Componentwise and Cartesian decompositions of linear relations

Let $A$ be a, not necessarily closed, linear relation in a Hilbert space \sH with a multivalued part \mul A. An operator $B$ in \sH with \ran B\perp\mul A^{**} is said to be an operator part of $A$ when A=B \hplus (\{0\}\times \mul A), where the sum is componentwise (i.e. span of the graphs). This decomposition provides a counterpart and an extension for the notion of closability of (unbounded) operators to the setting of linear relations. Existence and uniqueness criteria for the existence of an operator part are established via the so-called canonical decomposition of $A$. In addition, conditions are developed for the decomposition to be orthogonal (components defined in orthogonal subspaces of the underlying space). Such orthogonal decompositions are shown to be valid for several classes of relations. The relation $A$ is said to have a Cartesian decomposition if A=U+\I V, where $U$ and $V$ are symmetric relations and the sum is operatorwise. The connection between a Cartesian decomposition of $A$ and the real and imaginary parts of $A$ is investigated

Unitary Colligations in Πκ-Spaces, Characteristic Functions and Štraus Extensions

The main result of this paper is the description of certain linear manifolds T(λ), associated with a symmetric operator, in terms of certain boundary values of the characteristic function of a unitary colligation

Complementation and Lebesgue-type decompositions of linear operators and relations

In this paper, a new general approach is developed to construct and study Lebesgue-type decompositions of linear operators or relations T in the Hilbert space setting. The new approach allows to introduce an essentially wider class of Lebesgue-type decompositions than what has been studied in the literature so far. The key point is that it allows a nontrivial interaction between the closable and the singular components of T. The motivation to study such decompositions comes from the fact that they naturally occur in the corresponding Lebesgue-type decomposition for pairs of quadratic forms. The approach built in this paper uses so-called complementation in Hilbert spaces, a notion going back to de Branges and Rovnyak.© 2024 The Authors. Journal of the London Mathematical Society is copyright © London Mathematical Society. This is an open access article under the terms of the Creative Commons Attribution License, which permits use, distribution and reproduction in any medium, provided the original work is properly cited.fi=vertaisarvioitu|en=peerReviewed

Sequences of Operators, Monotone in the Sense of Contractive Domination

A sequence of operators Tn from a Hilbert space H to Hilbert spaces Kn which is nondecreasing in the sense of contractive domination is shown to have a limit which is still a linear operator T from H to a Hilbert space K. Moreover, the closability or closedness of Tn is preserved in the limit. The closures converge likewise and the connection between the limits is investigated. There is no similar way of dealing directly with linear relations. However, the sequence of closures is still nondecreasing and then the convergence is governed by the monotonicity principle. There are some related results for nonincreasing sequences.© The Author(s) 2024. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.fi=vertaisarvioitu|en=peerReviewed

Passive systems with a normal main operator and quasi-selfadjoint systems

Passive systems $\tau={T,M,N,H}$ with $M$ and $N$ as an input and output space and $H$ as a state space are considered in the case that the main operator on the state space is normal. Basic properties are given and a general unitary similarity result involving some spectral theoretic conditions on the main operator is established. A passive system $\tau$ with $M=N$ is said to be quasi-selfadjoint if $ran(T-T^*)\subset N$. The subclass $S^{qs}$ of the Schur class $S$ is the class formed by all transfer functions of quasi-selfadjoint passive systems. The subclass $S^{qs}$ is characterized and minimal passive quasi-selfadjoint realizations are studied. The connection between the transfer function belonging to the subclass $S^{qs}$ and the $Q$-function of $T$ is given.Comment: 29 page

Akkerranden in Nederland

VakpublicatieInstitute of Environmental Science

Multi-Objective and Multidisciplinary Design Optimisation of Unmanned Aerial Vehicle Systems using Hierarchical Asynchronous Parallel Multi-Objective Evolutionary Algorithms

The overall objective of this research was to realise the practical application of Hierarchical Asynchronous Parallel Evolutionary Algorithms for Multi-objective and Multidisciplinary Design Optimisation (MDO) of UAV Systems using high fidelity analysis tools. The research looked at the assumed aerodynamics and structures of two production UAV wings and attempted to optimise these wings in isolation to the rest of the vehicle. The project was sponsored by the Asian Office of the Air Force Office of Scientific Research under contract number AOARD-044078. The two vehicles wings which were optimised were based upon assumptions made on the Northrop Grumman Global Hawk (GH), a High Altitude Long Endurance (HALE) vehicle, and the General Atomics Altair (Altair), Medium Altitude Long Endurance (MALE) vehicle. The optimisations for both vehicles were performed at cruise altitude with MTOW minus 5% fuel and a 2.5g load case. The GH was assumed to use NASA LRN 1015 aerofoil at the root, crank and tip locations with five spars and ten ribs. The Altair was assumed to use the NACA4415 aerofoil at all three locations with two internal spars and ten ribs. Both models used a parabolic variation of spar, rib and wing skin thickness as a function of span, and in the case of the wing skin thickness, also chord. The work was carried out by integrating the current University of Sydney designed Evolutionary Optimiser (HAPMOEA) with Computational Fluid Dynamics (CFD) and Finite Element Analysis (FEA) tools. The variable values computed by HAPMOEA were subjected to structural and aerodynamic analysis. The aerodynamic analysis computed the pressure loads using a Boeing developed Morino class panel method code named PANAIR. These aerodynamic results were coupled to a FEA code, MSC.Nastran® and the strain and displacement of the wings computed. The fitness of each wing was computed from the outputs of each program. In total, 48 design variables were defined to describe both the structural and aerodynamic properties of the wings subject to several constraints. These variables allowed for the alteration of the three aerofoil sections describing the root, crank and tip sections. They also described the internal structure of the wings allowing for variable flexibility within the wing box structure. These design variables were manipulated by the optimiser such that two fitness functions were minimised. The fitness functions were the overall mass of the simulated wing box structure and the inverse of the lift to drag ratio. Furthermore, six penalty functions were added to further penalise genetically inferior wings and force the optimiser to not pass on their genetic material. The results indicate that given the initial assumptions made on all the aerodynamic and structural properties of the HALE and MALE wings, a reduction in mass and drag is possible through the use of the HAPMOEA code. The code was terminated after 300 evaluations of each hierarchical level due to plateau effects. These evolutionary optimisation results could be further refined through a gradient based optimiser if required. Even though a reduced number of evaluations were performed, weight and drag reductions of between 10 and 20 percent were easy to achieve and indicate that the wings of both vehicles can be optimised