Tanya Tarnopolskaya

Wave propagation on helices and hyperhelices: a fractal regression

A hyperhelix of order N is defined to be a self‐similar object consisting of a thin elastic rod wound into a helix, which is itself wound into a larger helix, until this process has been repeated N times. Wave propagation on such a structure can be discussed in a hierarchical manner, ultimately in terms of the wavenumber k defining propagation on the elementary rod. It is found that the dispersion curve expressing the wave frequency ω as a function of the elementary wavenumber k on the rod making up the initial helix is also a fractal object, with all the macroscopically observable wave phenomena for a hyperhelix of arbitrarily large order being compressed into a small wavenumber range of width about 2R2‐1α centred on the value k = R‐1, where R1 is the radius, α is the helical pitch angle of the smallest helix in the progression, and R2 is the radius of the next‐larger helix.

Synthesis of Optimal Bang–Bang Control for Cooperative Collision Avoidance for Aircraft (Ships) with Unequal Linear Speeds

Close proximity encounters most often occur for situations in which participants have unequal linear speeds. Cooperative collision avoidance strategies for such situations are investigated. We show that, unlike the encounters of participants with equal linear speeds, bang–bang collision avoidance strategies are not always optimal when the linear speeds are unequal, and we establish the conditions for which no optimal bang–bang controls exist near the terminal time. Nevertheless, under certain conditions, we demonstrate that bang–bang collision avoidance strategies remain optimal for encounters of participants with unequal linear speeds. Such conditions are established, and it appears that they cover a wide range of important practical situations. The synthesis of bang–bang control is constructed, and its optimality is established.

Synthesis of Optimal Control for Cooperative Collision Avoidance in a Close Proximity Encounter: Special Cases

Abstract The paper studies the optimal cooperative collision avoidance strategies in a planar close proximity encounter, with turning rates of the participants as the control functions. The maximization of the terminal miss distance is adopted as a performance criterion. This paper extends earlier analyses to the important special case when participants have unequal linear speeds but equal turn capabilities. The analysis is based on the Pontryagin maximum principle and the study of the properties of the extremals. The analysis is outlined in a unified manner that covers all special cases of the problem, including the cases of identical participants and the participants with unequal turn capabilities but equal linear speeds. The distinctive features of the mathematical structure of the problem and the optimal control solutions for different special cases of the problem are identified. The results of this paper are useful for setting and validating air traffic rules and for benchmarking and validating automated proximity management and collision avoidance systems.

Optimal bang-bang and singular controls in collision avoidance for participants with unequal linear speeds

We study optimal cooperative collision avoidance strategies for two participants with unequal linear speeds, but equal turn capabilities, in a planar close proximity encounter. Previous research showed that bang-bang strategies are optimal for participants with equal linear speeds. However, for participants with unequal linear speeds, bang-bang strategies are not necessarily optimal and, hence, singular control arcs may occur. We present a theoretical and numerical study of the structure of optimal controls with bang-bang and singular arcs. We prove that both controls can not be singular simultaneously and that the only possible singular control is a zero control. We derive formulas for the singular surfaces and verify that sufficient conditions hold for the computed extremal solutions. Different types of structural changes of the control strategies are identified.