Calculation of the Stability Index in Parameter-Dependent Calculus of Variations Problems: Buckling of a Twisted Elastic Strut

We consider the problem of minimizing the energy of an inextensible elastic strut with length 1 subject to an imposed twist angle and force. In a standard calculus of variations approach, one first locates equilibria by solving the Euler--Lagrange ODE with boundary conditions at arclength values 0 and 1. Then one classifies each equilibrium by counting conjugate points, with local minima corresponding to equilibria with no conjugate points. These conjugate points are arclength values $\sigma \le 1$ at which a second ODE (the Jacobi equation) has a solution vanishing at $0$ and $\sigma$. Finding conjugate points normally involves the numerical solution of a set of initial value problems for the Jacobi equation. For problems involving a parameter $\lambda$, such as the force or twist angle in the elastic strut, this computation must be repeated for every value of $\lambda$ of interest. Here we present an alternative approach that takes advantage of the presence of a parameter $\lambda$. Rather than search for conjugate points $\sigma \le 1$ at a fixed value of $\lambda$, we search for a set of special parameter values $\lambda_m$ (with corresponding Jacobi solution \bfzeta^m) for which $\sigma=1$ is a conjugate point. We show that, under appropriate assumptions, the index of an equilibrium at any $\lambda$ equals the number of these \bfzeta^m for which \langle \bfzeta^m, \Op \bfzeta^m \rangle < 0, where \Op is the Jacobi differential operator at $\lambda$. This computation is particularly simple when $\lambda$ appears linearly in \Op. We apply this approach to the elastic strut, in which the force appears linearly in \Op, and, as a result, we locate the conjugate points for any twisted unbuckled rod configuration without resorting to numerical solution of differential equations. In addition, we numerically compute two-dimensional sheets of buckled equilibria (as the two parameters of force and twist are varied) via a coordinated family of one-dimensional parameter continuation computations. Conjugate points for these buckled equilibria are determined by numerical solution of the Jacobi ODE

Multi-Objective Reinforcement Learning for Cognitive Radio-Based Satellite Communications

Previous research on cognitive radios has addressed the performance of various machine-learning and optimization techniques for decision making of terrestrial link properties. In this paper, we present our recent investigations with respect to reinforcement learning that potentially can be employed by future cognitive radios installed onboard satellite communications systems specifically tasked with radio resource management. This work analyzes the performance of learning, reasoning, and decision making while considering multiple objectives for time-varying communications channels, as well as different cross-layer requirements. Based on the urgent demand for increased bandwidth, which is being addressed by the next generation of high-throughput satellites, the performance of cognitive radio is assessed considering links between a geostationary satellite and a fixed ground station operating at Ka-band (26 GHz). Simulation results show multiple objective performance improvements of more than 3.5 times for clear sky conditions and 6.8 times for rain conditions

Multi-Objective Reinforcement Learning-based Deep Neural Networks for Cognitive Space Communications

Future communication subsystems of space exploration missions can potentially benefit from software-defined radios (SDRs) controlled by machine learning algorithms. In this paper, we propose a novel hybrid radio resource allocation management control algorithm that integrates multi-objective reinforcement learning and deep artificial neural networks. The objective is to efficiently manage communications system resources by monitoring performance functions with common dependent variables that result in conflicting goals. The uncertainty in the performance of thousands of different possible combinations of radio parameters makes the trade-off between exploration and exploitation in reinforcement learning (RL) much more challenging for future critical space-based missions. Thus, the system should spend as little time as possible on exploring actions, and whenever it explores an action, it should perform at acceptable levels most of the time. The proposed approach enables on-line learning by interactions with the environment and restricts poor resource allocation performance through virtual environment exploration. Improvements in the multiobjective performance can be achieved via transmitter parameter adaptation on a packet-basis, with poorly predicted performance promptly resulting in rejected decisions. Simulations presented in this work considered the DVB-S2 standard adaptive transmitter parameters and additional ones expected to be present in future adaptive radio systems. Performance results are provided by analysis of the proposed hybrid algorithm when operating across a satellite communication channel from Earth to GEO orbit during clear sky conditions. The proposed approach constitutes part of the core cognitive engine proof-of-concept to be delivered to the NASA Glenn Research Center SCaN Testbed located onboard the International Space Station