Geometric existence theory for the control-affine nonlinear optimal regulator

AbstractFor infinite horizon nonlinear optimal control problems in which the control term enters linearly in the dynamics and quadratically in the cost, well-known conditions on the linearised problem guarantee existence of a smooth globally optimal feedback solution on a certain region of state space containing the equilibrium point. The method of proof is to demonstrate existence of a stable Lagrangian manifold M and then construct the solution from M in the region where M has a well-defined projection onto state space. We show that the same conditions also guarantee existence of a nonsmooth viscosity solution and globally optimal set-valued feedback on a much larger region. The method of proof is to extend the construction of a solution from M into the region where M no-longer has a well-defined projection onto state space

Contamination of histology biopsy specimen - a potential source of error for surgeons: a case report

Tissue contamination is a common occurrence in pathology, but surgeons are relatively unaware of this. We present the case of a 45-year-old man with Barrett's oesophagus, in which the histology of routine biopsies of an asymptomatic patient, were reported as 'carcinoma in situ'. Further biopsies were taken over a three month period but showed no evidence of malignancy. Tissue contamination or 'cross over' was identified as the likely cause of the abnormal result. This case report highlights the importance of the correlation of the clinical and histopathological findings and tissue contamination should be considered when both of these findings are not consistent

Protocol for a systematic review of preference-based instruments for measuring care-related outcomes and their suitability for the palliative care setting

Published by the BMJ Publishing Group Limited. For permission to use (where not already granted under a licence) please go to http://www.bmj.com/company/products-services/rights-and-licensing/ INTRODUCTION: Despite informal caregivers' integral role in supporting people affected by disease or disability, economic evaluations often ignore the costs and benefits experienced by this group, especially in the palliative setting. The purpose of this systematic review is to identify preference-based instruments for measuring care-related outcomes and provide guidance on the selection of instrument in palliative care economic evaluations.METHODS AND ANALYSIS: A comprehensive search of the literature will be conducted from database inception (ASSIA; CINAHL; Cochrane library including DARE, NHS EED, HTA; Econlit; Embase; PsychINFO; PubMed). Published peer-reviewed, English-language articles reporting preference-based instruments for measuring care-related outcomes in any clinical area will be included. One researcher will complete the searches and screen the results for potentially eligible studies. A randomly selected subset of 10% citations will be independently screened by two researchers. Any disagreement will be resolved by consensus among the research team. Subsequently, a supplementary search will identify studies detailing the development, valuation, validation and application of the identified instruments. The degree of suitability of the instruments for palliative economic evaluations will be assessed using criteria in the International Society for Quality of Life Research minimum standards for patient-reported outcome measures, the checklist for reporting valuation studies of multiattribute utility-based instruments and information on the development of the instrument in the palliative setting. A narrative summary of the included studies and instruments will be provided; similarities and differences will be described and possible reasons for variations explored. Recommendations for practice on selection of instruments in palliative care economic analyses will be provided.ETHICS AND DISSEMINATION: This is a planned systematic review of published literature. Therefore, ethics approval to conduct this research is not required. Findings will be presented at leading palliative care and health economic conferences and published in a peer-reviewed journal.TRIAL REGISTRATION NUMBER: CRD42016034188

A Topological Interpretation of Stability in Linear Optimal Control Problems

A simple linear optimal control problem is interpreted as that of finding Jacobi fields along a geodesic on a Riemannian manifold. In this context, stability can be interpreted as a Morse index condition on the initial manifold for the geodesic problem. It is indicated how this stability condition van be generalised to non-linear problems via the Maslov index

Lie Algebras, Structure of Nonlinear Systems and Chaotic Motion

The structure theory of Lie algebras is used to classify nonlinear systems according to a Levi decomposition and the solvable and semisimple parts of a certain Lie algebra associated with the system. An approximation theory is developed and a new class of chaotic systems is introduced

Lagrangian Manifolds, Viscosity Solutions and Maslov Index

This paper considers a geometrical construction for stationary viscosity solutions based on Lagrangian manifolds. This has been proposed recently in the literature by M.V. Day. The construction involves a key assumption of Lipschitz continuity. We discuss in this paper how this assumption follows from the vanishing of the Maslov index on closed curves on the manifold, at least foe low dimensional manifolds. This is naturally satisfied for large portions of stable and unstable manifolds corresponding to hyperbolic equilibrium points. We discuss examples where this situation arises including infinite time control and nonlinear filtering. We also give an introduction to the Maslov index and outline the connections with canonical tunnelling operators and idempotent analysis approaches to deriving stationary solutions to Hamilton-Jacobi-Bellman equations

Lagrangian Manifolds and Asymptotically Optimal Stabilizing Feedback Control

In this paper, under some local controllability hypotheses, we present an algorithm for finding an approximation to the optimal control for an infinite horizon linear analytic optimal regulator problem. This approximation is asymptotically optimal, so that it converges to the optimal control close to the origin and is extremely simple to to implement as it involves only the solution of algebraic Riccati equations. We demonstrate its effectiveness on a simulated inverted pendulum and present a test for determining in advance a stability region