Filippov's selection theorem and the existence of solutions for optimal control problems in time scales

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

On some extension of optimal control theory

Some problems of Calculus of Variations do not have solutions in the class of classic continuous and smooth arcs. This suggests the need of a relaxation or extension of the problem ensuring the existence of a solution in some enlarged class of arcs. This work aims at the development of an extension for a more general optimal control problem with nonlinear control dynamics in which the control function takes values in some closed, but not necessarily bounded, set. To achieve this goal, we exploit the approach of R.V. Gamkrelidze based on the generalized controls, but related to discontinuous arcs. This leads to the notion of generalized impulsive control. The proposed extension links various approaches on the issue of extension found in the literature.

Explicit Construction of Optimal Exact Regenerating Codes for Distributed Storage

Erasure coding techniques are used to increase the reliability of distributed storage systems while minimizing storage overhead. Also of interest is minimization of the bandwidth required to repair the system following a node failure. In a recent paper, Wu et al. characterize the tradeoff between the repair bandwidth and the amount of data stored per node. They also prove the existence of regenerating codes that achieve this tradeoff. In this paper, we introduce Exact Regenerating Codes, which are regenerating codes possessing the additional property of being able to duplicate the data stored at a failed node. Such codes require low processing and communication overheads, making the system practical and easy to maintain. Explicit construction of exact regenerating codes is provided for the minimum bandwidth point on the storage-repair bandwidth tradeoff, relevant to distributed-mail-server applications. A sub-space based approach is provided and shown to yield necessary and sufficient conditions on a linear code to possess the exact regeneration property as well as prove the uniqueness of our construction. Also included in the paper, is an explicit construction of regenerating codes for the minimum storage point for parameters relevant to storage in peer-to-peer systems. This construction supports a variable number of nodes and can handle multiple, simultaneous node failures. All constructions given in the paper are of low complexity, requiring low field size in particular.

Mean first passage time approach to the problem of optimal barrier subdivision for Kramer's escape rate

We consider the effect of subdividing the potential barrier along the reaction coordinate on Kramer's escape rate for a model potential, Using the known supersymmetric potential approach, we show the existence of an optimal number of subdivisions that maximizes the rate, We cast the problem as a mean first passage time problem of a biased random walker and obtain equivalent results, We briefly summarize the results of our investigation on the increase in the escape rate by placing a blow-torch in the unstable part of one of the potential wells. (C) 1999 Elsevier Science B.V. All rights reserved.

The geometry of optimal control solutions on some six dimensional lie groups

This paper examines optimal solutions of control systems with drift defined on the orthonormal frame bundle of particular Riemannian manifolds of constant curvature. The manifolds considered here are the space forms Euclidean space E-3, the spheres S-3 and the hyperboloids H-3 with the corresponding frame bundles equal to the Euclidean group of motions SE(3), the rotation group SO(4) and the Lorentz group SO(1,3). The optimal controls of these systems are solved explicitly in terms of elliptic functions. In this paper, a geometric interpretation of the extremal solutions is given with particular emphasis to a singularity in the explicit solutions. Using a reduced form of the Casimir functions the geometry of these solutions are illustrated.

Analysis of optimal control problems for the incompressible MHD equations and implementation in a finite element multiphysics code

This thesis deals with the study of optimal control problems for the incompressible Magnetohydrodynamics (MHD) equations. Particular attention to these problems arises from several applications in science and engineering, such as fission nuclear reactors with liquid metal coolant and aluminum casting in metallurgy. In such applications it is of great interest to achieve the control on the fluid state variables through the action of the magnetic Lorentz force. In this thesis we investigate a class of boundary optimal control problems, in which the flow is controlled through the boundary conditions of the magnetic field. Due to their complexity, these problems present various challenges in the definition of an adequate solution approach, both from a theoretical and from a computational point of view. In this thesis we propose a new boundary control approach, based on lifting functions of the boundary conditions, which yields both theoretical and numerical advantages. With the introduction of lifting functions, boundary control problems can be formulated as extended distributed problems. We consider a systematic mathematical formulation of these problems in terms of the minimization of a cost functional constrained by the MHD equations. The existence of a solution to the flow equations and to the optimal control problem are shown. The Lagrange multiplier technique is used to derive an optimality system from which candidate solutions for the control problem can be obtained. In order to achieve the numerical solution of this system, a finite element approximation is considered for the discretization together with an appropriate gradient-type algorithm. A finite element object-oriented library has been developed to obtain a parallel and multigrid computational implementation of the optimality system based on a multiphysics approach. Numerical results of two- and three-dimensional computations show that a possible minimum for the control problem can be computed in a robust and accurate manner.

Exchange controls and the transmission of equity market volatility:the case of the UK

The paper investigates the impact that the relaxation of UK exchange controls in October 1979, had on the transmission of equity market volatility from the UK to other major equity markets. It is suggested that the existence of exchange controls in the UK was an important source of market segmentation which disturbed the transmission of shocks from one country to another, even when shocks contained global information. It is found that when a spillover GARCH(1,1) model is estimated for the five years before and after the removal of exchange controls, volatility shocks spill over from the UK to other markets much more strongly after the removal of exchange controls. This appears to suggest that volatility as well as returns have become more closely related since the UK removed exchange controls.

Determination of optimal preventive maintenance strategy for serial production lines

A Split System Approach (SSA) based methodology is presented to assist in making optimal Preventive Maintenance decisions for serial production lines. The methodology treats a production line as a complex series system with multiple PM actions over multiple intervals. Both risk related cost and maintenance related cost are factored into the methodology as either deterministic or random variables. This SSA based methodology enables Asset Management (AM) decisions to be optimized considering a variety of factors including failure probability, failure cost, maintenance cost, PM performance, and the type of PM strategy. The application of this new methodology and an evaluation of the effects of these factors on PM decisions are demonstrated using an example. The results of this work show that the performance of a PM strategy can be measured by its Total Expected Cost Index (TECI). The optimal PM interval is dependent on TECI, PM performance and types of PM strategies. These factors are interrelated. Generally it was found that a trade-off between reliability and the number of PM actions needs to be made so that one can minimize Total Expected Cost (TEC) for asset maintenance.

A Stochastic View of Optimal Regret through Minimax Duality

We study the regret of optimal strategies for online convex optimization games. Using von Neumann's minimax theorem, we show that the optimal regret in this adversarial setting is closely related to the behavior of the empirical minimization algorithm in a stochastic process setting: it is equal to the maximum, over joint distributions of the adversary's action sequence, of the difference between a sum of minimal expected losses and the minimal empirical loss. We show that the optimal regret has a natural geometric interpretation, since it can be viewed as the gap in Jensen's inequality for a concave functional--the minimizer over the player's actions of expected loss--defined on a set of probability distributions. We use this expression to obtain upper and lower bounds on the regret of an optimal strategy for a variety of online learning problems. Our method provides upper bounds without the need to construct a learning algorithm; the lower bounds provide explicit optimal strategies for the adversary. Peter L. Bartlett, Alexander Rakhlin

Inequitable Provision of Optimal Services for Patients with Chronic Heart failure: A National Geo-Mapping Study

Objective: To compare the location and accessibility of current Australian chronic heart failure (CHF) management programs and general practice services with the probable distribution of the population with CHF. Design and setting: Data on the prevalence and distribution of the CHF population throughout Australia, and the locations of CHF management programs and general practice services from 1 January 2004 to 31 December 2005 were analysed using geographic information systems (GIS) technology. Outcome measures: Distance of populations with CHF to CHF management programs and general practice services. Results: The highest prevalence of CHF (20.3–79.8 per 1000 population) occurred in areas with high concentrations of people over 65 years of age and in areas with higher proportions of Indigenous people. Five thousand CHF patients (8%) discharged from hospital in 2004–2005 were managed in one of the 62 identified CHF management programs. There were no CHF management programs in the Northern Territory or Tasmania. Only four CHF management programs were located outside major cities, with a total case load of 80 patients (0.7%). The mean distance from any Australian population centre to the nearest CHF management program was 332 km (median, 163 km; range, 0.15–3246 km). In rural areas, where the burden of CHF management falls upon general practitioners, the mean distance to general practice services was 37 km (median, 20 km; range, 0–656 km). Conclusion: There is an inequity in the provision of CHF management programs to rural Australians.

Sing & Grow : the co-existence of evaluation research and clinical practice in an early intervention music therapy project

Sing & Grow is a short term early intervention music therapy program for at risk families. Sing & Grow uses music to strengthen parent-child relationships by increasing positive parent-child interactions, assisting parents to bond with their children, and extending the repertoire of parents’ skills in relating to their child through interactive . Both the Australian and New Zealand governments are looking for evidence based research to highlight the effectiveness of funded programs in early childhood. As a government funded program, independent evaluation is a requirement of the delivery of the service. This paper explains the process involved in setting up and managing this large scale evaluation from engaging the evaluators and designing the project, to the data gathering stage. It describes the various challenges encountered and concludes that a highly collaborative and communicative partnership bet en researchers and clinicians is essential to ensure data can be gathered with minimal disturbance to clinical music therapy practice.