Showing papers on "Bounded function published in 1986"

Compact sets in the spaceL p (O,T; B)

Abstract: A characterization of compact sets in Lp (0, T; B) is given, where 1⩽P⩾∞ and B is a Banach space. For the existence of solutions in nonlinear boundary value problems by the compactness method, the point is to obtain compactness in a space Lp (0,T; B) from estimates with values in some spaces X, Y or B where X⊂B⊂Y with compact imbedding X→B. Using the present characterization for this kind of situations, sufficient conditions for compactness are given with optimal parameters. As an example, it is proved that if {fn} is bounded in Lq(0,T; B) and in L loc 1 (0, T; X) and if {∂fn/∂t} is bounded in L loc 1 (0, T; Y) then {fn} is relatively compact in Lp(0,T; B), ∀p

Reaching approximate agreement in the presence of faults

Abstract: This paper considers a variant of the Byzantine Generals problem, in which processes start with arbitrary real values rather than Boolean values or values from some bounded range, and in which approximate, rather than exact, agreement is the desired goal. Algorithms are presented to reach approximate agreement in asynchronous, as well as synchronous systems. The asynchronous agreement algorithm is an interesting contrast to a result of Fischer et al, who show that exact agreement with guaranteed termination is not attainable in an asynchronous system with as few as one faulty process. The algorithms work by successive approximation, with a provable convergence rate that depends on the ratio between the number of faulty processes and the total number of processes. Lower bounds on the convergence rate for algorithms of this form are proved, and the algorithms presented are shown to be optimal.

Decentralized adaptive control of interconnected systems

Abstract: The adaptive control of a class of large-scale systems formed of an arbitrary interconnection of subsystems with unknown parameters, nonlinearities, and bounded disturbances is investigated. It is first shown that no matter how weak the interconnections are, a decentralized adaptive control scheme can become unstable. Approaches are then developed for stabilization and tracking using new decentralized adaptive controllers. In the case where the relative degree n^{*} of the transfer function of each decoupled subsystem is less than or equal to two, sufficient conditions are established which guarantee boundedness and exponential convergence of the state and parameter errors to bounded residual sets. In the absence of disturbances and interconnections, the decentralized adaptive control schemes guarantee exact convergence of the tracking errors to zero. The effectiveness of the proposed adaptive schemes is demonstrated using a simple example.

An "analytical centre" for polyhedrons and new classes of global algorithms for linear (smooth, convex) programming

Abstract: For an arbitrary, finite intersection of halfspaces bi≥ i=1,...,m, xeRn, i.e. a bounded, convex polyhedron P(am, bm) we define a "central" point x(am,bm)eP, which has the following properties: x depends on (am,bm) analytically (i.e. rather smoothly); x is affinely invariant; there exist ellipsoids containing P and contained in P, centered at x with similarity ratio (m-1); x(am,bm) can be computed effectively by maximizing a strongly concave, analytic function over P. New methods are presented for globalizing (globally accelerating) the convergence of Newton'-s method for finding saddle points of convex concave, smooth functions based on homotopy (i.e. continuation) ideas. The above results are applied to outline new approaches to linear (smooth, convex) programming by constructing affine invariant relatives of Karmarkar'-s interior point, projective invariant method.

A New Adaptive Law for Robust Adaptation without Persistent Excitation

Abstract: A new adaptive law motivated by that given in [1] is proposed for the robust adaptive control of plants with unknown parameters. In this adaptive law the output error e l plays a dual role in the adjustment of the control parameter vector. In the ideal case the adaptive system has bounded solutions; in addition the error equations are uniformly asymptotically stable in the large when the reference input is sufficiently persistently exciting. The adaptive system is also shown to be robust under bounded external disturbances. Finally it is shown that, by suitably modifying the adaptive law, the overall system can be made robust in the presence of unmodeled dynamics of the plant.

Sequence spaces defined by a modulus

Abstract: Ruckle[4] used the idea of a modulus function ƒ (see Definition 1 below) to construct the sequence spaceThis space is an FK space, and Ruckle proved that the intersection of all such L(f) spaces is o, the space of finite sequences, thereby answering negatively a question of A. Wilansky: ‘Is there a smallest FK-space in which the set {e1, e2, …} of unit vectors is bounded?’

Solution of an inverse problem for fractals and other sets.

Abstract: The problem of providing succinct approximate descriptions of given bounded subsets of Rn can be solved by application of the contraction mapping principle.

A regularized decomposition method for minimizing a sum of polyhedral functions

Abstract: A problem of minimizing a sum of many convex piecewise-linear functions is considered. In view of applications to two-stage linear programming, where objectives are marginal values of lower level problems, it is assumed that domains of objectives may be proper polyhedral subsets of the space of decision variables and are defined by piecewise-linear induced feasibility constraints. We propose a new decomposition method that may start from an arbitrary point and simultaneously processes objective and feasibility cuts for each component. The master program is augmented with a quadratic regularizing term and comprises an a priori bounded number of cuts. The method goes through nonbasic points, in general, and is finitely convergent without any nondegeneracy assumptions. Next, we present a special technique for solving the regularized master problem that uses an active set strategy and QR factorization and exploits the structure of the master. Finally, some numerical evidence is given.

The compressible Euler equations in a bounded domain: existence of solutions and the incompressible limit

Abstract: A short-time existence theorem is proven for the Euler equations for nonisentropic compressible fluid flow in a bounded domain, and solutions with low Mach number and almost incompressible initial data are shown to be close to corresponding solutions of the equations for incompressible flow.

The Simplex Method: A Probabilistic Analysis

Abstract: 0 Introduction.- Formulation of the problem and basic notation.- 1 The problem.- A Historical Overview.- 2 The gap between worst case and practical experience.- 3 Alternative algorithms.- 4 Results of stochastic geometry.- 5 The results of the author.- 6 The work of Smale.- 7 The paper of Haimovich.- 8 Quadratic expected number of steps for sign-invariance model.- Discussion of different stochastic models.- 9 What is the "Real World Model"?.- Outline of Chapters 1-5.- 10 The basic ideas and the methods of this book.- 11 The results of this book.- 12 Conclusion and conjectures.- 1 The Shadow-Vertex Algorithm.- 1 Primal interpretation.- 2 Dual interpretation.- 3 Numerical realization of the algorithm.- 4 The algorithm for Phase I.- 2 The Average Number of Pivot Steps.- 1 The probability space.- 2 An integral formula for the expected number of S.- 3 A transformation of coordinates.- 4 Generalizations.- 3 The Polynomiality of the Expected Number of Steps.- 1 Comparison of two integrals.- 2 An application of Cavalieri's Principle.- 3 The influence of the distribution.- 4 Evaluation of the quotient.- 5 The average number of steps in our complete Simplex-Method.- 4 Asymptotic Results.- 1 An asymptotic upper bound in integral form.- 2 Asymptotic results for certain classes of distributions.- 3 Special distributions with bounded support.- 4 Asymptotic bounds under uniform distributions.- 5 Asymptotic bounds under Gaussian distribution.- 5 Problems with Nonnegativity Constraints.- 1 The geometry.- 2 The complete solution method.- 3 A simplification of the boundary-condition.- 4 Explicit formulation of the intersection-condition.- 5 Componentwise sign-independence and the intersection condition.- 6 The average number of pivot steps.- 6 Appendix.- 1 Gammafunction and Betafunction.- 2 Unit ball and unit sphere.- 3 Estimations under variation of the weights.- References.

Admissible sets and feedback control for discrete-time linear dynamical systems with bounded controls and states

Abstract: For linear discrete-time dynamical systems, necessary and sufficient conditions are given for a control sequence belonging to a polyhedral constraint set, to stabilize the system under the condition that the state is restricted to a polyhedral state constraint set. A variable structure linear state feedback controller, given in terms of the controls at the vertices of the polyhedral state constraint set, is presented.

Robust adaptive control in the presence of bounded disturbances

Abstract: The model reference adaptive control of a linear plant subjected to bounded disturbances is considered. By analyzing a set of nonlinear differential equations, it is shown that the global behavior of the adaptive system depends upon the persistent excitation of the reference input as well as the amplitude of the external disturbances. The principal contribution of the paper is the derivation of sufficient conditions on the persistent excitation of the reference input, given the maximum amplitude of disturbance, for the signals in the adaptive system to be globally bounded.

Optimal isoparametric finite elements and error estimates for domains involving curved boundaries

Abstract: We describe a practical procedure for the triangulation of arbitrary n-dimensional regular bounded domains by isoparametric simplicial elements preserving the optimal order of accuracy. Along similar lines we prove general error estimates for the finite element solution of second order elliptic problems. Various kinds of nonhomogeneous boundary conditions can be dealt with, such as Dirichlet, Neumann and Robin boundary conditions.

A singular integral

Abstract: In this paper we show that if K(x) = O(x)/IxjXj is a CalderonZygmund kernel, where Q E L(Sn-l) for some 1 2. This paper is related to boundedness properties of a variation of CalderonZygmund operators. Let H(x) = b(IxI)Q(X)/IxIn be a kernel, where iSn-l Q(x) da(x') = 0, Q(Ax) = Q(x) for x E Rn, A > 0, and b is radial. Define (1) f(x) =imj H(y)f(x y) dy = P.V. H * f(x). If b -1 and Q E Lq(Sn-1) for some q > 1, then T is a Calder6n-Zygmund operator which is bounded on LP(Rn) for 1 1; see [4]. In [1] R. Fefferman showed that if Q satisfies a Lipschitz condition and b is bounded, then T is bounded on LP(Rn), 1 2. This is not true when n = 1; for instance, if H(x) = sin(lxl)/x, then its Fourier transform H(s) is unbounded, indicating that T cannot be bounded on L2(R'). It turns out that the smoothness condition on Q can be relaxed a great deal. THEOREM. If Q E Lq(Sn-l) for some 1 2. PROOF. First we show that the Fourier transform of H is bounded; thereby T is bounded on L2(Rn). Using polar coordinates and letting x = px', p = xl, we get = lim l ()dp( 0 (x) ePx" (do ( /) ?-+ s

Weighted inequalities for the one-sided Hardy-Littlewood maximal functions

Abstract: Let M+f(x) = SUPh , 0(l/h)Jx+h (t)1 Idt denote the one-sided maximal function of Hardy and Littlewood. For w(x) > 0 on R and 1 0 and a = v-ll(P -), then M+ is bounded from LP(v) to LP(w) if and only if [ [M (X,a)] w 0. The corresponding weak type inequality is also characterized. Further properties of A weights, such as A + A e and A p = (A +)(A1 -P, are established.

On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients

Abstract: : Locally bounded weak solutions of degenerate parabolic equations are proven to be locally hoelder continuous. Hoelder estimates are also derived up to the boundary for both Dirichlet data and (non-linear) variational data. Via a counterexample it is shown that non-negative solutions, in general, do not satisfy the parabolic version of the Harnack inequality.

Third law of black-hole dynamics: A formulation and proof.

Abstract: It is shown that no continuous process in which the energy tensor of accreted matter remains bounded and satisfies the weak energy condition in a neighborhood of the apparent horizon can reduce the surface gravity of a black hole to zero within a finite advanced time. This gives a precise expression to the third law of black-hole mechanics.

Robust compliant motion for manipulators, part II: Design method

Abstract: A controller design methodology to develop a robust compliant motion for robot manipulators is described. The achievement of the target dynamics (the target impedance is introduced in Part I) and preservation of stability robustness in the presence of bounded model uncertainties are the key issues in the design method. State-feedback and force-feedforward gains are chosen to guarantee the achievement of the target dynamics, while preserving stability in the presence of the model uncertainties. In general, the closed-loop behavior of a system cannot be shaped arbitrarily over an arbitrarily wide frequency range. It is proved that a special class of impedances that represent our set of performance specifications are mathematically achievable asymptotically through state-feedback and interaction-force feedforward as actuator bandwidths become large, and we offer a geometrical design method for achieving them in the presence of model uncertainties. The design method reveals a classical trade-off between a system's performance over a bounded frequency range and its stability relative to model uncertainties via multivariable Nyquits criteria. Two classes of such uncertainties are dealt with. While the first class of model uncertainties is formed from the uncertainties in the parameters of the modeled dynamics, the high-frequency unmodeled dynamics form the second class of model uncertainties. The multivariable Nyquist criterion is used to examine trade-offs in stability robustness against approximation of desired target impedances over bounded frequency ranges.

A bound for global solutions of semilinear heat equations

Abstract: We show that global positive solutions of the initial-boundary value problem foru t =Δu+u p are bounded, provided thatp>1 is subcritical. Our bound depends only on sup norm of the initial data and is useful to classify initial data by the asymptotic behavior of the solutions as time tend to infinity.

Can One Hear the Dimension of a Fractal

Abstract: We consider the spectrum of the Laplacian in a bounded open domain of ℝ n with a rough boundary (ie with possibly non-integer dimension) and we discuss a conjecture by M V Berry generalizing Weyl's conjecture Then using ideas Mark Kac developed in his famous study of the drum, we give upper and lower bounds for the second term of the expansion of the partition function The main thesis of the paper is to show that the relevant measure of the roughness of the boundary should be based on Minkowski dimensions and on Minkowski measures rather than on Haussdorff ones

Some applications of Hausdorff dimension inequalities for ordinary differential equations

Abstract: Upper bounds are obtained for the Hausdorff dimension of compact invariant sets of ordinary differential equations which are periodic in the independent variable. From these are derived sufficient conditions for dissipative analytic n-dimensional ω-periodic differential equations to have only a finite number of ω-periodic solutions. For autonomous equations the same conditions ensure that each bounded semi-orbit converges to a critical point. These results yield some information about the Lorenz equation and the forced Duffing equation.

Constructing Belts in Two-Dimensional Arrangements with Applications

Abstract: For H a set of lines in the Euclidean plane, $A(H)$ denotes the induced dissection, called the arrangement of H. We define the notion of a belt in $A(H)$, which is bounded by a subset of the edges in $A(H)$, and describe two algorithms for constructing belts. All this is motivated by applications to a host of seemingly unrelated problems including a type of range search and finding the minimum area triangle with the vertices taken from some finite set of points.

Supra-convergent schemes on irregular grids

Abstract: As Tikhonov and Samarskil showed for k = 2, it is not essential that k th-order compact difference schemes be centered at the arithmetic mean of the stencil's points to yield second-order convergence (although it does suffice). For stable schemes and even k, the main point is seen when the k th difference quotient is set equal to the value of the k th derivative at the middle point of the stencil; the proof is particularly transparent for k = 2. For any k, in fact, there is a ( k/2J -parameter family of symmetric averages of the values of the k th derivative at the points of the stencil which, when similarly used, yield second-order convergence. The result extends to stable compact schemes for equations with lower-order terms under general boundary conditions. Although the extension of Numerov's tridiagonal scheme (approximating D2y = f with third-order truncation error) yields fourth-order con- vergence on meshes consisting of a bounded number of pieces in which the mesh size changes monotonically, it yields only third-order convergence to quintic polynomials on any three- periodic mesh with unequal adjacent mesh sizes and fixed adjacent mesh ratios. A result of some independent interest is appended (and applied): it characterizes, simply, those functions of k variables which possess the property that their average value, as one translates over one period of an arbitrary periodic sequence of arguments, is zero; i.e., those bounded functions whose average value, as one translates over arbitrary finite sequences of arguments, goes to zero as the length of the sequences increases.

Factorization problems and operator identities

Abstract: CONTENTS Introduction Chapter I. Factorization of the transfer operator-valued function § 1. Realization of operator-valued functions § 2. A factorization method § 3. Factorization of rational operator-valued functions Chapter II. Operator identities and S-nodes § 1. Simplest properties of an S-node § 2. Symmetric S-nodes § 3. The operator Bezoutiant Chapter III. Continual factorizations and inverse problems § 1. The continual factorization of W(z) § 2. The inverse problem for bounded operator-valued functions § 3. The direct and the inverse spectral problems Chapter IV. Applications and examples § 1. An effective solution of the inverse problem § 2. Examples § 3. Interpolation problems § 4. On a class of extremal problems References

Observation of dynamical systems in the presence of bounded nonlinearities/uncertainties

Abstract: This paper proposes new types of observers for dynamical systems subjected to bounded nonlinearities or uncertainties. The design of these observers utilizes techniques related to variable structure theory. A measure for the rate at which the estimates converge to the actual states is derived. Simulation results exhibiting the performance of these observers are included.

A Theorem of I. Schur and Its Impact on Modern Signal Processing

Abstract: An algorithm of Schur for characterizing power series that are bounded in the unit circle is shown to have applications to a variety of problems in science and engineering. These include speech analysis and synthesis, inverse scattering, decoding of error-correcting codes, synthesis of digital filters, modeling of random signals, Pade approximation for linear systems, and zero location of polynomials.