Sensitivity interpretations of the costate function of optimal control

It is well known that, for optimal control problems with freeright endpoints, the costate function P() of the maximum principlehas a sensitivity interpretation in terms of the value functionV. Specifically, if () is an optimal state trajectory and h(t) is the hamiltonian function evaluated alongp(t), (t), and the optimal control, we have where V is the Clarke generalized gradient.Recently Subbotina has improved this result, replacing the Clarkegeneralized gradient by the tighter ‘strict differential’.We provide a simple proof of this refinement, weaken the hypothesesunder which it is known to apply, and extend it to cover alsoproblems involving right endpoint constraints.     

Reducing Subspaces for a Class of Multiplication Operators

Let D be the open unit disk in the complex plane C. The Bergmanspace is the Hilbert space of analytic functions f in D such that where dA is the normalized area measure on D. If are two functions in , then the inner product of f and g is given by We study multiplication operators on induced by analytic functions. Thus for H (D), the space ofbounded analytic functions in D, we define by It is easy to check that M is a bounded linear operator on with ||M||=||||=sup{|(z)|:zD}.     

A Twisted Sampling Theorem

A generalized sampling theorem is proved for a set of functionsf that are well behaved in the frequency domain. Specifically,let W be a positive bounded integrable function, let be a reconstructionfunction, and let A denote the operation of multiplication inthe frequency domain by a bounded function A(s). Then coefficientsh_n (–<n<) are determined to minimize the maximalL²-error in an approximation scheme for functions f ranging over the set is the Fourier of f. Formulae are given for the minimal obtainablemaximal error, and the optimal choice (choices) of h_n (–<n<)is characterized through its discrete Fourier transform .     

Link Polynomials of Higher Order

In this paper, we study certain polynomial invariants of links(singular or non-singular) that are related to the Homfly polynomialand Vassiliev's invariants. The Homfly polynomial H_L [3] (alsoknown as the Flypmoth polynomial) satisfies the well-known skeinrelation The Vassiliev invariants [1, 2] (of order 1) satisfy the relations and The invariants that we study satisfy the skein relations      

Maximum Precision in Elliott-Donaldson Formulae

This paper considers the approximation of by quadratures which are constructed inthe manner described by Elliott & Donaldson. Results areestablished for choosing the nodes so that the quadrature is exact on as wide a range of positive and negativepowers of z as possible.     

The Complex Plank Problem

It is shown that if is a sequence of norm 1 vectors in a complex Hilbert space and is a sequence of non-negative numbers satisfying then there is a unit vector z for which for every j. The result is a strong,complex analogue of the author's real plank theorem.     

Approximate parametric optimization of infinite-dimensional systems

We consider the problem of finding g M_n such that where M_n is the n-dimensional subspace of the complexHilbert space L²(0, ) spanned by an n-tuple of normalized eigenvectoesof the operator , corresponding to eigenvalues. The solution is g = P_nf and P_n denotesthe orthoprojector onto M_n. From Grabowski (1991) we know thatP_n can be expressed in terms of the Malmquist functions. Wegive an alternative approach, more convenient for applicationof the standard mathematical software. The problem of convergenceas n is discussed from both theoretical and numerical viewpoint.The reslts are illustrated by the problems of finding the optimaladjustment of the proportional controller stabilizing a distributedplant. Email: pgrab{at}ia.agh.edu.pl     

One-Sided BMO Spaces

In this paper we introduce the one-sided sharp functions dennedby and where z⁺ = max(z, 0). We study the BMO spaces associatedto and and their relation with the good weights for theone-sided Hardy-Littlewood maximal functions. Finally, as anapplication of our results, we characterize the weights forone-sided fractional integrals and one-sided fractional maximaloperators.     

Paley's Inequality for the Jacobi Expansions

Let be an analytic function in the unit disc satisfying Then, which is familiar as Paley's inequality. In this paper, an analogue of this inequality withrespect to the Jacobi expansions is established.     

Bernstein-Type Inequalities for Linear Combinations of Shifted Gaussians

Let P_n be the collection of all polynomials of degree at mostn with real coefficients. A subtle Bernstein-type extremal problemis solved by establishing the inequality for all , and m= 1, 2, ..., where c is an absolute constant and Some related inequalities and direct and inversetheorems about the approximation by elements of in L_q (R) are also discussed. 2000 Mathematics SubjectClassification 41A17 (primary).     

On Ramanujan's Inequalities for exp(k)

Ramanujan claimed in his first letter to Hardy (16 January 1913)that where (k) lies between 2/21 and 8/45. This conjecture was provedin 1995 by Flajolet et al. The paper establishes the followingrefinement. where Both bounds for ^*(k) are sharp.     

Oscillation Criteria for First-Order Delay Equations

This paper is concerned with the oscillatory behaviour of first-orderdelay differential equations of the form (1) where is non-decreasing, (t)< t for t t₀ and . Let the numbers k andL be defined by It is proved here that when L < 1 and 0 < k 1/e all solutionsof equation (1) oscillate in several cases in which the condition holds, where ₁ is the smaller root of the equation = e^k. 2000Mathematics Subject Classification 34K11 (primary); 34C10 (secondary).     

An Entire Function Defined by a Nonlinear Recurrence Relation

For the nonlinear recurrence relation it is proved that the limit exists and defines an entire function of ₂ = ₁(1-₁).     

Global Branch of Solutions for Non-Linear Schrodinger Equations with Deepening Potential Well

We consider the stationary non-linear Schrödinger equation where > 0 and the functionsf and g are such that and for some bounded open set R^N. We use topological methods to establish the existenceof two connected sets D^± of positive/negative solutionsin R x W², ^p R^N where that cover the interval (, ()) in the sense that and furthermore, The number () is characterized as the unique value of in theinterval (, ) for which the asymptotic linearization has a positiveeigenfunction. Our work uses a degree for Fredholm maps of indexzero. 2000 Mathematics Subject Classification 35J60, 35B32,58J55.     

On the Best Possible Constants in the Khintchine Inequality for P >= 3

In the paper the best constants in the Khintchine inequalityfor p 3 are found, namely where . This gives stronger estimates than the hitherto Known ones forthe constants depend on n.     

A Radial Uniqueness Theorem for Sobolev Functions

We show that continuous functions u in the Sobolev space , 1 < p n, which have the limitzero in a certain weak sense in a set of positive p-capacityon B with where B is the open unit ball of Rⁿ and for 0 > > , are identically zero. Conversely, we produce for each 1 > p n and each positive a non-constant function u in , continuous in , and a compact set EB of positive p-capacity such that u = 0 in E and the aboveinequality holds with exponent p – l + .     

The Sector of Analyticity of the Ornstein-Uhlenbeck Semigroup on Lp Spaces with Respect to Invariant Measure

The sector of analyticity of the Ornstein–Uhlenbeck semigroupis computed on the space := L^p (R^N; µ) with respect to its invariant measure µ.If A= + Bx· denotes the generator of the Ornstein–Uhlenbecksemigroup, then the angle ₂ of the sector of analyticity in is /2 minus the spectral angleof BQ, Q being the matrix determining the Gaussian measure µ.The angle of analyticity in is then given by the formula      

Near-minimax Interpolation by a Polynomial in z and z-1 on a Circular Annulus

A polynomial of degree n in z^–1 and n–1 in z isdefined by an interpolation projection from the space A(N_p) of functions f analytic in the circular annulusp^–1 < <p and continuous on itsboundaries = p^–1, p. The points ofinterpolation are chosen to be spaced at equal angles aroundthe two boundaries, with arguments on the inner boundary midwaybetween those on the outer boundary. By calculating the Lebesgueconstants numerically, is found to be close to a minimax approximation for all p 1and all degrees n in the range 1 n 15. In the limiting casesp = 1 and, it is proved that is asymptotic to 2^–1 log n. More specifically and , where _nis the Lebesgueconstant of Gronwall for equally spaced interpolation on a circleby a polynomial of degree n. It is also demonstrated that is not in general monotonic in p, and that is not everywhere differentiable in p.     

The Weiss Conjecture for Bounded Analytic Semigroups

New results concerning the so-called Weiss conjecture on admissibleoperators for bounded analytic semigroups are given. Let be a bounded analytic semigroup withgenerator –A on some Banach space X. It is proved thatif A^1/2 is admissible for A, that is, if there is an estimate then any continuous mappingC : D(A) Y valued in a Banach space Y is admissible for A providedthat there is an estimate .for , Re()<0. This holds in particular if is a contractive (analytic) semigroup on Hilbertspace. In the converse direction, it is shown that this mayhappen for a bounded analytic semigroup on Hilbert space thatis not similar to a contractive one. Applications in non-HilbertianBanach spaces are also given.     

Weighted Hardy-Type Inequalities for Differences and the Extension Problem for Spaces with Generalized Smoothness

It is well known that there are bounded domains Rⁿ whose boundaries are not smooth enough for there to exist a bounded linear extensionfor the Sobolev space into , but the embedding is nevertheless compact. For the Lipboundaries (0<<1) studied in [3, 4], there does not existin general an extension operator of into but there is a bounded linear extension of into and the smoothness retained by thisextension is enough to ensure that the embedding is compact. It is natural to ask if this is typicalfor bounded domains which are such that is compact, that is, that there exists a boundedextension into a space of functions in Rⁿ which enjoy adequatesmoothness. This is the question which originally motivatedthis paper. Specifically we study the ‘extension by zero’operator on a space of functions with given ‘generalized’smoothness defined on a domain with an irregular boundary, anddetermine the target space with respect to which it is bounded.