Optimal Navigation Functions for Nonlinear Stochastic Systems

This paper presents a new methodology to craft navigation functions for nonlinear systems with stochastic uncertainty. The method relies on the transformation of the Hamilton-Jacobi-Bellman (HJB) equation into a linear partial differential equation. This approach allows for optimality criteria to be incorporated into the navigation function, and generalizes several existing results in navigation functions. It is shown that the HJB and that existing navigation functions in the literature sit on ends of a spectrum of optimization problems, upon which tradeoffs may be made in problem complexity. In particular, it is shown that under certain criteria the optimal navigation function is related to Laplace's equation, previously used in the literature, through an exponential transform. Further, analytical solutions to the HJB are available in simplified domains, yielding guidance towards optimality for approximation schemes. Examples are used to illustrate the role that noise, and optimality can potentially play in navigation system design.Comment: Accepted to IROS 2014. 8 Page

ADAM: a general method for using various data types in asteroid reconstruction

We introduce ADAM, the All-Data Asteroid Modelling algorithm. ADAM is simple and universal since it handles all disk-resolved data types (adaptive optics or other images, interferometry, and range-Doppler radar data) in a uniform manner via the 2D Fourier transform, enabling fast convergence in model optimization. The resolved data can be combined with disk-integrated data (photometry). In the reconstruction process, the difference between each data type is only a few code lines defining the particular generalized projection from 3D onto a 2D image plane. Occultation timings can be included as sparse silhouettes, and thermal infrared data are efficiently handled with an approximate algorithm that is sufficient in practice due to the dominance of the high-contrast (boundary) pixels over the low-contrast (interior) ones. This is of particular importance to the raw ALMA data that can be directly handled by ADAM without having to construct the standard image. We study the reliability of the inversion by using the independent shape supports of function series and control-point surfaces. When other data are lacking, one can carry out fast nonconvex lightcurve-only inversion, but any shape models resulting from it should only be taken as illustrative global-scale ones.Comment: 11 pages, submitted to A&

Wave-Packet Scattering off the Kink-Solution

We investigate the propagation of a wave--packet in the $\phi^4$ model. We solve the time-dependent equation of motion for two distinct initial conditions: The wave-packet in a trivial vacuum background and in the background of the kink soliton solution. We extract the scattering matrix from the wave-packet in the kink background at very late times and compare it with the result from static potential scattering in the small amplitude approximation. We vary the size of the initial wave-packet to identify non-linear effects as, for example, the replacement of the center of the kink.Comment: 15 pages, 7 figures (from 14 eps files), 4 tables, Int. J. Mod. Phys. A, in prin

Scalable Approach to Uncertainty Quantification and Robust Design of Interconnected Dynamical Systems

Development of robust dynamical systems and networks such as autonomous aircraft systems capable of accomplishing complex missions faces challenges due to the dynamically evolving uncertainties coming from model uncertainties, necessity to operate in a hostile cluttered urban environment, and the distributed and dynamic nature of the communication and computation resources. Model-based robust design is difficult because of the complexity of the hybrid dynamic models including continuous vehicle dynamics, the discrete models of computations and communications, and the size of the problem. We will overview recent advances in methodology and tools to model, analyze, and design robust autonomous aerospace systems operating in uncertain environment, with stress on efficient uncertainty quantification and robust design using the case studies of the mission including model-based target tracking and search, and trajectory planning in uncertain urban environment. To show that the methodology is generally applicable to uncertain dynamical systems, we will also show examples of application of the new methods to efficient uncertainty quantification of energy usage in buildings, and stability assessment of interconnected power networks

Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer

A digital computer is generally believed to be an efficient universal computing device; that is, it is believed able to simulate any physical computing device with an increase in computation time of at most a polynomial factor. This may not be true when quantum mechanics is taken into consideration. This paper considers factoring integers and finding discrete logarithms, two problems which are generally thought to be hard on a classical computer and have been used as the basis of several proposed cryptosystems. Efficient randomized algorithms are given for these two problems on a hypothetical quantum computer. These algorithms take a number of steps polynomial in the input size, e.g., the number of digits of the integer to be factored.Comment: 28 pages, LaTeX. This is an expanded version of a paper that appeared in the Proceedings of the 35th Annual Symposium on Foundations of Computer Science, Santa Fe, NM, Nov. 20--22, 1994. Minor revisions made January, 199