 Metric Graphs
		2.2 Schrödinger Operators on Metric Graphs
	3 Properties of the Ground State
		3.1 Positivity of the First Eigenfunction
		3.2 Existence of Negative Eigenvalues
	4 Graph Manipulations Changing the Total Length of the Graph
		4.1 Increasing the Length of One Edge
		4.2 Increasing the Length of all Edges
		4.3 Attaching an Edge Between Two Vertices
		4.4 Attaching a Pendant Graph
		4.5 Inserting a Graph at a Vertex
		4.6 Shrinking Edge Lengths to Zero
	5 Graph Manipulations Preserving the Total Length
		5.1 Changing the Strength of a Coupling
		5.2 Changing delta Couplings to delta\' Couplings
		5.3 Joining Two Vertices
		5.4 Unfolding Parallel and Pendant Edges
	6 Hadamard-Type Formulas
	7 Bounds for the Lowest Eigenvalue
	References
Asymptotic Stability of Port-Hamiltonian Systems
	1 Introduction
	2 Characterisation of Asymptotic/Semi-uniform Stability
		Proof of Theorem 1.3
	3 Exponential Stability Revisited
	4 Example: Network of Vibrating Strings
	5 Vibrating Strings and Exponential Stability
	6 Conclusion
	Appendix A Well-Posedness of Ordinary Differential  Equations with Measurable Coefficients
	References
Port-Hamiltonian Formulation of Oseen Flows
	1 Introduction
	2 Formal Derivation of Energy Balance: Flow Through a Cylinder
	3 Port-Hamiltonian System Nodes
	4 Oseen Equations as Port-Hamiltonian System
		4.1 Assumptions on the Domain
		4.2 Spaces and Traces
		4.3 Dissipation Node Corresponding to the Oseen System
	5 Conclusion
	References
On the Equivalence of Geometric and Descriptor Representations of Linear Port-Hamiltonian Systems
	1 Introduction
	2 Preliminaries from Multivalued Linear Algebra
	3 Two Formulations of pH Systems
		3.1 Geometric Representation of pH Systems
		3.2 Port-Hamiltonian Descriptor Systems
	4 Equivalence of the Two Formulations
		4.1 From Geometric pH to Descriptor pH
		4.2 From Descriptor pH to Geometric pH
	5 Conclusion, Extensions, and Open Problems
	References
On Differential-Algebraic Equations with Bounded Spectrum in Banach Spaces
	1 Introduction
	2 Setting
		2.1 Reduction to Bounded Coefficient Operators
		2.2 Reduction to Operators
		2.3 Spectral Sets
	3 DAEs of Finite Index
	4 DAEs with Bounded Spectrum and Infinite Index
		4.1 Essentially Bounded Solutions on the Half-Axis
		4.2 Square Integrable Solutions on Compact Intervals
		4.3 Analytic Solutions
	Appendix
	References
BIBO Stability for Funnel Control: Semilinear Internal Dynamics with Unbounded Input and Output Operators
	1 Introduction
	2 BIBO Stability of Semilinear State Space Systems
	3 Linear Systems with Nonlinear Feedback and BIBO Stability
	4 Different Types of Uniformly Lipschitz Nonlinearities
	5 Locally Lipschitz Nonlinearities with Internal Control
	6 An Application to Funnel Control
		6.1 General Considerations on Funnel Control for Systems with Relative Degree One
		6.2 Dynamical Analysis of a Coupled ODE–PDE System for Funnel Control
		6.3 Numerical simulations
	7 Conclusion and Perspectives
	References
On Checking Lp-Admissibility for Parabolic Control Systems
	1 Introduction
		1.1 State Space Systems and Admissibility
		1.2 Boundary Control Systems
		1.3 Current State of Lp-Admissibility Theory
		1.4 Aims
	2 Interpolation Spaces, Extrapolation Spaces, and Admissibility
		2.1 Mapping Properties and Their Relation to Admissibility
		2.2 Example: Admissibility for the Heat Equation via Interpolation
	3 Characterizations for Multiplier-Like Systems
		3.1 Non-admissibility for a Dirichlet Controlled Heat Equation
		3.2 Normal Semigroups
		3.3 Conditions for Multiplication Semigroups
	4 Examples with Generators Having Non-discrete Spectra
		4.1 The Laplacian on the Full Space
		4.2 The Laplacian on a Half Line
	5 Applications to Infinite-Dimensional Input Spaces
	References