Lipschitz modulus of linear and convex inequality systems with the Hausdorff metric

This paper analyzes the Lipschitz behavior of the feasible set mapping associated with linear and convex inequality systems in Rn. To start with, we deal with the parameter space of linear (finite/semi-infinite) systems identified with the corresponding sets of coefficient vectors, which are assumed to be closed subsets of Rn+1. In this framework the size of perturbations is measured by means of the (extended) Hausdorff distance. A direct antecedent, extensively studied in the literature, comes from considering the parameter space of all linear systems with a fixed index set, T, where the Chebyshev (extended) distance is used to measure perturbations. In the present work we propose an appropriate indexation strategy which allows us to establish the equality of the Lipschitz moduli of the feasible set mappings in both parametric contexts, as well as to benefit from existing results in the Chebyshev setting for transferring them to the Hausdorff one. In a second stage, the possibility of perturbing directly the set of coefficient vectors of a linear system leads to new contributions on the Lipschitz behavior of convex systems via linearization techniques. © 2020, Springer-Verlag GmbH Germany, part of Springer Nature and Mathematical Optimization Society. Correction: The article “Lipschitz modulus of linear and convex inequality systems with the Hausdorff metric”, written by Beer,G., Cánovas, M.J., López, M.A., Parra, J.was originally published Online First without Open Access. After publication in volume 189, issue 1–2, page 75–98 the author decided to opt for Open Choice and to make the article an Open Access publication. Therefore, the copyright of the article has been changed to © The Author(s) 2020 and the article is forthwith distributed under the terms of the Creative Commons Attribution 4.0 International License. https://doi.org/10.1007/s10107-021-01751-

Solving trajectory optimization problems in the presence of probabilistic constraints

The objective of this paper is to present an approximation-based strategy for solving the problem of nonlinear trajectory optimization with the consideration of probabilistic constraints. The proposed method defines a smooth and differentiable function to replace probabilistic constraints by the deterministic ones, thereby converting the chance-constrained trajectory optimization model into a parametric nonlinear programming model. In addition, it is proved that the approximation function and the corresponding approximation set will converge to that of the original problem. Furthermore, the optimal solution of the approximated model is ensured to converge to the optimal solution of the original problem. Numerical results, obtained from a new chance-constrained space vehicle trajectory optimization model and a 3-D unmanned vehicle trajectory smoothing problem, verify the feasibility and effectiveness of the proposed approach. Comparative studies were also carried out to show the proposed design can yield good performance and outperform other typical chance-constrained optimization techniques investigated in this paper

A review of optimization techniques in spacecraft flight trajectory design

For most atmospheric or exo-atmospheric spacecraft flight scenarios, a well-designed trajectory is usually a key for stable flight and for improved guidance and control of the vehicle. Although extensive research work has been carried out on the design of spacecraft trajectories for different mission profiles and many effective tools were successfully developed for optimizing the flight path, it is only in the recent five years that there has been a growing interest in planning the flight trajectories with the consideration of multiple mission objectives and various model errors/uncertainties. It is worth noting that in many practical spacecraft guidance, navigation and control systems, multiple performance indices and different types of uncertainties must frequently be considered during the path planning phase. As a result, these requirements bring the development of multi-objective spacecraft trajectory optimization methods as well as stochastic spacecraft trajectory optimization algorithms. This paper aims to broadly review the state-of-the-art development in numerical multi-objective trajectory optimization algorithms and stochastic trajectory planning techniques for spacecraft flight operations. A brief description of the mathematical formulation of the problem is firstly introduced. Following that, various optimization methods that can be effective for solving spacecraft trajectory planning problems are reviewed, including the gradient-based methods, the convexification-based methods, and the evolutionary/metaheuristic methods. The multi-objective spacecraft trajectory optimization formulation, together with different class of multi-objective optimization algorithms, is then overviewed. The key features such as the advantages and disadvantages of these recently-developed multi-objective techniques are summarised. Moreover, attentions are given to extend the original deterministic problem to a stochastic version. Some robust optimization strategies are also outlined to deal with the stochastic trajectory planning formulation. In addition, a special focus will be given on the recent applications of the optimized trajectory. Finally, some conclusions are drawn and future research on the development of multi-objective and stochastic trajectory optimization techniques is discussed

Trajectory planning for hypersonic reentry vehicle satisfying deterministic and probabilistic constraints

The present work explores the optimal flight of aero-assisted reentry vehicles during the atmospheric entry flight phase with the consideration of both deterministic and control chance constraints. To describe the mission profile, a chance-constrained optimal control model is established. Due to the existence of probabilistic constraints (chance constraints), standard numerical trajectory planning algorithms cannot be directly applied to address the considered problem. Hence, we firstly present an approximation-based strategy to replace the probabilistic constraint by a deterministic version. In this way, the transformed optimal control model becomes solvable for standard trajectory optimization methods. In order to obtain enhanced computational performance, an alternative convex-relaxed optimal control formulation is also given. This is achieved by convexifying the vehicle nonlinear dynamics/constraints and by introducing a convex probabilistic constraint handling strategy. Numerical simulations are provided to demonstrate the effectiveness of these two chance-constrained optimization approaches and the corresponding probabilistic constraint handling strategies