Georg Propst

On nonconvergence of adjoint semigroups for control systems with delays

It is shown that the adjoints of a spline-based approximation scheme for delay equations do not converge strongly.

Piecewise linear approximation for hereditary control problems

This paper presents finite-dimensional approximations for linear retarded functional differential equations by use of discontinuous piecewise linear functions. The approximation scheme is applied to optimal control problems, when a quadratic cost integral must be minimjzed subject to the controlled retarded system. It is shown that the approximate optimal feedback operators converge to the true ones both in the case where the cost integral ranges over a finite time interval, as well as in the case where it ranges over an infinite time interval. The arguments in the last case rely on the fact that the piecewise linear approximations to stable systems are stable in a uniform sense. This feature is established using a vector-component stability criterion in the state space

A mathematical model for the deformation of the eyeball by an elastic band

\mathbb{R}^n \times L^2

Local well-posedness of a class of hyperbolic PDE–ODE systems on a bounded interval

and the favorable eigenvalue behavior of the piecewise linear approximations.

Computation of equilibria in models of flue gas washer plants

In a certain kind of eye surgery, the human eyeball is deformed sustainably by the application of an elastic band. This article presents a mathematical model for the mechanics of the combined eye/band structure along with an algorithm to compute the model solutions. These predict the immediate and the lasting indentation of the eyeball. The model is derived from basic physical principles by minimizing a potential energy subject to a volume constraint. Assuming spherical symmetry, this leads to a two-point boundary-value problem for a non-linear second-order ordinary differential equation that describes the minimizing static equilibrium. By comparison with laboratory data, a preliminary validation of the model is given.

On the truncation of the infinite sum of exponentials in a memory kernel of the linear wave equation

The well-posedness theory for hyperbolic systems of first-order quasilinear PDEs with ODEs boundary conditions (on a bounded interval) is discussed. Such systems occur in multi-scale blood flow models, as well as valveless pumping and fluid mechanics. The theory is presented in the setting of Sobolev spaces Hm (m ≥ 3 being an integer), which is an appropriate set-up when it comes to proving existence of smooth solutions using energy estimates. A blow-up criterion is also derived, stating that if the maximal time of existence is finite, then the state leaves every compact subset of the hyperbolicity region, or its first-order derivatives blow-up. Finally, we discuss physical examples which fit in the general framework presented.

Legendre-Tau-Padé approximations to the one-dimensional wave equation with boundary oscillators

A flue gas washer is a plant for the absorption of noxious components of industrial gas output. The type of flue gas washers that is considered here is a spray tower. In the tower gas is raising upward and water drops with suspended limestone fall downward. Thereby sulfur dioxide is washed out of the gas. In industrial practice, the plant is operated under long term constant gas and water input. In this article, a mathematical model for equilibrium states of the plant is set up from basic principles. Its discretized version consists of a large system of algebraic equations for the relevant chemical concentrations in the two phases. An efficient iterative algorithm for solving this system is proposed and shown to converge in a wide range of examples. A satisfactory validation of the model is given by comparison with experimental data from existing literature. It is demonstrated that the model along with the algorithm is applicable for the design of spray towers.

Well-posedness and regularity of linear hyperbolic systems with dynamic boundary conditions

We consider the one-dimensional wave equation, with a damping term that convolies the past history with an infmite sum of exponentials. The system is reformulated, as an abstract Cauchy problem. By means of semigroup theory it is shown that the systems with truncated sum of exponentials approximate the original one in L 2 sense. On the other hand it is demonstrated that the system and the approximations have different spectral propertie Wir untersuchen die eindimensionale Wellengleichung mit einem Reibungsterm, der die vergangene Entwicklung mit einer unendlichen Samme ton Exponentiolfunktionen faltet. Das System wird als abstraktes Cauchy-Problem reformu liert. Mit Hilje der Theorie der C 0 -Halbgruppen wird gezeigt, dass die Systeme mit abgebrochenen Summen das Ausgangssystem im L d -Sinn approximieren. Es wird bemiesen. dass das Ausgangssystem und dic Approximationen unterschiedliche spektrale Eigenschaften besitzen.

A dynamic two-phase flow model for a purification plant

The one-dimensional wave equation that is coupled to oscillators at the boundaries is approximated using Legendre polynomials. The convergence of the corresponding semigroups is proven by means of Trotter-Kato theory. By expansion of the characteristic determinants it is shown that the transfer functions of the approximate systems arc those of the exact system with the exponential functions replaced by Pade fractions.

Discretized energy minimization in a wave guide with point sources

We consider first-order hyperbolic systems on an interval with dynamic boundary conditions. These systems occur when the ordinary differential equation dynamics on the boundary interact with the waves in the interior. The well-posedness for linear systems is established using an abstract Friedrichs theorem. Due to the limited regularity of the coefficients, we need to introduce the appropriate space of test functions for the weak formulation. It is shown that the weak solutions exhibit a hidden regularity at the boundary as well as at interior points. As a consequence, the dynamics of the boundary components satisfy an additional regularity. Neither result can be achieved from standard semigroup methods. Nevertheless, we show that our weak solutions and the semigroup solutions coincide. For illustration, we give three particular physical examples that fit into our framework.