Feedback exact null controllability for unbounded control problems in Hilbert space

Terminal constraint optimal control problems with unbounded control operators are considered. It is shown that the optimal solutions can be represented in a feedback form via a solution of an appropriate Riccati equation. In particular, it is proved that, for systems described by partial differential equations with infinite speed of propagation, boundary exact null controllability can be realized in feedback form.This work was partially supported by the National Science Foundation, Grant No. DMS-89-02811, and by the Air Force Office of Scientific Research, Grant No. AFOSR-89-0511 DEF.     

Occupation times in systems of null recurrent Markov processes

Summary We study asymptotic properties of differences of occupation times for infinite systems of noninteracting Markovian particles. Under a suitable normalisation we prove convergence in law to a nondegerate Gaussian field. We also obtain large deviations properties. These results generalise previous results obtained separately by both authors.Supported in part by the Office of Graduate Studies and Research (1990), University of Maryland and by NSF Grant No. DMS 9207928Supported in part by the Fonds Institutionnel de Recherche, Université du Québec à Trois-Rivières and by the Natural Sciences and Engineering Research Council of Canada, Grant No. OGP0042137     

On distributionally chaotic and null systems

By a topological dynamical system, we mean a pair (X,f), where X is a compactum and f is a continuous self-map on X. A system is said to be null if its topological sequence entropies are zero along all strictly increasing sequences of natural numbers. We show that there exists a null system which is distributionally chaotic. This system admits open distributionally scrambled sets, and its collection of all maximal distributionally scrambled sets has the same cardinality as the collection of all subsets of the phase space. Finally such system can even exist on continua.     

Continuity of the null space basis and constrained optimization

Many constrained optimization algorithms use a basis for the null space of the matrix of constraint gradients. Recently, methods have been proposed that enable this null space basis to vary continuously as a function of the iterates in a neighborhood of the solution. This paper reports results from topology showing that, in general, there is no continuous function that generates the null space basis of all full rank rectangular matrices of a fixed size. Thus constrained optimization algorithms cannot assume an everywhere continuous null space basis. We also give some indication of where these discontinuities must occur. We then propose an alternative implementation of a class of constrained optimization algorithms that uses approximations to the reduced Hessian of the Lagrangian but is independent of the choice of null space basis. This approach obviates the need for a continuously varying null space basis.Research supported by NSF grant MCS 81-15475 and DCR-8403483Research supported by ARO contracts DAAG 29-81-K-0108 and DAAG 29-84-K-0140     

Methods for computing the local null controllability time of linear control systems

We present methods for computing the local null controllability time in the class of linear control systems (not necessarily autonomous) whose families of null controllability sets are monotone with respect to inclusion.     

Nonparametric tests for bounds on the derivative of a regression function

We consider two tests of the null hypothesis that the k-th derivative of a regression function is uniformly bounded by a specified constant. These tests can be used to study the shape of the regression function. For instance, we can test for convexity of the regression function by setting k=2 and the constant equal to zero. Our tests are based on k-th order divided difference of the observations. The asymptotic distribution and efficacies of these tests are computed and simulation results presented.Research supported by Natural Sciences and Engineering Research Council of Canada Grant OGP0007969.Research supported by National Science Foundation Grant DMS-9306738.     

Some Geometrical Aspects of Efficient Points in Vector Optimization

We study efficient point sets in terms of extreme points, positive support points and strongly positive exposed points. In the case when the ordering cone has a bounded base, we prove that the efficient point set of a weakly compact convex set is contained in the closed convex hull of its strongly positive exposed points, thereby extending the Phelps theorem. We study also the density of positive proper efficient point sets. This research was supported by a Central Research Grant of Hong Kong Polytechnic University, Grant G-T 507. Research of the first author was also supported by the National Natural Science Foundation of P.R. China, Grant 10361008, and the Natural Science Foundation of Yunnan Province, China, Grant 2003A002M. Research of the second author was also supported by the Natural Science Foundation of Chongqing. Research of the third author was supported by a research grant from Australian Research Counsil.     

Perfectly meager sets and universally null sets

We will show that there is no example of a set distinguishing between universally null and perfectly meager sets.

     

The null field equations for flexural oscillations of elastic plates

A null field method is constructed to solve the exterior Dirichlet, Neumann, and Robin boundary value problems associated with the high‐frequency harmonic oscillations of Mindlin‐type plates. The case of an infinite plate with a bounded elastic inclusion is also considered. Additionally, the completeness of certain sets of wavefunctions is investigated. Copyright © 2012 John Wiley & Sons, Ltd.     

Size of subsets of groups and Haar null sets

This is a study of several notions of size of subsets of groups. The first part (sections 3–5) concerns a purely algebraic setting with the underlying group discrete. The various notions of size considered there are similar to each other in that each of them assesses the size of a set using a family of measures and translations of the set; they differ in the type of measures used and the type of translations allowed. The way these various notions relate to each other is tightly and, perhaps, unexpectedly connected with the algebraic structure of the group. An important role is played by amenable, ICC (infinite conjugacy class), and FC (finite conjugacy class) groups.The second part of the paper (section 6), which was the original motivation for the present work, deals with a well-studied notion of smallness of subsets of Polish, not necessarily locally compact, groups – Haar null sets. It contains applications of the results from the first part in solving some problems posed by Christensen and by Mycielski. These applications are the first results detecting connections between properties of Haar null sets and algebraic properties (amenability, FC) of the underlying group.Received: October 2003 Revision: January 2004 Accepted: January 2004     

Suggested research topics in sensitivity and stability analysis for semi-infinite programming problems

We suggest several important research topics for semi-infinite programs whose problem functions and index sets contain parameters that are subject to perturbation. These include optimal value and optimal solution sensitivity and stability properties and penalty function approximation techniques. The approaches proposed are a natural carryover from parametric nonlinear programming, with emphasis on practical applicability and computability.Research supported by National Science Foundation Grant SES 8722504 and Grant ECS-86-19859 and Grant N00014-89-J-1537, Office of Naval Research.     

Some remarks on Teichmüller spaces and modular groups

We study the fixed point sets of subgroups of modular groups acting on Teichmiiller spaces.     

Countable homogeneous partially ordered sets

It is shown that there are only countably many countable homogeneous partially ordered sets, thereby affirming a conjecture of Henson [2]. A classification of these partially ordered sets is given. Research partially supported by NSF Grant MCS76-07258. Presented by B. Jónsson.     

Minimizing pseudoconvex functions on convex compact sets

An algorithm is presented which minimizes continuously differentiable pseudoconvex functions on convex compact sets which are characterized by their support functions. If the function can be minimized exactly on affine sets in a finite number of operations and the constraint set is a polytope, the algorithm has finite convergence. Numerical results are reported which illustrate the performance of the algorithm when applied to a specific search direction problem. The algorithm differs from existing algorithms in that it has proven convergence when applied to any convex compact set, and not just polytopal sets.This research was supported by the National Science Foundation Grant ECS-85-17362, the Air Force Office Scientific Research Grant 86-0116, the Office of Naval Research Contract N00014-86-K-0295, the California State MICRO program, and the Semiconductor Research Corporation Contract SRC-82-11-008.     

On null Lagrangians

We consider multiple-integral variational problems where the Lagrangian function, defined on a frame bundle, is homogeneous. We construct, on the corresponding sphere bundle, a canonical Lagrangian form with the property that it is closed exactly when the Lagrangian is null. We also provide a straightforward characterization of null Lagrangians as sums of determinants of total derivatives. We describe the correspondence between Lagrangians on frame bundles and those on jet bundles: under this correspondence, the canonical Lagrangian form becomes the fundamental Lepage equivalent. We also use this correspondence to show that, for a single-determinant null Lagrangian, the fundamental Lepage equivalent and the Carathéodory form are identical.     

Local Linear Convergence for Alternating and Averaged Nonconvex Projections

The idea of a finite collection of closed sets having “linearly regular intersection” at a point is crucial in variational analysis. This central theoretical condition also has striking algorithmic consequences: in the case of two sets, one of which satisfies a further regularity condition (convexity or smoothness, for example), we prove that von Neumann’s method of “alternating projections” converges locally to a point in the intersection, at a linear rate associated with a modulus of regularity. As a consequence, in the case of several arbitrary closed sets having linearly regular intersection at some point, the method of “averaged projections” converges locally at a linear rate to a point in the intersection. Inexact versions of both algorithms also converge linearly. Research of A.S. Lewis supported in part by National Science Foundation Grant DMS-0504032. Research of D.R. Luke supported in part by National Science Foundation Grant DMS-0712796.     

The central limit problem for,infinite products of,and Lévy processes of renewal sequences

Summary Necessary and sufficient conditions are obtained for: (i) convergence of row products from a null triangular array of renewal sequences to a particular renewal sequence and (ii) convergence of an infinite product of renewal sequences to a non-trivial limit. These products correspond to intersections of regenerative phenomena of integers. Lévy processes of such regenerative phenomena are constructed.Research partially supported by National Science Foundation Grant MCS 78-01168     

Criteria for quasi-convexity and pseudo-convexity: Relationships and comparisons

A first order criterion for pseudo-convexity and second order criteria for quasi-convexity and pseudo-convexity are given for twice differentiable functions on open convex sets. The relationships between these second order criteria and other known criteria are also analysed. Finally, the numbers of operations required to verify these criteria are calculated and compared.This research was supported by the Natural Sciences and Engineering Research Council Canada, Grant no. A8312, and by NATO Research Grant no. 1934.     

Orthogonal projection regularization operators

Tikhonov regularization often is applied with a finite difference regularization operator that approximates a low-order derivative. This paper proposes the use of orthogonal projections as regularization operators, e.g., with the same null space as commonly used finite difference operators. Applications to iterative and SVD-based methods for Tikhonov regularization are described. Truncated iterative and SVD methods are also considered. Research of L. Reichel was supported in part by an OBR Research Challenge Grant. Research of F. Sgallari was supported in part by PRIN 2004 grant 2004014411-005.