Approximation of infinite delay and Volterra type equations

Summary We consider a class of infinite delay equations of neutral type which includes Volterrra type integral und integrodifferential equations. Using abstract approximation results (cf. Trotter-Kato-type) for strongly continuous semigroups we develop an approximation scheme which is based on approximation of the system state by Laguerre (and Legendre) polynomials. Numerical examples demonstrate the feasibility of the scheme and show infinite order convergence for smooth data.Supported in part by the Air Force Office of Scientific Research under Contracts AFORS-84-0398 and AFORS-85-0303 and the National Aeronautics and Space Administration under NASA Grant NAG-1-517 and by NSF under Grant UINT-8521208Supported in part by the Air Force Office of Scientific Research under Contract AFORS-84-0398 and in part by the Fonds zur Förderung der wissenschaftlichen Forschung, Austria, under project No. S3206     

The synthesis of optimal controls for linear,time-optimal problems with retarded controls

Optimization problems involving linear systems with retardations in the controls are studied in a systematic way. Some physical motivation for the problems is discussed. The topics covered are: controllability, existence and uniqueness of the optimal control, sufficient conditions, techniques of synthesis, and dynamic programming. A number of solved examples are presented.This research was supported in part by the National Aeronautics and Space Administration under Grant No. NGL-40-002-015, in part by the Air Force Office of Scientific Research under Grant No. AF-AFOSR-693-67B, and in part by the National Science Foundation under Grant No. GP-15132.     

An adaptive numerical method for solving linear Fredholm integral equations of the first kind

A general procedure is presented for numerically solving linear Fredholm integral equations of the first kind. The approximate solution is expressed as a continuous piecewise linear (spline) function. The method involves collocation followed by the solution of an appropriately scaled stabilized linear algebraic system. The procedure may be used iteratively to improve the accuracy of the approximate solution. Several numerical examples are given.Supported in part by the Office of Naval Research under Contract No. NR 044-457.Supported in part by the National Science Foundation under Grant No. GJ-31827.     

Inverse parabolic problems and discrete orthogonality

Summary This paper considers a discrete sampling scheme for the approximate recovery of initial data for one dimensional parabolic initial boundary value problems on a bounded interval. To obtain a given approximate, data is sampled at a single time and at a finite number of spatial points. The significance of this inversion scheme is the ability to accurately predict the error in approximation subject to choice of sample time and spatial sensor locations. The method is based on a discrete analogy of the continuous orthogonality for Sturm-Liouville systems. This property, which is of independent mathematical interest, is the notion of discrete orthogonal systems, which loosely speaking provides an exact (or approximate) Gauss-type quadrature for the continuous biorthogonality conditions.Supported in part by NSF Grant #DMS8905-344. Texas Advanced Research Program Grant #0219-44-5195 and AFOSR Grant #88-0309Visiting at Texas Tech University, Fall 1989Supported in part by NSF Grant #DMS8905-344, NSA grant #MDA904-85-H009 and NASA Grant #NAQ2-89     

Approximation of nonlinear functional differential equation control systems

We develop a general approximation framework for use in optimal control problems governed by nonlinear functional differential equations. Our approach entails only the use of linear semigroup approximation results, while the nonlinearities are treated as perturbations of a linear system. Numerical results are presented for several simple nonlinear optimal control problem examples.This research was supported in part by the US Air Force under Contract No. AF-AFOSR-76-3092 and in part by the National Science Foundation under Grant No. NSF-GP-28931x3.     

Extrapolation methods for dynamic partial differential equations

Summary Several extrapolation procedures are presented for increasing the order of accuracy in time for evolutionary partial differential equations. These formulas are based on finite difference schemes in both the spatial and temporal directions. One of these schemes reduces to a Runge-Kutta type formula when the equations are linear. On practical grounds the methods are restricted to schemes that are fourth order in time and either second, fourth or sixth order in space. For hyperbolic problems the second order in space methods are not useful while the fourth order methods offer no advantage over the Kreiss-Oliger method unless very fine meshes are used. Advantages are first achieved using sixth order methods in space coupled with fourth order accuracy in time. The averaging procedure advocated by Gragg does not increase the efficiency of the scheme. For parabolic problems severe stability restrictions are encountered that limit the applicability to problems with large cell Reynolds number. Computational results are presented confirming the analytic discussions.This report was prepared as a result of work performed under NASA Contract No. NAS1-14101 while the author was in residence at ICASE, NASA Langley Research Center, Hampton, VA 23665, USA, and under ERDA Grant No. E(11-1)-3077-III while he was at Courant Institute of Mathematical Sciences, New York, NY 10012, USA     

Second-order finite element approximations and a posteriori error estimation for two-dimensional parabolic systems

Summary We discuss an adaptive local refinement finite element method of lines for solving vector systems of parabolic partial differential equations on two-dimensional rectangular regions. The partial differential system is discretized in space using a Galerkin approach with piecewise eight-node serendipity functions. An a posteriori estimate of the spatial discretization error of the finite element solution is obtained using piecewise fifth degree polynomials that vanish on the edges of the rectangular elements of a grid. Ordinary differential equations for the finite element solution and error estimate are integrated in time using software for stiff differential systems. The error estimate is used to control a local spatial mesh refinement procedure in an attempt to keep a global measure of the error within prescribed limits. Examples appraising the accuracy of the solution and error estimate and the computational efficiency of the procedure relative to one using bilinear finite elements are presented.Dedicated to Prof. Ivo Babuka on the occasion of his 60th birthdayThis research was partially supported by the U.S. Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR 85-0156 and the U.S. Army Research Office under Contract Number DAAL 03-86-K-0112     

The treatment of nonhomogeneous Dirichlet boundary conditions by thep-version of the finite element method

Summary The paper addresses the problem of the implementation of nonhomogeneous essential Dirichlet type boundary conditions in thep-version of the finite element method.Partially supported by the Office of Naval Research under Grant N-00014-85-K-0169Research partially supported by the Air Force Office of Scientific Research, Air Force Systems Command, USAF, under Grant Number AFOSR 85-0322     

Solution manifolds and submanifolds of parametrized equations and their discretization errors

Summary The paper concerns solution manifolds of nonlinear parameterdependent equations (1)F(u, )=y₀ involving a Fredholm operatorF between (infinite-dimensional) Banach spacesX=Z× andY, and a finitedimensional parameter space . Differntial-geometric ideas are used to discuss the connection between augmented equations and certain onedimensional submanifolds produced by numerical path-tracing procedures. Then, for arbitrary (finite) dimension of , estimates of the error between the solution manifold of (1) and its discretizations are developed. These estimates are shown to be applicable to rather general nonlinear boundaryvalue problems for partial differential equations.This work was in part supported by the U.S. Air Force Office of Scientific Research under Grant 80-0176, the National Science Foundation under Grant MCS-78-05299, and the Office of Naval Research under Contract N-00014-80-C-0455     

The multiplier method of Hestenes and Powell applied to convex programming

For nonlinear programming problems with equality constraints, Hestenes and Powell have independently proposed a dual method of solution in which squares of the constraint functions are added as penalties to the Lagrangian, and a certain simple rule is used for updating the Lagrange multipliers after each cycle. Powell has essentially shown that the rate of convergence is linear if one starts with a sufficiently high penalty factor and sufficiently near to a local solution satisfying the usual second-order sufficient conditions for optimality. This paper furnishes the corresponding method for inequality-constrained problems. Global convergence to an optimal solution is established in the convex case for an arbitrary penalty factor and without the requirement that an exact minimum be calculated at each cycle. Furthermore, the Lagrange multipliers are shown to converge, even though the optimal multipliers may not be unique.This work was supported in part by the Air Force Office of Scientific Research under Grant No. AF-AFOSR-72-2269.     

Conditions for finite moments of waiting times inG/G/1 queues

We find conditions for E(W ) to be finite whereW is the stationary waiting time random variable in a stableG/G/1 queue with dependent service and inter-arrival times.Supported in part by KBN under grant 640/2/9, and at the Center for Stochastic Processes, Department of Statistics at the University of North Carolina Chapel Hill by the Air Force Office of Scientific Research Grant No. 91-0030 and the Army Research Office Grant No. DAAL09-92-G-0008.     

Existence theorems for multidimensional Lagrange problems

Existence theorems are proved for multidimensional Lagrange problems of the calculus of variations and optimal control. The unknowns are functions of several independent variables in a fixed bounded domain, the cost functional is a multiple integral, and the side conditions are partial differential equations, not necessarily linear, with assigned boundary conditions. Also, unilateral constraints may be prescribed both on the space and the control variables. These constraints are expressed by requiring that space and control variables take their values in certain fixed or variable sets wich are assumed to be closed but not necessarily compact.This research was partially supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-942-65.     

Integration of the primer vector in a central force field

This paper examines the primer vector which governs optimal solutions for orbital transfer when the central force field has a more general form than the usual inverse-square-force law. Along a null-thrust are that connects two successive impulses, the two sets of state and adjoint equations are decoupled. This allows the reduction of the problem to the integration of a linear first-order differential equation, and hence the solution of the optimal coasting are in the most general central force field can be obtained by simple quadratures. Immediate applications of the results can be seen in solving problems of escape in the equatorial plane of an oblate planet, satellite swing by, or station keeping around Lagrangian points in the three-body problem.This work has been sponsored by the Air Force Office of Scientific Research, under Grant No. AF-AFOSR-71-2129.     

Existence,uniqueness and continuity of solutions of integral equations. An addendum

Summary This note corrects au error in the continuous dependence theorem of an earlier paper on the subject stated in the title. Here it is shows that the topology of one of the spaces of functions can be modified in order to obtain the desired continuity results. This research was support in part by the National Aeronautics Space Administration under Grant No. NGL-40-002-015, and in part by the U. S. Air Force under Grant No. AF-AFOSR-67-0693A. This research was supported in part by the National Science Foundation under Grant No. GP-3904 and GP-7041 and in part by the U.S. Army under Grant No. D1-31-124-ARO-D-265. Entrata in Redazione il 10 aprile 1970.     

On the existence of noncontinuous irreducible representations of a locally compact group

It is shown that a non-discrete locally compact group has non-continuous irreducible representations.Research sponsored by the Air Force Office of Scientific Research under Grant AF-AFOSR-903-67 and by the National Aeronautics and Space Administration under Grant NGR 44-004-042 and Grant NSG-269-62.     

On a new computing technique in optimal control and its application to minimal-time flight profile optimization

A new constructive approach to optimization problems for dynamic systems that avoids having to solve dynamic equations and has both computational and theoretical advantages is presented. On the theoretical level, it yields a simple direct method for the Hamiltonian and the maximum principle and a constructive derivation of the Lagrange parameters. The computational aspects are illustrated by results obtained for a minimal-time flight profile optimization problem.This research was supported in part by the Applied Mathematics Division, Office of Scientific Research, Office of Aerospace Research, United States Air Force, under Grant No. 69-1408. All the calculations presented in this paper were carried out under the direction of Mr. L. W. Taylor, NASA Flight Research Center, Edwards, California, who also contributed many essential ideas throughout this work. This paper was presented at the Colloquium on Optimization held in Novosibirsk, USSR, 1968.     

Anchoring conditions for the sequential gradient-restoration algorithm and the modified quasilinearization algorithm for optimal control problems with bounded state

In Refs. 1–2, the sequential gradient-restoration algorithm and the modified quasilinearization algorithm were developed for optimal control problems with bounded state. These algorithms have a basic property: for a subarc lying on the state boundary, the state boundary equations are satisfied at every iteration, if they are satisfied at the beginning of the computational process. Thus, the subarc remains anchored on the state boundary. In this paper, the anchoring conditions employed in Refs. 1–2 are derived.This research was supported by the Office of Scientific Research, Office of Aerospace Research, United States Air Force, Grant No. AF-AFOSR-72-2185.     

On finite element subspaces over quadrilateral and hexahedral meshes for incompressible viscous flow problems

Summary Optimal rates of convergence for the approximate solution of the stationary Stokes equations are obtained for finite element schemes which use piecewise constants to approximate the pressure and piecewise linear or piecewise bilinear or trilinear polynomials to approximate the velocity over fairly general quadrilateral or hexahedral meshes.This research was supported by NSF Grant MC80-16532     

Hopf-algebraic structure of combinatorial objects and differential operators

A Hopf-algebraic structure on a vector space which has as basis a family of trees is described, and we survey some applications of this structure to combinatorics and to differential operators. Some possible future directions for this work are indicated. Supported in part by NASA Grant NAG 2-513. Supported in part by NSF Grant DMS 870-1085.     

Distribution of product and quotient of Bessel function variates

Summary In this paper, the technique of Mellin transforms is employed to obtain the distribution of the product and quotient of two independent Bessel function random variables. Two different types of Bessel function variates are considered. The results are then specialized to yield a wide variety of classical distributions of importance in applications. Supported by Air Force Office of Scientific Research Grant AF-AFOSR-68-1411.