C0C_0C0-semigroups for hyperbolic partial differential equations on a one-dimensional spatial domain (original) (raw)

FA ] 1 4 N ov 2 01 8 Well-posedness of systems of 1-D hyperbolic partial differential equations

2018

We consider the well-posedness of a class of hyperbolic partial differential equations on a one dimensional spatial domain. This class includes in particular infinite networks of transport, wave and beam equations, or even combinations of these. Equivalent conditions for contraction semigroup generation are derived. We consider these equations on a finite interval as well as on a semi-axis.

Well-posedness of systems of 1-D hyperbolic partial differential equations

Journal of Evolution Equations, 2018

We consider the well-posedness of a class of hyperbolic partial differential equations on a one dimensional spatial domain. This class includes in particular infinite networks of transport, wave and beam equations, or even combinations of these. Equivalent conditions for contraction semigroup generation are derived. We consider these equations on a finite interval as well as on a semi-axis.

Well-posedness of a class of hyperbolic partial differential equations on the semi-axis

Journal of Evolution Equations, 2019

In this article we study a class of hyperbolic partial differential equations of order one on the semi-axis. The so-called port-Hamiltonian systems cover for instance the wave equation and the transport equation, but also networks of the aforementioned equations fit into this framework. Our main results firstly characterize the boundary conditions which turn the corresponding linear operator into the generator of a strongly continuous semigroup. Secondly, we equip the equation with inputs (control) and outputs (observation) at the boundary and prove that this leads to a well-posed boundary control system. We illustrate our results via an example of coupled transport equations on a network, that allows to model transport from and to infinity. Moreover, we study a vibrating string of infinite length with one endpoint. Here, we show that our results allow to treat cases where the physical constants of the string tend to zero at infinity.

Well-posedness and regularity of hyperbolic boundary control systems on a one-dimensional spatial domain

Esaim-control Optimisation and Calculus of Variations, 2009

We study hyperbolic partial differential equations on a one dimensional spatial domain with control and observation at the boundary. Using the idea of feedback we show these systems are well-posed in the sense of Weiss and Salamon if and only if the state operator generates a C 0 -semigroup. Furthermore, we show that the corresponding transfer function is regular, i.e., has a limit for s going to infinity. keywords: Infinite-dimensional systems, hyperbolic boundary control systems, C 0 -semigroup, well-posedness. * file:artikel/Dirac/well-posedness 4.tex

Linear neutral partial differential equations: a semigroup approach

International Journal of Mathematics and Mathematical Sciences, 2003

We study linear neutral PDEs of the form (∂/∂t)F u t = BFu t + Φu t , t ≥ 0; u 0 (t) = ϕ(t), t ≤ 0, where the function u(·) takes values in a Banach space X. Under appropriate conditions on the difference operator F and the delay operator Φ, we construct a C 0 -semigroup on C 0 (R − ,X) yielding the solutions of the equation.

Feedback semigroups and cosine operators for boundary feedback parabolic and hyperbolic equations

Journal of Differential Equations, 1983

General second-order parabolic and hyperbolic equations on a bounded domain are considered. The input is applied in the Neumann or mixed boundary condition and is expressed as a finite-dimensional feedback. In the parabolic case, the feedback acts, in particular, on the Dirichlet frace of the solution: here it is shown that the resulting closed loop syslem defines a (feedback) C,,-semigroup on L>(D) (in fact, on H3'2~2'(sZ), p > 0), that is both analytic and compact for positive times, and whose generator has compact resolvent. In the hyperbolic case, the feedback acts on the position vector only, or on its Dirichlet trace in a special case: here a similar result is established regarding the existence of a feedback C,,-cosine operator. Moreover, an example is given, which hints that the class of prescribed feedbacks acting on the Dirichlet trace cannot be substantially enlarged. Functional analytic techniques are employed, in particular perturbation theory. However, perturbation theory for the original variable fails on Lz(0), the space in which the final result is sought. Therefore, our approach employs perturbation theory, after a suitable continuous extension, on the larger space [H""'"(a)]'.

Hyperbolic Partial Differential Equations

We begin our study of finite difference methods for partial differential equations by considering the important class of partial differential equations called hyperbolic equations. In later chapters we consider other classes of partial differential equations, especially parabolic and elliptic equations. For each of these classes of equations we consider prototypical equations, with which we illustrate the important concepts and distinguishing features associated with each class. The reader is referred to other textbooks on partial differential equations for alternate approaches, e.g., Folland [18], Garabedian [22], and Weinberger [68]. After introducing each class of differential equations we consider finite difference methods for the numerical solution of equations in the class.

Results of Semigroup of Linear Equation Generating a Wave Equation

Earthline Journal of Mathematical Sciences, 2022

In this paper, we present results of ω-order preserving partial contraction mapping generating a wave equation. We use the theory of semigroup to generate a wave equation by showing that the operator 0 I ∆ 0 , which is A, is the infinitesimal generator of a C 0-semigroup of operators in some appropriately chosen Banach of functions. Furthermore we show that the operator A is closed, unique and that operator A is the infinitesimal generator of a wave equation.