Numerical Solution of the Dynamic Programming Equation for the Optimal Control of Quantum Spin Systems

The purpose of this paper is to describe the numerical solution of the Hamilton-Jacobi-Bellman (HJB) for an optimal control problem for quantum spin systems. This HJB equation is a first order nonlinear partial differential equation defined on a Lie group. We employ recent extensions of the theory of viscosity solutions from Euclidean space to Riemannian manifolds to interpret possibly non-differentiable solutions to this equation. Results from differential topology on the triangulation of manifolds are then used to develop a finite difference approximation method, which is shown to converge using viscosity solution techniques. An example is provided to illustrate the method.Comment: 11 pages, 5 figure

Towards adaptive multi-robot systems: self-organization and self-adaptation

Dieser Beitrag ist mit Zustimmung des Rechteinhabers aufgrund einer (DFG geförderten) Allianz- bzw. Nationallizenz frei zugänglich.This publication is with permission of the rights owner freely accessible due to an Alliance licence and a national licence (funded by the DFG, German Research Foundation) respectively.The development of complex systems ensembles that operate in uncertain environments is a major challenge. The reason for this is that system designers are not able to fully specify the system during specification and development and before it is being deployed. Natural swarm systems enjoy similar characteristics, yet, being self-adaptive and being able to self-organize, these systems show beneficial emergent behaviour. Similar concepts can be extremely helpful for artificial systems, especially when it comes to multi-robot scenarios, which require such solution in order to be applicable to highly uncertain real world application. In this article, we present a comprehensive overview over state-of-the-art solutions in emergent systems, self-organization, self-adaptation, and robotics. We discuss these approaches in the light of a framework for multi-robot systems and identify similarities, differences missing links and open gaps that have to be addressed in order to make this framework possible

Linear Hamilton Jacobi Bellman Equations in High Dimensions

The Hamilton Jacobi Bellman Equation (HJB) provides the globally optimal solution to large classes of control problems. Unfortunately, this generality comes at a price, the calculation of such solutions is typically intractible for systems with more than moderate state space size due to the curse of dimensionality. This work combines recent results in the structure of the HJB, and its reduction to a linear Partial Differential Equation (PDE), with methods based on low rank tensor representations, known as a separated representations, to address the curse of dimensionality. The result is an algorithm to solve optimal control problems which scales linearly with the number of states in a system, and is applicable to systems that are nonlinear with stochastic forcing in finite-horizon, average cost, and first-exit settings. The method is demonstrated on inverted pendulum, VTOL aircraft, and quadcopter models, with system dimension two, six, and twelve respectively.Comment: 8 pages. Accepted to CDC 201

Differential-Algebraic Equations and Beyond: From Smooth to Nonsmooth Constrained Dynamical Systems

The present article presents a summarizing view at differential-algebraic equations (DAEs) and analyzes how new application fields and corresponding mathematical models lead to innovations both in theory and in numerical analysis for this problem class. Recent numerical methods for nonsmooth dynamical systems subject to unilateral contact and friction illustrate the topicality of this development.Comment: Preprint of Book Chapte