An efficient method for multiobjective optimal control and optimal control subject to integral constraints

We introduce a new and efficient numerical method for multicriterion optimal control and single criterion optimal control under integral constraints. The approach is based on extending the state space to include information on a "budget" remaining to satisfy each constraint; the augmented Hamilton-Jacobi-Bellman PDE is then solved numerically. The efficiency of our approach hinges on the causality in that PDE, i.e., the monotonicity of characteristic curves in one of the newly added dimensions. A semi-Lagrangian "marching" method is used to approximate the discontinuous viscosity solution efficiently. We compare this to a recently introduced "weighted sum" based algorithm for the same problem. We illustrate our method using examples from flight path planning and robotic navigation in the presence of friendly and adversarial observers.Comment: The final version accepted by J. Comp. Math. : 41 pages, 14 figures. Since the previous version: typos fixed, formatting improved, one mistake in bibliography correcte

Multiobjective Optimization of Non-Smooth PDE-Constrained Problems

Multiobjective optimization plays an increasingly important role in modern applications, where several criteria are often of equal importance. The task in multiobjective optimization and multiobjective optimal control is therefore to compute the set of optimal compromises (the Pareto set) between the conflicting objectives. The advances in algorithms and the increasing interest in Pareto-optimal solutions have led to a wide range of new applications related to optimal and feedback control - potentially with non-smoothness both on the level of the objectives or in the system dynamics. This results in new challenges such as dealing with expensive models (e.g., governed by partial differential equations (PDEs)) and developing dedicated algorithms handling the non-smoothness. Since in contrast to single-objective optimization, the Pareto set generally consists of an infinite number of solutions, the computational effort can quickly become challenging, which is particularly problematic when the objectives are costly to evaluate or when a solution has to be presented very quickly. This article gives an overview of recent developments in the field of multiobjective optimization of non-smooth PDE-constrained problems. In particular we report on the advances achieved within Project 2 "Multiobjective Optimization of Non-Smooth PDE-Constrained Problems - Switches, State Constraints and Model Order Reduction" of the DFG Priority Programm 1962 "Non-smooth and Complementarity-based Distributed Parameter Systems: Simulation and Hierarchical Optimization"

State-of-the-art in aerodynamic shape optimisation methods

Aerodynamic optimisation has become an indispensable component for any aerodynamic design over the past 60 years, with applications to aircraft, cars, trains, bridges, wind turbines, internal pipe flows, and cavities, among others, and is thus relevant in many facets of technology. With advancements in computational power, automated design optimisation procedures have become more competent, however, there is an ambiguity and bias throughout the literature with regards to relative performance of optimisation architectures and employed algorithms. This paper provides a well-balanced critical review of the dominant optimisation approaches that have been integrated with aerodynamic theory for the purpose of shape optimisation. A total of 229 papers, published in more than 120 journals and conference proceedings, have been classified into 6 different optimisation algorithm approaches. The material cited includes some of the most well-established authors and publications in the field of aerodynamic optimisation. This paper aims to eliminate bias toward certain algorithms by analysing the limitations, drawbacks, and the benefits of the most utilised optimisation approaches. This review provides comprehensive but straightforward insight for non-specialists and reference detailing the current state for specialist practitioners

International Conference on Continuous Optimization (ICCOPT) 2019 Conference Book

The Sixth International Conference on Continuous Optimization took place on the campus of the Technical University of Berlin, August 3-8, 2019. The ICCOPT is a flagship conference of the Mathematical Optimization Society (MOS), organized every three years. ICCOPT 2019 was hosted by the Weierstrass Institute for Applied Analysis and Stochastics (WIAS) Berlin. It included a Summer School and a Conference with a series of plenary and semi-plenary talks, organized and contributed sessions, and poster sessions. This book comprises the full conference program. It contains, in particular, the scientific program in survey style as well as with all details, and information on the social program, the venue, special meetings, and more

Bayesian multi-objective optimisation with mixed analytical and black-box functions: application to tissue engineering

Tissue engineering and regenerative medicine looks at improving or restoring biological tissue function in humans and animals. We consider optimising neotissue growth in a three-dimensional scaffold during dynamic perfusion bioreactor culture, in the context of bone tissue engineering. The goal is to choose design variables that optimise two conflicting objectives: (i) maximising neotissue growth and (ii) minimising operating cost. We make novel extensions to Bayesian multi-objective optimisation in the case of one analytical objective function and one black-box, i.e. simulation-based, objective function. The analytical objective represents operating cost while the black-box neotissue growth objective comes from simulating a system of partial differential equations. The resulting multi-objective optimisation method determines the trade-off in the variables between neotissue growth and operating cost. Our method outperforms the most common approach in literature, genetic algorithms, in terms of data efficiency, on both the tissue engineering example and standard test functions. The resulting method is highly applicable to real-world problems combining black-box models with easy-to-quantify objectives like cost

Space-time POD-Galerkin approach for parametric flow control

In this contribution we propose reduced order methods to fast and reliably solve parametrized optimal control problems governed by time dependent nonlinear partial differential equations. Our goal is to provide a tool to deal with the time evolution of several nonlinear optimality systems in many-query context, where a system must be analysed for various physical and geometrical features. Optimal control can be used in order to fill the gap between collected data and mathematical model and it is usually related to very time consuming activities: inverse problems, statistics, etc. Standard discretization techniques may lead to unbearable simulations for real applications. We aim at showing how reduced order modelling can solve this issue. We rely on a space-time POD-Galerkin reduction in order to solve the optimal control problem in a low dimensional reduced space in a fast way for several parametric instances. The proposed algorithm is validated with a numerical test based on environmental sciences: a reduced optimal control problem governed by viscous Shallow Waters Equations parametrized not only in the physics features, but also in the geometrical ones. We will show how the reduced model can be useful in order to recover desired velocity and height profiles more rapidly with respect to the standard simulation, not losing accuracy

On the generation of environmentally efficient flight trajectories

To achieve a sustainable future for air transport, the International Civil Aviation Organization has proposed goals for reductions in community noise impact, local air quality and climate impacting emissions. The goals are intended to be achieved through advances in engine design, aircraft design and through improvements in aircraft operational procedures. This thesis focuses on operational procedures, and considers how trajectory generation methods can be used to support flight and airspace planners in the planning and delivery of environmentally efficient flight operations. The problem of planning environmentally efficient trajectories is treated as an optimal control problem that is solved through the application of a direct method of trajectory optimisation combined with a stochastic Non Linear Programming (NLP) solver. Solving the problem in this manner allows decision makers to explore the relationships between how aircraft are operated and the consequent environmental impacts of the flights. In particular, this thesis describes a multi-objective optimisation methodology intended to support the planning of environmentally efficient climb and descent procedures. The method combines environmental, trajectory and NLP methods to generate Pareto fronts between several competing objectives. It is shown how Pareto front information can then be used to allow decision makers to make informed decisions about potential tradeoffs between different environmental goals. The method is demonstrated through its application to a number of real world, many objective procedure optimisation studies. The method is shown to support in depth analysis of the case study problems and was used to identify best balance procedure characteristics and procedures in an objective, data driven approach not achievable through existing methods. Driven by operator specific goals to reduce CO2 emissions, work in this thesis also looks at trajectory based flight planning of CO2 efficient trajectories. The results are used to better understand the impacts of ATM constraints and recommended procedures on both the energy management and fuel efficiency of flights. Further to this, it is shown how trajectory optimisation methods can be applied to the analysis of conventional assumptions on fuel efficient aircraft operations. While the work within is intended to be directly relevant to the current air traffic management system, both consideration and discussion is given over to the evolution and continued relevance of the work to the Single European Sky trajectory based concept of operation