Impulsive perturbations to differential equations: stable/unstable pseudo-manifolds, heteroclinic connections, and flux

State-dependent time-impulsive perturbations to a two-dimensional autonomous flow with stable and unstable manifolds are analysed by posing in terms of an integral equation which is valid in both forwards- and backwards-time. The impulses destroy the smooth invariant manifolds, necessitating new definitions for stable and unstable pseudo-manifolds. Their time-evolution is characterised by solving a Volterra integral equation of the second kind with discontinuous inhomogeniety. A criteria for heteroclinic trajectory persistence in this impulsive context is developed, as is a quantification of an instantaneous flux across broken heteroclinic manifolds. Several examples, including a kicked Duffing oscillator and an underwater explosion in the vicinity of an eddy, are used to illustrate the theory

A New Computational Method for Optimizing Nonlinear Impulsive Systems

In this paper, we consider a system that evolves by switching between several subsystems of ordinary differential equations. The switching mechanism is this system induces an instantaneous change in the system's state, which can be controlled through a set of decision parameters. We develop a new computational method, based on nonlinear programming, for optimizing the system parameters and the subsystem switching times. We then successfully apply this method to two interesting examples

Theoretical Foundation of Copernicus: A Unified System for Trajectory Design and Optimization

The fundamental methods are described for the general spacecraft trajectory design and optimization software system called Copernicus. The methods rely on a unified framework that is used to model, design, and optimize spacecraft trajectories that may operate in complex gravitational force fields, use multiple propulsion systems, and involve multiple spacecraft. The trajectory model, with its associated equations of motion and maneuver models, are discussed

Analytic Guidance Strategies for Passively Safe Rendezvous and Proximity Operations

A safety ellipse is a type of relative motion trajectory that is commonly used for unmanned rendezvous and proximity operations. As the name suggests, safety ellipses are passively safe relative motion trajectories, which means that their natural motion inherently maintains a low collision risk. The focus of this dissertation is the derivation, analysis, and application of guidance strategies that reconfigure, establish, and exit a safety ellipse. The guidance strategies consist of a set of ∆v vectors and impulse times, all written in closed form. Through applications of optimal control theory and parameter optimization, it is shown that these maneuver sequences are fuel optimal for a range of practical safety ellipse reconfiguration, establishment, and exit scenarios. It is also shown that the resulting transfer trajectories remain passively safe across the same range of scenarios

Finite dimensional approximation to fractional stochastic integro-differential equations with non-instantaneous impulses

This manuscript proposes a class of fractional stochastic integro-differential equation (FSIDE) with non-instantaneous impulses in an arbitrary separable Hilbert space. We use a projection scheme of increasing sequence of finite dimensional subspaces and projection operators to define approximations. In order to demonstrate the existence and convergence of an approximate solution, we utilize stochastic analysis theory, fractional calculus, theory of fractional cosine family of linear operators and fixed point approach. Furthermore, we examine the convergence of Faedo-Galerkin(F-G) approximate solution to the mild solution of our given problem. Finally, a concrete example involving partial differential equation is provided to validate the main abstract results