On guaranteed stability of uncertain linear systems via linear control

This paper addresses the problem of stabilizing an uncertain linear system. The uncertaintyq(·) which enters the dynamics is nonstistical in nature. That is, noa priori statistics forq(·) are assumed; only boundsQ on the admissible variations ofq(·) are taken as given. The results given here applied to so-called matched systems differ from previous results in two ways. Firstly, the stabilizing control in this paper is linear; for this same class of problems, many of the existing results would require a nonlinear control. Furthermore, those results which do in fact yield linear controls are only valid when a certain matrix (q) (formed using the given data) is negative definite for allq Q. In contrast, the theory given here only requires compactness of the bounding setQ. Secondly, we show that the so-called matching conditions (used in earlier work) can be generalized so as to encompass a larger class of dynamical systems.This research was supported by the US Department of Energy under Contract No. ET-78-S-01-3390.     

Uniformly non-l n (1) Musielak-Orlicz sequence spaces

We give a necessary and sufficient condition for the uniformly non-l _n ⁽¹⁾ property of Musielak-Orlicz sequence spacesl ^Φ generated by a sequence Φ=(ϕ_n:n⩾l) of finite Orlicz functions such that for eachn∈ℕ. As a result, forn ₀⩾2, there exist spacesl ^Φ which are only uniformly non-l _n ⁽¹⁾ forn⩾n ₀. Moreover we obtain a characterization of uniformly non-l _n ⁽¹⁾ and reflexive Orlicz sequence spaces over a wide class of purely atomic measures and of uniformly non-l _n ⁽¹⁾ Nakano sequence spaces. This extends a result of Luxemburg in [19]. Submitted in memory of Professor W. Orlicz     

Properties of the random search in global optimization

From theorems which we prove about the behavior of gaps in a set ofN uniformly random points on the interval [0, 1], we determine properties of the random search procedure in one-dimensional global optimization. In particular, we show that the uniform grid search is better than the random search when the optimum is chosen using the deterministic strategy, that a significant proportion of large gaps are contained in the uniformly random search, and that the error in the determination of the point at which the optimum occurs, assuming that it is unique, will on the average be twice as large using the uniformly random search compared with the uniform grid. In addition, some of the properties of the largest gap are verified numerically, and some extensions to higher dimensions are discussed. The latter show that not all of the conclusions derived concerning the inadequacies of the one-dimensional random search extend to higher dimensions, and thaton average the random search is better than the uniform grid for dimensions greater than 6.This paper is based on work started in the Statistics Department of Princeton University when the first author was visiting as a Research Associate. Part of this research was supported by the Office of Naval Research, Contract No. 0014-67-A-0151-0017, and by the US Army Research Office—Durham, Contract No. DA-31-124-ARO-D-215.2 The authors wish to thank B. Omodei for his careful work in preparing the programs for the results of Figs. 1–2 and Table 1. The computations were performed on the IBM 360/50 of the Australian National University's Computer Centre. Thanks are also due to R. Miles for suggestions regarding the extension of the results to multidimensional regions, and to P. A. P. Moran and R. Brent for suggestions regarding the evaluation of the integral ₀ ¹ ... ^0/1 (x ₁ ² + ... +x _n ^/2 )^1/2 dx ₁ ...dx _n.     

Harnack's inequality for parabolic nonlocal equations

The main result of this paper is a nonlocal version of Harnack's inequality for a class of parabolic nonlocal equations. We additionally establish a weak Harnack inequality as well as local boundedness of solutions. None of the results require the solution to be globally positive.     

Local boundedness of weak solutions to the Diffusive Wave Approximation of the Shallow Water equations

In this paper we prove that weak solutions to the Diffusive Wave Approximation of the Shallow Water equations

${?}_{t}u?\mathrm{?}?({(u?z)}^{\alpha }|\mathrm{?}u{|}^{\gamma ?1}\mathrm{?}u)=f$

are locally bounded. Here, u describes the height of the water, z is a given function that represents the land elevation and f is a source term accounting for evaporation, infiltration or rainfall.     

Strong Convergence Theorems for Mixed Typ e Asymptotically Nonexpansive Mappings

The purpose of this paper is to study a new two-step iterative scheme with mean errors of mixed type for two asymptotically nonexpansive self-mappings and two asymptotically nonexpansive nonself-mappin...     

On the ‐boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer

In this paper, we prove the ‐boundedness of solution operator families of the generalized Stokes resolvent problem in an infinite layer with resolvent parameter , where , and our boundary conditions are nonhomogeneous Neumann on upper boundary and Dirichlet on lower boundary. We want to emphasize that we can choose 0 < ? < π ∕ 2 and γ₀ > 0 arbitrarily, although usual parabolic theorem tells us that we must choose a large γ₀ > 0 for given 0 < ? < π ∕ 2. We also prove the maximal L_p ? L_q regularity theorem of the nonstationary Stokes problem as an application of the ‐boundedness. The key of our approach is to apply several technical lemmas to the exact solution formulas of a resolvent problem. The formulas are obtained through the solutions of the ODEs, in the Fourier space, driven by the partial Fourier transform with respect to tangential space variable . Copyright © 2014 John Wiley & Sons, Ltd.     

A Quantitative Regularity Estimate for Nonnegative Supersolutions of Uniformly Parabolic Equations

This note establishes an interior quantitative lower bound for nonnegative supersolutions of fully nonlinear uniformly parabolic equations. The result may be interpreted as a quantitative version of a growth lemma established by Krylov and Safonov for nonnegative supersolutions of linear uniformly parabolic equations in nondivergence form. Our approach is different, and follows from an application of a reverse Holder inequality. The result is the parabolic analogue of an elliptic regularity estimate established by Caffarelli, Souganidis, and Wang in the stochastic homogenization of fully nonlinear uniformly elliptic equations.     

格林互易定理在求解均匀带电圆环的静电势分布中的应用

对于静电场,由普遍的格林互易定理得出特殊情形下的格林互易定理.应用该定理推导出均匀带电圆环的数种在形式上互不相同的静电势分布解式.