Stability of Stochastic Reaction-Diffusion Systems with Markovian Switching and Impulsive Perturbations

This paper is devoted to investigating mean square stability of a class of stochastic reactiondiffusion systems with Markovian switching and impulsive perturbations. Based on Lyapunov functions and stochastic analysis method, some new criteria are established. Moreover, a class of semilinear stochastic impulsive reaction-diffusion differential equations with Markovian switching is discussed and a numerical example is presented to show the effectiveness of the obtained results

Stability and Stabilization of Impulsive Stochastic Delay Differential Equations

We consider the stability and stabilization of impulsive stochastic delay differential equations (ISDDEs). Using the Lyapunov-Razumikhin method, we obtain the sufficient conditions to guarantee the pth moment exponential stability of ISDDEs. Then the almost sure exponential stability is considered and the sufficient conditions of the almost sure exponential stability are obtained. Moreover, the stabilization problem of ISDDEs is studied and the criterion on impulsive stabilization of ISDDEs is established. At last, examples are presented to illustrate the correctness of our results

Formation Flight of Earth Satellites on Low-Eccentricity KAM Tori

The problem of Earth satellite constellation and formation flight is investigated in the context of Kolmogorov-Arnold-Moser (KAM) theory. KAM tori are constructed utilizing Wiesel’s Low-Eccentricity Earth Satellite Theory, allowing numerical representation of the perturbed tori describing Earth orbits acted upon by geopotential perturbations as sets of Fourier series. A maneuvering strategy using the local linearization of the KAM tangent space is developed and applied, demonstrating the ability to maneuver onto and within desired torus surfaces. Constellation and formation design and maintenance on KAM tori are discussed, along with stability and maneuver error concerns. It is shown that placement of satellites on KAM tori results in virtually no secular relative motion in the full geopotential to within computational precision. The effects of maneuver magnitude errors are quantified in terms of a singular value decomposition of the modal system for several orbits of interest, introducing a statistical distribution in terms of torus angle drift rates due to mismatched energies. This distribution is then used to create expectations of the steady-state station-keeping costs, showing that these costs are driven by operational and spacecraft limitations, and not by limitations of the dynamics formulation. A non-optimal continuous control strategy for formations based on Control Lyapunov Functions is also outlined and demonstrated in the context of formation reconfiguration

Trajectory Control and Optimization for Responsive Spacecraft

The concept of responsive space has been gaining interest, and growing to include systems that can be re-tasked to complete multiple missions within their lifetime. The purpose of this study is to develop an algorithm that produces a maneuver trajectory that will cause a spacecraft to arrive at a particular location within its orbit earlier than expected. The time difference, delta t, is used as a metric to quantify the effects of the maneuver. Two separate algorithms are developed. The first algorithm is an optimal control method and is developed through Optimal Control Theory. The second algorithm is a feedback control method and is developed through Lyapunov Theory. It is shown that the two algorithms produce equivalent results for the maneuvers discussed. In-plane maneuver results are analyzed analytically, and an algebraic expression for delta t is derived. Examples are provided of how the analytic expression can be used for mission planning purposes. The feedback control algorithm is then further developed to demonstrate the simplicity of implementing additional capabilities. Finally, a set of simulations is analyzed to show that in order to maximize the amount of delta t achieved, a spacecraft must be allowed as much lead time as possible, and begin thrusting as early as possible

Trajectory Control and Optimization for Responsive Spacecraft

The concept of responsive space has been gaining interest, and growing to include systems that can be re-tasked to complete multiple missions within their lifetime. The purpose of this study is to develop an algorithm that produces a maneuver trajectory that will cause a spacecraft to arrive at a particular location within its orbit earlier than expected. The time difference, delta t, is used as a metric to quantify the effects of the maneuver. Two separate algorithms are developed. The first algorithm is an optimal control method and is developed through Optimal Control Theory. The second algorithm is a feedback control method and is developed through Lyapunov Theory. It is shown that the two algorithms produce equivalent results for the maneuvers discussed. In-plane maneuver results are analyzed analytically, and an algebraic expression for delta t is derived. Examples are provided of how the analytic expression can be used for mission planning purposes. The feedback control algorithm is then further developed to demonstrate the simplicity of implementing additional capabilities. Finally, a set of simulations is analyzed to show that in order to maximize the amount of delta t achieved, a spacecraft must be allowed as much lead time as possible, and begin thrusting as early as possible

Minimum-Fuel Trajectory Design in Multiple Dynamical Environments Utilizing Direct Transcription Methods and Particle Swarm Optimization

Particle swarm optimization is used to generate an initial guess for designing fuel-optimal trajectories in multiple dynamical environments. Trajectories designed in the vicinity of Earth use continuous or finite low-thrust burning and transfer from an inclined or equatorial circular low-Earth-orbit to a geostationary orbit. In addition, a trajectory from near-Earth to a periodic orbit about the cislunar Lagrange point with minimized impulsive burn costs is designed within a multi-body dynamical environment. Direct transcription is used in conjunction with a nonlinear optimizer to find locally-optimal trajectories given the initial guess. The near-Earth transfers are propagated at low-level thrust where neither the very-low-thrust spiral solution nor the impulsive transfer is an acceptable starting point. The very-high-altitude transfer is designed in a multi-body dynamical environment lacking a closed-form analytical solution. Swarming algorithms excel given a small number of design parameters.When continuous control time histories are needed, employing a polynomial parameterization facilitates the generation of feasible solutions. For design in a circular restricted three-body system, particle swarm optimization gains utility due to a more global search for the solution, but may be more sensitive to boundary constraints. Computation time and constraint weighting are areas where a swarming algorithm is weaker than other approaches

What is the temperature of a granular medium?

In this paper we discuss whether thermodynamical concepts and in particular the notion of temperature could be relevant for the dynamics of granular systems. We briefly review how a temperature-like quantity can be defined and measured in granular media in very different regimes, namely the glassy-like, the liquid-like and the granular gas. The common denominator will be given by the Fluctuation-Dissipation Theorem, whose validity is explored by means of both numerical and experimental techniques. It turns out that, although a definition of a temperature is possible in all cases, its interpretation is far from being obvious. We discuss the possible perspectives both from the theoretical and, more importantly, from the experimental point of view