Franz Kappel

Spline approximations for functional differential equations

Abstract We develop an approximation framework for linear hereditary systems which includes as special cases approximation schemes employing splines of arbitrary order. Numerical results for first- and third-order spline-based methods are presented and compared with results obtained using a previously developed scheme based on averaging ideas.

Some considerations to the fundamental theory of infinite delay equations

In recent years we see an increasing interest in infinite delay equations. The main reason is that equations of this type become more and more important for different applications. Regardless of the specific problem one has to deal with it is in most cases necessary to establish some fundamental theory as, for instance, existence, uniqueness of solutions and continuous dependence on initial data. Also in the last few years we find an increasing number of papers where the general theory of linear and nonlinear semigroups or evolution operators is applied to functional differential equations. Of fundamental importance for all approaches is the right choice of the state space which in most cases is a Banach space of functions or of equivalence classes of functions. For equations with bounded delays this in general is not a difficult problem. But for infinite delay equations the choice of an appropriate state space is no more trivial. It was natural to investigate which properties of the state space are sufficient in order to establish the fundamental theory for infinite delay equations. Up to now the most thorough discussion of this problem is contained in [12]. The set of axioms given there seems to be in more or less final form. Other systems of axioms are just slight modifications of those given in [12] (cf., for instance, [18, 211) or consider more special cases (as in [6, 17). Section 1 of this paper can be considered as a discussion of the axioms given in [12] which results in a somewhat streamlined version of these axioms. It should be mentioned here that state spaces for equations with bounded retardation are included as special case, of course. In Section 2 we prove existence, uniqueness and continuous dependence of solutions concentrating on the nonstandard situation where the right-hand side of the equation is not defined for elements in the state space but only for representatives of elements in the state space. Such situations occur for instance if difference-differential equations are considered in a space like UP

An Implementation of Shor's r-Algorithm

Here we introduce a new implementation of well-known Shors r-algorithm with space dilations along the difference of two successive (sub)gradients for minimization of a nonlinear (non-smooth) function (N.Z. Shor, Minimization methods for Non-Differentiable Functions, Springer-Verlag: Berlin, 1985. Springer Series in Computational Mathematics, vol. 3). The modifications made to Shors algorithm are heuristic. They mostly concern the termination criteria and the line search strategy. A large number of test runs indicate that this implementation is very robust, efficient and accurate. We hope that this implementation of Shors r-algorithm will prove to be useful for solving a wide class of non-smooth optimization problems.

Cardiovascular and respiratory systems : modeling, analysis, and control

Preface 1. The cardiovascular system under an ergometric workload 2. Respiratory modeling 3. Cardio-Respiratory Modeling 4. Blood volume and the venous system 5. Future directions Appendix A. Supplemental calculations B. A Nonlinear feedback law C. Retarded functional differential equations: Basic theory Bibliography Index.

Control and Estimation of Distributed Parameter Systems

The volume here presented contains the Proceedings of the International Conference on Control of Distributed Parameter Systems, held in Graz (Austria) from July 15–21, 2001. It was the one eighth in a series of conferences that began in 1982. The book includes are a broad variety of topics related to partial differential equations, ranging from abstract functional analytic framework to aspects of modelling, with the main emphasis, however, on theory and numerics of optimal control for nonlinear distributed parameter systems. The proceedings contain 16 articles written by 27 authors, each of the papers containing new research results, not published before. They give a very useful overview to many of the current theoretical and industrial problems. The upto-date references at the end of the articles are also very helpful, and the nice, uniform TeX style of the book will be appreciated by the readers. In what follows, I describe briefly the papers contained in this collection. 1 H.T. Banks, S.C. Beeler and H.T. Tran, State estimations and tracking control of nonlinear dynamical systems. Based on the ”state-dependent Riccati equation”, nonlinear estimators and nonlinear feedback tracking controls are constructed for a wide class of systems. An application to a flight dynamics simulation shows that the corresponding computational methods are easily implementable and efficient. H.T. Banks, H. Tran and S. Wynne, The well-posedness results for a shear wave propagation model. Existence and uniqueness results are established for a nonlinear model for propagation of shear waves in viscoelastic tissue. R. Becker and B. Wexler, Mesh adaptation for parameter identification problems. The authors consider automatic mesh refinement for parameter identification problems involving PDEs. The idea is to solve the inverse problem on a ”cheap” discrete model, which still captures the ”essential” features of the physical model. To this end, a posteriori error estimator is used to successively

A mathematical model for fundamental regulation processes in the cardiovascular system.

Based on the four compartment model by Grodins we develop a model for the response of the cardiovascular system to a short term submaximal workload. Basic mechanisms included in the model are Starlings law of the heart, the Bowditch effect and autoregulation in the peripheral regions. A fundamental assumption is that the action of the feedback control is represented by the baroceptor loop and minimizes a quadratic cost functional. Simulation results show that the model provides a satisfactory description of data obtained in bicycle ergometer tests.

Evolution equations and approximations

Dissipative and maximal monotone operators linear semigroups analytic semigroups approximation of C-0 semigroups nonlinear semigroups of contractions locally quasi-dissipative evolution equations the Crandall-Pazy class variational formulations and Gelfand triples applications to concrete systems approximation of solutions for evolution equations semilinear evolution equations. Appendices: some inequalities convergence of Steklov means some technical results needed in Section 9.2.

An approximation theorem for the algebraic Riccati equation

For an infinite-dimensional linear quadratic control problem in Hilbert space, approximation of the solution of the algebraic Riccati operator equation in the strong operator topology is considered under conditions weaker than uniform exponential stability of the approximating systems. As an application, strong onvergence of the approximating Riccati operators in case of a previously developed spline approximation scheme for delay systems is established. Finally, convergence of the transfer-functions of the approximating systems is investigated.

Spline approximation for retarded systems and the Riccati equation

The purpose of this paper is to introduce a new spline approximation scheme for retarded functional differential equations. The special feature of this approximation scheme is that it preserves the product space structure of retarded systems and approximates the adjoint semigroup in a strong sense. These facts guarantee the convergence of the solution operators for the differential Riccati equation in a strong sense. Numerical findings indicate a significant improvement in the convergence behaviour over both the averaging and the previous spline approximation scheme.

A fully discretized approximation scheme for size-structured population models

An efficient algorithm for computing solutions to a class of models for size-structured populations is presented. Furthermore, some numerical examples are discussed.