On the necessary conditions for optimal control of retarded systems

In this paper we derive necessary conditions, in the form of a maximum principle, for the optimal control of nonlinear, finitely retarded functional differential equations with function-space boundary conditions. We establish these conditions in a setting which guarantees the existence of regular multipliers, admits pointwise control constraints, and, with added restrictions, ensures nontriviality of the multipliers.The majority of this work was done while the first author was guest at the Institute für Numerische und Angewandte Mathematik der Universität Göttingen, Göttingen, BRD.     

The controllability problem for nonlinear Volterra systems

We discuss the controllability of systems whose dynamics are governed by a large class of nonlinear Volterra integral equations. The property of controllability is shown to be equivalent to the existence of a fixed point of a certain set-valued map. We show that convexity and seminormality conditions intimately related to those assumed in proofs of existence theorems for optimal controls are sufficient to guarantee controllability. Approximate controllability results are obtained by first introducing generalized solutions and then showing, under only mild additional restrictions, that ordinary solutions are dense in this broader class of trajectories.Dedicated to L. CesariTu se' lo mio maestro e il mio autore: Tu se' solo colui, da cui io tolsi Lo bello stile che m' ha fatto onore. Dante, Canto I: 85–87This work was written while the author was on leave to the Institut für Numerische und Angewandte Mathematik, Universität Göttingen, Göttingen, West Germany.     

On the solution of Maxwell's equations in polygonal domains

This paper is concerned with the structure of the singular and regular parts of the solution of time‐harmonic Maxwell's equations in polygonal plane domains and their effective numerical treatment. The asymptotic behaviour of the solution near corner points of the domain is studied by means of discrete Fourier transformation and it is proved that the solution of the boundary value problem does not belong locally to H² when the boundary of the domain has non‐acute angles. A splitting of the solution into a regular part belonging to the space H², and an explicitly described singular part is presented. For the numerical treatment of the boundary value problem, we propose a finite element discretization which combines local mesh grading and the singular field methods and derive a priori error estimates that show optimal convergence as known for the classical finite element method for problems with regular solutions. Copyright © 2006 John Wiley & Sons, Ltd.     

On the condition number of boundary integral operators for the exterior Dirichlet problem for the Helmholtz equation

Summary Brakhage and Werner, Leis and Panich suggested to reduce the exterior Dirichlet boundary value problem for the Helmholtz equation to an integral equation of the second kind which is uniquely solvable for all frequencies by seeking the solution in the form of a combined double- and single-layer potential. We present an analysis of the appropriate choice of the parameter coupling the double- and single-layer potential in order to minimize the condition number of the integral operator.This research was carried out while the second author was visiting the University of Göttingen on a DAAD-stipendium     

Invariance principles for von Mises and U-statistics

Summary The almost sure approximation of von Mises-statistics and U-statistics by appropriate stochastic integrals with respect to Kiefer processes is obtained. In general these integrals are non-Gaussian processes. As applications we get almost sure versions for the estimator of the variance and for the ²-test of goodness of fit.This work was done while the last author was a visiting professor at the Institut für Mathematische Stochastik at the University of Göttingen during the Spring of 1982. He thanks the Institut and its members for their hospitality     

The Third Problem for the Laplace Equation with a Boundary Condition from L p

The third problem for the Laplace equation is studied on an open set with Lipschitz boundary. The boundary condition is in L_p and it is fulfilled in the sense of the nontangential limit. The existence and the uniqueness of a solution is proved and the solution is expressed in the form of a single layer potential. For domains with C¹ boundary the explicit solution of the problem is calculated.     

Near-optimal data-independent point locations for radial basis function interpolation

The goal of this paper is to construct data-independent optimal point sets for interpolation by radial basis functions. The interpolation points are chosen to be uniformly good for all functions from the associated native Hilbert space. To this end we collect various results on the power function, which we use to show that good interpolation points are always uniformly distributed in a certain sense. We also prove convergence of two different greedy algorithms for the construction of near-optimal sets which lead to stable interpolation. Finally, we provide several examples. AMS subject classification 41A05, 41063, 41065, 65D05, 65D15This work has been done with the support of the Vigoni CRUI-DAAD programme, for the years 2001/2002, between the Universities of Verona and Göttingen.     

A Priori Error Estimates of Crank–Nicolson Finite Volume Element Method for a Hyperbolic Optimal Control Problem

In this article, a Crank–Nicolson linear finite volume element scheme is developed to solve a hyperbolic optimal control problem. We use the variational discretization technique for the approximation of the control variable. The optimal convergent order O(h² + k²) is proved for the numerical solution of the control, state and adjoint‐state in a discrete L²‐norm. To derive this result, we also get the error estimate (convergent order O(h² + k²)) of Crank–Nicolson finite volume element approximation for the second‐order hyperbolic initial boundary value problem. Numerical experiments are presented to verify the theoretical results.© 2016 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 32: 1331–1356, 2016     

Local mesh refinement for the discretization of Neumann boundary control problems on polyhedra

This paper deals with optimal control problems constrained by linear elliptic partial differential equations. The case where the right‐hand side of the Neumann boundary is controlled, is studied. The variational discretization concept for these problems is applied, and discretization error estimates are derived. On polyhedral domains, one has to deal with edge and corner singularities, which reduce the convergence rate of the discrete solutions, that is, one cannot expect convergence order two for linear finite elements on quasi‐uniform meshes in general. As a remedy, a local mesh refinement strategy is presented, and a priori bounds for the refinement parameters are derived such that convergence with optimal rate is guaranteed. As a by‐product, finite element error estimates in the H¹(Ω)‐norm, L²(Ω)‐norm and L²(Γ)‐norm for the boundary value problem are obtained, where the latter one turned out to be the main challenge. Copyright © 2015 John Wiley & Sons, Ltd.     

The three dimensional inverse scattering problem for acoustic waves

We consider the inverse scattering problem for an acoustically soft obstacle in R³. By assuming a priori that the unknown scattering obstacle is starlike and has its boundary lying in a compact family of Hölder continuously differentiable surfaces, it is shown that an optimal solution can be constructed which depends continuously on the measured far field data. Remarks are made on the numerical approximation of the optimal solution.     

An existence and uniqueness result of a nonlinear two-dimensional elliptic boundary value problem

We consider a boundary value problem for the generalized two-dimensional flow equation Δφ = Δφ · h for h a C^α vector field, where the speed is prescribed on a part of the boundary. By using Bers theory combined with elliptic operator theory in nonsmooth domains, we show existence and uniqueness of a C^2,α solution with nonvanishing gradient, and we find positive lower and upper bounds for |Δφ| along with C^2,α estimates of φ, in terms of the C^α and L^∞ norms of h. ©1995 John Wiley & Sons, Inc.     

Piecewise optimal distributed Controls for 2D Boussinesq equations

In this article, we study the dynamics of a piecewise (in time) distributed optimal control problem for the Boussinesq equations which model velocity tracking over time coupled to thermal dynamics. We also study the dynamics of semidiscrete approximation of this problem. We prove that the rates of velocity tracking coupled to thermal dynamics are exponential and that the difference between the solution of the semi‐discrete piecewise optimal control problem and the desired states in L² and H¹ norms decay to zero exponentially as n→∞. Copyright © 2000 John Wiley & Sons, Ltd.     

Über Lambertsche Reihen

Ohne ZusammenfassungHans Späth ist am 27. November 1928 wenige Stunden vor der Vollendung seiner Habilitation in Göttingen unerwartet verstorben. Unsere Wissenschaft verliert in ihm einen ungewöhnlich reich begabten Forscher, dessen bisherige Leistungen zu den schönsten Hoffnungen berechtigten. Trotz seiner Jugend—er wurde am 8. September 1901 zu Haigerloch geboren—liegt schon eine größere Anzahl wertvoller Arbeiten von ihm vor, andere waren im Druck oder fanden sich unter seinen nachgelassenen Papieren. Alle diese Arbeiten sind aus dem Trieb entstanden, die mathematischen Tatsachen nicht einfach zu übernehmen, sondern durch selbständiges Nachdenken sich zu erobern. So ist er unbefangen und mit gutem Erfolg auch an längst erledigte Fragen wie die Transzendenz vone und , die Irreduzibilität der Kreistelungsgleichung u. a. herangegangen und hat dort neue Wege und neue Resultate gefunden.—Nach Ableistung eines Referendarjahres und nach seiner Promotion (Göttingen 1927) war er Assistent in Tübingen; im Oktober 1928 ging er nach Göttingen, um sich dort zu habilitieren. An allen Stellen, an denen er gewesen, gedenkt man gerne dieses wertvollen Menschen, der der mathematischen Wissenschaft noch viel hätte sein können.     

Optimal intensity control of a multi-class queue

We study a mixed problem of optimal scheduling and input and output control of a single server queue with multi-classes of customers. The model extends the classical optimal scheduling problem by allowing the general point processes as the arrival and departure processes and the control of the arrival and departure intensities. The objective of our scheduling and control problem is to minimize the expected discounted inventory cost over an infinite horizon, and the problem is formulated as an intensity control. We find the well-knownc is the optimal solution to our problem.Supported in part by NSF under grant ECS-8658157, by ONR under contract N00014-84-K-0465, and by a grant from AT&T Bell Laboratories.The work was done while the author was a postdoctoral fellow in the Division of Applied Sciences, Harvard University, Cambridge, Massachusetts 02138.     

Weak solutions of the Vlasov–Poisson initial boundary value problem

This paper deals with existence results for a Vlasov-Poisson system, equipped with an absorbing-type law for the Vlasov equation and a Dirichlet-type boundary condition for the Poisson part. Using the ideas of Lions and Perthame [21], we prove the existence of a weak solution having good L^p estimates for moment and electric field, by a good control on the higher moments of the initial data. As an application, we establish a homogenization result in the Hilbertian framework for this type of problem in non-homogeneous media, following the work by Alexandre and Hamdache [2] for general kinetic equations, and Cioranescu and Mural [11] for the Laplace problem.     

Modified quasilinearization algorithm for optimal control problems with nondifferential constraints and general boundary conditions

This paper considers the numerical solution of the problem of minimizing a functionalI, subject to differential constraints, nondifferential constraints, and general boundary conditions. It consists of finding the statex(t), the controlu(t), and the parameter so that the functionalI is minimized while the constraints are satisfied to a predetermined accuracy.The modified quasilinearization algorithm (MQA) is extended, so that it can be applied to the solution of optimal control problems with general boundary conditions, where the state is not explicitly given at the initial point.The algorithm presented here preserves the MQA descent property on the cumulative error. This error consists of the error in the optimality conditions and the error in the constraints.Three numerical examples are presented in order to illustrate the performance of the algorithm. The numerical results are discussed to show the feasibility as well as the convergence characteristics of the algorithm.This work was supported by the Electrical Research Institute of Mexico and by CONACYT, Consejo Nacional de Ciencia y Tecnologia, Mexico City, Mexico.     

Finite element solution of vector Poisson equation with a coupling boundary condition

The vector Poisson equation is sometimes supplemented by conditions that include the specification of the boundary value of the divergence of the unknown. A rigorous analysis of such a vector Poisson problem and uncoupled solution methods have been presented for domains of C^1,1 and Lipschitz regularity in [1] and [2], respectively. In this work, the finite element approximation of the two uncoupled solution methods is studied, and a convergence analysis of the numerical schemes is provided together with some numerical results. © 2000 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 16: 71–83, 2000     

Optimal Control of the Obstacle for a Parabolic Variational Inequality

An optimal control problem for a parabolic obstacle variational inequality is considered. The obstacle in L²(0, T; H²(Ω) ∩ H¹₀(Ω)) with ψ_t ∈ L²(Q) is taken as the control, and the solution to the obstacle problem is taken as the state. The goal is to find the optimal control so that the state is close to the desired profile while the norm of the obstacle is not too large. Existence and necessary conditions for the optimal control are established.     

Analytic Solutions of L∞ Optimal Control Problems for the Wave Equation

There are very few results about analytic solutions of problems of optimal control with minimal L norm. In this paper, we consider such a problem for the wave equation, where the derivative of the state is controlled at both boundaries. We start in the zero position and consider a problem of exact control, that is, we want to reach a given terminal state in a given finite time. Our aim is to find a control with minimal L norm that steers the system to the target.We give the analytic solution for certain classes of target points, for example, target points that are given by constant functions. For such targets with zero velocity, the analytic solution has been given by Bennighof and Boucher in Ref. 1.     

A STUDY OF THE PROFITABILITY FOR AN OPTIMAL CONTROL PROBLEM WHEN THE SIZE OF THE DOMAIN CHANGES

ABSTRACT. In this work we consider the increase in benefit for a control problem when the size of domain increases. Our control problem involves the study of the profitability of a biological growing species whose growth is confined to a bounded domain Ω? R^N and is modeled by a logistic elliptic equation with different boundary conditions (Dirichlet or Neumann). The payoff-cost functional considered, J, is of quadratic type. We prove that, under Dirichlet boundary conditions, the optimal benefit (sup J) increases when the domain ? increases. This is not true under Neumann boundary conditions.