37 research outputs found
Effect-specific analysis of propagation parameters
Get PDFResults of propagation measurements with the satellite OLYMPUS carried out at 12.5, 20, and 30 GHz at the Research Center of the Deutsche Bundespost Telekom are discussed. In particular, attenuation, scintillation, and depolarization measurements are analyzed with special emphasis on frequency scaling of the various effects
On the L_p-solvability of higher order parabolic and elliptic systems with BMO coefficients
Full text linkWe prove the solvability in Sobolev spaces for both divergence and
non-divergence form higher order parabolic and elliptic systems in the whole
space, on a half space, and on a bounded domain. The leading coefficients are
assumed to be merely measurable in the time variable and have small mean
oscillations with respect to the spatial variables in small balls or cylinders.
For the proof, we develop a set of new techniques to produce mean oscillation
estimates for systems on a half space.Comment: 44 pages, introduction revised, references expanded. To appear in
Arch. Rational Mech. Ana
Second order optimality conditions and their role in PDE control
Get PDFIf f : Rn R is twice continuously differentiable, f’(u) = 0 and f’’(u) is positive definite, then u is a local minimizer of f. This paper surveys the extension of this well known second order suffcient optimality condition to the case f : U R, where U is an infinite-dimensional linear normed space. The reader will be guided from the case of finite-dimensions via a brief discussion of the calculus of variations and the optimal control of ordinary differential equations to the control of nonlinear partial differential equations, where U is a function space. In particular, the following questions will be addressed: Is the extension to infinite dimensions straightforward or will unexpected difficulties occur? How second order sufficient optimality conditions must be modified, if simple inequality constraints are imposed on u? Why do we need second order conditions and how can they be applied? If they are important, are we able to check if they are fulfilled order sufficient optimality condition to the case f : U R, where U is an infinite-dimensional linear normed space. The reader will be guided from the case of finite-dimensions via a brief discussion of the calculus of variations and the optimal control of ordinary differential equations to the control of nonlinear partial differential equations, where U is a function space. In particular, the following questions will be addressed: Is the extension to infinite dimensions straightforward or will unexpected difficulties occur? How second order sufficient optimality conditions must be modified, if simple inequality constraints are imposed on u? Why do we need second order conditions and how can they be applied? If they are important, are we able to check if they are fulfilled?
It turns out that infinite dimensions cause new difficulties that do not occur in finite dimensions. We will be faced with the surprising fact that the space, where f’’(u) exists can be useless to ensure positive definiteness of the quadratic form v f’’(u)v2. In this context, the famous two-norm discrepancy, its consequences, and techniques for overcoming this difficulty are explained. To keep the presentation simple, the theory is developed for problems in function spaces with simple box constraints of the form a = u = ß. The theory of second order conditions in the control of partial differential equations is presented exemplarily for the nonlinear heat equation. Different types of critical cones are introduced, where the positivity of f’’(u) must be required. Their form depends on whether a so-called Tikhonov regularization term is part of the functional f or not. In this context, the paper contains also new results that lead to quadratic growth conditions in the strong sense.
As a first application of second-order sufficient conditions, the stability of optimal solutions with respect to perturbations of the data of the control problem is discussed. Second, their use in analyzing the discretization of control problems by finite elements is studied. A survey on further related topics, open questions, and relevant literature concludes the paper.The first author was partially supported by the Spanish Ministerio de EconomĂa y Competitividad under project MTM2011-22711, the second author by DFG in the framework of the Collaborative Research Center SFB 910, project B6
