A convergence result for a sequence of distributed-parameter optimal control problems

In this paper, we consider a sequence of abstract optimal control problems by allowing the cost integrand, the partial differential operator, and the control constraint set all to vary simultaneously. Using the notions of -convergence of functions,G-convergence of operators, and Kuratowski-Mosco convergence of sets, we show that the values of the approximating problems converge to that of the limit problem. Also we show that a convergent sequence of optimal pairs for the approximating problems has a limit which is optimal for the limit problem. A concrete example of parabolic optimal control problems is worked out in detail.This research was supported by NSF Grant No. DMS-88-02688.     

Γ-convergence and optimal control problems

In this paper, we give some applications ofG-convergence and -convergence to the study of the asymptotic limits of optimal control problems. More precisely, given a sequence (P _h) of optimal control problems and a control problem (P), we determine some general conditions, involvingG-convergence and -convergence, under which the sequence of the optimal pairs of the problems (P _h) converges to the optimal pair of problem (P).The authors wish to thank Professor E. De Giorgi for many stimulating discussions.     

Asymptotic behavior of optimal solutions to control problems for systems described by differential inclusions corresponding to partial differential equations

In the paper, we consider differential inclusions related to PDEs of parabolic type and some control problems with integral cost functionals associated to them. Given a sequence of such problems, we investigate first the asymptotic behavior of solution sets (mild solutions or more precisely selection-trajectory pairs) for differential inclusions, and we get some semicontinuity or continuity results (Kuratowski convergence of solution sets). Then, we prove the -convergence of cost functionals, related to the above Kuratowski convergence of solution sets. Finally, applying the Buttazzo-Dal Maso abstract scheme, based on the sequential -convergence, we obtain results concerning the asymptotic behavior (hence, also stability results) for optimal solutions to control problems as well as the convergence of minimal values.The authors would like to thank Professors G. Dal Maso and S. Spagnolo for helpful conversations.This work was done when the first author was visiting ICTP and ISAS in Trieste in 1990/91.     

Control problems for parabolic and hyperbolic equations via the theory ofG and Γ convergence

Summary A sequence of optimal control problems for systems governed by PDE'sis considered. The parameter (index of an element of the sequence) appears in the cost functionals which have integral form, as well as in the state equations which are of parabolic or hyperbolic type. It is proved that, under some -convergence of the cost functionals and some convergence of the indicator functions of sets of admissible solutions, the optimal solutions exist and converge to an optimal solution of the limit problem.     

Convergence of Discrete Dirichlet Forms to Continuous Dirichlet Forms on Fractals

It is well known that a Dirichlet form on a fractal structure can be defined as the limit of an increasing sequence of discrete Dirichlet forms, defined on finite subsets which fill the fractal. The initial form is defined on V ⁽⁰⁾, which is a sort of boundary of the fractal, and we have to require that it is an eigenform, i.e., an eigenvector of a particular nonlinear renormalization map for Dirichlet forms on V ⁽⁰⁾. In this paper, I prove that, provided an eigenform exists, even if the form on V ⁽⁰⁾ is not an eigenform, the corresponding sequence of discrete forms converges to a Dirichlet form on all of the fractal, both pointwise and in the sense of -convergence (but these two limits can be different). The problem of -convergence was first studied by S. Kozlov on the Gasket.     

Fiber Bundles and Regular Approximation of Codimension-One Cycles

We derive an approximation of codimension-one integral cycles(and cycles modulo p) in a compact Riemannian manifold bymeans of piecewise regular cycles: we obtain both flat convergence andconvergence of the masses. The theorem is proved by using suitableprincipal bundles with a discrete group. As a byproduct, we give analternative proof of the main results, which does not use the regularitytheory for homology minimizers in a Riemannian manifold. This also givesa result of -convergence.     

Regulator convergence in commutativel-groups

In the theory of lattice ordered groups there are considered several types of convergence. In this work it is shown that for nets (r)-convergence is essentially stronger than (o)-convergence, while for sequences these notions are not comparable (as is known, in K-lineals, (r)-convergence for sequences as well as for nets is stronger than (o)-convergence); in K-groups (r)-convergence of sequences is stronger than (o)-convergence. (A sequence is considered (o)-convergent if it is compressed by monotone sequences to a common limit.)Translated from Matematicheskie Zametki, Vol. 3, No. 3, pp. 279–284, March, 1968.     

The existence of sensitive optimal policies in two multi-dimensional queueing models

Recently Dekker and Hordijk [3,4] introduced conditions for the existence of deterministic Blackwell optimal policies in denumerable Markov decision chains with unbounded rewards. These conditions include- uniform geometric recurrence.The-uniform geometric recurrence property also implies the existence of average optimal policies, a solution to the average optimality equation with explicit formula's and convergence of the value iteration algorithm for average rewards. For this reason, the verification of-uniform geometric convergence is also useful in cases where average and-discounted rewards are considered.On the other hand,-uniform geometric recurrence is a heavy condition on the Markov decision chain structure for negative dynamic programming problems. The verification of-uniform geometric recurrence for the Markov chain induced by some deterministic policy together with results by Sennott [14] yields the existence of a deterministic policy that minimizes the expected average cost for non-negative immediate cost functions.In this paper-uniform geometric recurrence will be proved for two queueing models: theK competing queues and the two centre open Jackson network with control of the service rates.The research of the author is supported by the Netherlands Organization for Scientific Research N.W.O.     

Application of the Augmented Lagrangian-SQP Method to Optimal Control Problems for the Stationary Burgers Equation

In this paper optimal control problems for the stationary Burgers equation are analyzed. To solve the optimal control problems the augmented Lagrangian-SQP method is applied. This algorithm has second-order convergence rate depending upon a second-order sufficient optimality condition. Using piecewise linear finite elements it is proved that the discretized augmented Lagrangian-SQP method is well-defined and has second-order rate of convergence. This result is based on the proof of a uniform discrete Babuka-Brezzi condition and a uniform second-order sufficient optimality condition.     

A blossoming approach to accuracy of the degree elevation process

It is known that the sequence of control polygons of a Bézier-De Casteljau curve or surface obtained by the degree elevation process converges towards the underlying curve and surface. The notion of blossoming or polar form associated with a polynomial allows to control the accuracy of this convergence and, as a by-product, to give a new and completer proof of convergence.     

On the relation between algebraic stability andB-convergence for Runge-Kutta methods

Summary This paper is concerned with the numerical solution of stiff initial value problems for systems of ordinary differential equations using Runge-Kutta methods. For these and other methods Frank, Schneid and Ueberhuber [7] introduced the important concept ofB-convergence, i.e. convergence with error bounds only depending on the stepsizes, the smoothness of the exact solution and the so-called one-sided Lipschitz constant . Spijker [19] proved for the case <0 thatB-convergence follows from algebraic stability, the well-known criterion for contractivity (cf. [1, 2]). We show that the order ofB-convergence in this case is generally equal to the stage-order, improving by one half the order obtained in [19]. Further it is proved that algebraic stability is not only sufficient but also necessary forB-convergence.This study was completed while this author was visiting the Oxford University Computing Laboratory with a stipend from the Netherlands Organization for Scientific Research (N.W.O.)     

Global convergence of Newton’s method on an interval

The solution of an equation f(x)= given by an increasing function f on an interval I and right-hand side , can be approximated by a sequence calculated according to Newtons method. In this article, global convergence of the method is considered in the strong sense of convergence for any initial value in I and any feasible right-hand side. The class of functions for which the method converges globally is characterized. This class contains all increasing convex and increasing concave functions as well as sums of such functions on the given interval. The characterization is applied to Keplers equation and to calculation of the internal rate of return of an investment project.An earlier version was presented at the Joint National Meeting of TIMS and ORSA, Las Vegas, May 7–9, 1990. Financial support from Økonomisk Forskningsfond, Bodø, Norway, is gratefully acknowledged. The author thanks an anonymous referee for helpful comments and suggestions.     

A Korovkin type approximation theorems via $$\mathcal{I}$$ -convergence

Using the concept of -convergence we provide a Korovkin type approximation theorem by means of positive linear operators defined on an appropriate weighted space given with any interval of the real line. We also study rates of convergence by means of the modulus of continuity and the elements of the Lipschitz class.     

Finite element analysis of transient electromagnetic scattering problems

In this paper, Newmark time-stepping scheme and edge elements are used to numerically solve the time-dependent scattering problem in a three-dimensional cavity. Finite element methods based on the variational formulation derived in [23] are considered. Due to the lack of regularity of _r , the existence and uniqueness of the discrete solutions and their convergence are proved by using the concept of collectively compact operators. An optimal convergence rate in the energy norm is also established.     

A linear quadratic optimal control problem with disturbances—An algebraic Riccati equation and differential games approach

A linear quadratic optimal control problem with coexisting initial and persistent disturbances is studied. Upper and lower values and relevant algebraic Riccati equations (ARE for short) are introduced. Various relations among these values are presented. The solvability of the resulting AREs is shown to be closely related to the solvability of the original optimal control problem. A formula is obtained for the solution of one of the AREs, which is of nonstandard form. Several unexpected features of the original problem are revealed from the standpoint of differential games. Some known results on the so-calledH optimal control problem are recovered.This work was partially supported by the Chinese NSF under Grant 19131050, the Chinese State Education Commission Science Foundation, the SEDC Foundation for Young Academics, and the Fok Ying Tung Education Foundation.     

μ-Statistically Convergent Function Sequences

In the present paper we are concerned with convergence in -density and -statistical convergence of sequences of functions defined on a subset D of real numbers, where is a finitely additive measure. Particularly, we introduce the concepts of -statistical uniform convergence and -statistical pointwise convergence, and observe that -statistical uniform convergence inherits the basic properties of uniform convergence.     

Convergence Properties of Dikin’s Affine Scaling Algorithm for Nonconvex Quadratic Minimization

We study convergence properties of Dikins affine scaling algorithm applied to nonconvex quadratic minimization. First, we show that the objective function value either diverges or converges Q-linearly to a limit. Using this result, we show that, in the case of box constraints, the iterates converge to a unique point satisfying first-order and weak second-order optimality conditions, assuming the objective function Hessian Q is rank dominant with respect to the principal submatrices that are maximally positive semidefinite. Such Q include matrices that are positive semidefinite or negative semidefinite or nondegenerate or have negative diagonals. Preliminary numerical experience is reported.     

Boundedly UC Spaces and Topologies on Function Spaces

A new interesting topology on graphs of partial maps is introduced. This topology can be considered as a natural extension to a non locally compact setting of former topologies defined by P. Brandi, R. Ceppitelli and K. Back, having applications in mathematical economics, differential equations and in the convergence of dynamic programming models. New characterizations of boundedly Atsuji spaces are given by the coincidence of and the topology τ _ucb of uniform convergence on bounded sets on C(X,Y) and by topological properties of .      

Soliton-Stripe Patterns in Charged Langmuir Monolayers

We consider a charged Langmuir monolayer problem where electrostatic interaction forces undulations in the molecular concentration of the monolayer. Using the -convergence theory in singular perturbative variational calculus, we prove the existence of soliton-stripe lamellar patterns as one-dimensional local minimizers of the free energy, which are characterized by sharp domain walls delineating fully segregated dense liquid and dilute gas regions of the monolayer.