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Scientific computation of conservation laws in the calculus of variations and optimal control

Gouveia, Paulo D.F.; Torres, Delfim F.M.
Fonte: Universidade de Aveiro Publicador: Universidade de Aveiro
Tipo: Relatório
ENG
Relevância na Pesquisa
75.82%
We present analytic computational tools that permit us to identify, in an automatic way, conservation laws in optimal control. The central result we use is the famous Noether’s theorem, a classical theory developed by Emmy Noether in 1918, in the context of the calculus of variations and mathematical physics, and which was extended recently to the more general context of optimal control. We show how a Computer Algebra System can be very helpful in finding the symmetries and corresponding conservation laws in optimal control theory, thus making useful in practice the theoretical results recently obtained in the literature. A Maple implementation is provided and several illustrative examples given.

L-splines - A manifestation of optimal control

Rodrigues, Rui C.; Leite, F. Silva
Fonte: Centro de Matemática da Universidade de Coimbra Publicador: Centro de Matemática da Universidade de Coimbra
Tipo: Pré-impressão
ENG
Relevância na Pesquisa
75.72%
We show how to generate a class of Euclidean splines, called L-splines, as solutions of a high-order variational problem. We also show connections between L-splines and optimal control theory, leading to the conclusion that L-splines are manifestations of an optimal behavior; ISR, project ERBFMRXCT970137

Fenômeno Fuller em problemas de controle ótimo: trajetórias em tempo mínino de veículos autônomos subaquáticos; Fuller Phenomenon in optimal control problems: minimum time path of autonomous underwater vehicles.

Oda, Eduardo
Fonte: Biblioteca Digitais de Teses e Dissertações da USP Publicador: Biblioteca Digitais de Teses e Dissertações da USP
Tipo: Dissertação de Mestrado Formato: application/pdf
Publicado em 03/06/2008 PT
Relevância na Pesquisa
75.84%
As equações do modelo bidimensional de veículos autônomos subaquáticos fornecem um exemplo de sistema de controle não linear com o qual podemos ilustrar propriedades da teoria de controle ótimo. Apresentamos, sistematicamente, como os conceitos de formalismo hamiltoniano e teoria de Lie aparecem de forma natural neste contexto. Para tanto, estudamos brevemente o Princípio do Máximo de Pontryagin e discutimos características de sistemas afins. Tratamos com cuidado do Fenômeno Fuller, fornecendo critérios para decidir quando ele está ou não presente em junções, utilizando para isso uma linguagem algébrica. Apresentamos uma abordagem numérica para tratar problemas de controle ótimo e finalizamos com a aplicação dos resultados ao modelo bidimensional de veículo autônomo subaquático.; The equations of the two-dimensional model for autonomous underwater vehicles provide an example of a nonlinear control system which illustrates properties of optimal control theory. We present, systematically, how the concepts of the Hamiltonian formalism and the Lie theory naturally appear in this context. For this purpose, we briefly study the Pontryagin's Maximum Principle and discuss features of affine systems. We treat carefully the Fuller Phenomenon...

Optimal and sub-optimal control in Dengue epidemics

Caetano, MAL; Yoneyama, T.
Fonte: Wiley-Blackwell Publicador: Wiley-Blackwell
Tipo: Artigo de Revista Científica Formato: 63-73
ENG
Relevância na Pesquisa
75.85%
This work concerns the application of the optimal control theory to Dengue epidemics. The dynamics of this insect-borne disease is modelled as a set of non-linear ordinary differential equations including the effect of educational campaigns organized to motivate the population to break the reproduction cycle of the mosquitoes by avoiding the accumulation of still water in open-air recipients. The cost functional is such that it reflects a compromise between actual financial spending (in insecticides and educational campaigns) and the population health (which can be objectively measured in terms of, for instance, treatment costs and loss of productivity). The optimal control problem is solved numerically using a multiple shooting method. However, the optimal control policy is difficult to implement by the health authorities because it is not practical to adjust the investment rate continuously in time. Therefore, a suboptimal control policy is computed assuming, as the admissible set, only those controls which are piecewise constant. The performance achieved by the optimal control and the sub-optimal control policies are compared with the cases of control using only insecticides when Breteau Index is greater or equal to 5 and the case of no-control. The results show that the sub-optimal policy yields a substantial reduction in the cost...

Optimal linear and nonlinear control design for chaotic systems

Rafikov, Marat; Balthazar, José Manoel
Fonte: Universidade Estadual Paulista Publicador: Universidade Estadual Paulista
Tipo: Conferência ou Objeto de Conferência Formato: 867-873
ENG
Relevância na Pesquisa
75.88%
In this work, the linear and nonlinear feedback control techniques for chaotic systems were been considered. The optimal nonlinear control design problem has been resolved by using Dynamic Programming that reduced this problem to a solution of the Hamilton-Jacobi-Bellman equation. In present work the linear feedback control problem has been reformulated under optimal control theory viewpoint. The formulated Theorem expresses explicitly the form of minimized functional and gives the sufficient conditions that allow using the linear feedback control for nonlinear system. The numerical simulations for the Rössler system and the Duffing oscillator are provided to show the effectiveness of this method. Copyright © 2005 by ASME.

On some extension of optimal control theory

Karamzin, Dmitry Yu; Oliveira, Valeriano A. de; Pereira, F.L.; Silva, Geraldo Nunes
Fonte: Universidade Estadual Paulista Publicador: Universidade Estadual Paulista
Tipo: Artigo de Revista Científica Formato: 284-291
POR
Relevância na Pesquisa
75.73%
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.

On the degeneracy phenomenon for nonlinear optimal control problems with higher index state constraints

Lopes, Sofia Oliveira; Fontes, Fernando A. C. C.
Fonte: Universidade do Minho Publicador: Universidade do Minho
Tipo: Relatório
Publicado em 04/06/2008 ENG
Relevância na Pesquisa
75.78%
Relatório Técnico do Núcleo de Investigação Officina Mathematica.; Necessary conditions of optimality (NCO) play an important role in optimization problems. They are the major tool to select a set of candidates to minimizers. In optimal control theory, the NCO appear in the form of a Maximum Principle (MP). For certain optimal control problems with state constraints, it might happen that the MP are unable to provide useful information --- the set of all admissible solutions coincides with the set of candidates that satisfy the MP. When this happens, the MP is said to degenerate. In the recent years, there has been some literature on fortified forms of the MP in such way that avoid degeneracy. These fortified forms involve additional hypotheses --- Constraint Qualifications. Whenever the state constraints have higher index (i.e. their first derivative with respect to time does not depend on control), the current constraint qualifications are not adequate. So, the main purpose here is fortify the maximum principle for optimal control problems with higher index constraints, for which there is a need to develop new constraint qualifications. The results presented here are a generalization to nonlinear problems of a previous work.; The financial support from Projecto FCT POSC/EEA-SRI/61831/2004 is gratefully acknowledged.

Optimal control for a tuberculosis model with reinfection and post-exposure interventions

Silva, C. J.; Torres, D. F. M.
Fonte: Elsevier Publicador: Elsevier
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
85.9%
We apply optimal control theory to a tuberculosis model given by a system of ordinary differential equations. Optimal control strategies are proposed to minimize the cost of interventions, considering reinfection and post-exposure interventions. They depend on the parameters of the model and reduce effectively the number of active infectious and persistent latent individuals. The time that the optimal controls are at the upper bound increase with the transmission coefficient. A general explicit expression for the basic reproduction number is obtained and its sensitivity with respect to the model parameters is discussed. Numerical results show the usefulness of the optimization strategies. © 2013 Elsevier Inc.

Optimal control strategies for tuberculosis treatment: a case study in Angola

Silva, C. J.; Torres, D. F. M.
Fonte: American Institute of Mathematical Sciences (AIMS) Publicador: American Institute of Mathematical Sciences (AIMS)
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
75.74%
We apply optimal control theory to a tuberculosis model given by a system of ordinary differential equations. Optimal control strategies are proposed to minimize the cost of interventions. Numerical simulations are given using data from Angola.

Multiobjective approach to optimal control for a tuberculosis model

Denysiuk, R.; Silva, C. J.; Torres, D. F. M.
Fonte: Taylor & Francis Publicador: Taylor & Francis
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
75.82%
Mathematical modelling can help to explain the nature and dynamics of infection transmissions, as well as support a policy for implementing those strategies that are most likely to bring public health and economic benefits. The paper addresses the application of optimal control strategies in a tuberculosis model. The model consists of a system of ordinary differential equations, which considers reinfection and post-exposure interventions. We propose a multiobjective optimization approach to find optimal control strategies for the minimization of active infectious and persistent latent individuals, as well as the cost associated to the implementation of the control strategies. Optimal control strategies are investigated for different values of the model parameters. The obtained numerical results cover a whole range of the optimal control strategies, providing valuable information about the tuberculosis dynamics and showing the usefulness of the proposed approach.

A TB-HIV/AIDS coinfection model and optimal control treatment

Silva, C. J.; Torres, D. F. M.
Fonte: American Institute of Mathematical Sciences (AIMS) Publicador: American Institute of Mathematical Sciences (AIMS)
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
75.76%
We propose a population model for TB-HIV/AIDS coinfection transmission dynamics, which considers antiretroviral therapy for HIV infection and treatments for latent and active tuberculosis. The HIV-only and TB-only sub-models are analyzed separately, as well as the TB-HIV/AIDS full model. The respective basic reproduction numbers are computed, equilibria and stability are studied. Optimal control theory is applied to the TB-HIV/AIDS model and optimal treatment strategies for co-infected individuals with HIV and TB are derived. Numerical simulations to the optimal control problem show that non intuitive measures can lead to the reduction of the number of individuals with active TB and AIDS.

Pseudospectral Methods for Optimal Motion Planning of Differentially Flat Systems

Ross, I. Michael; Fahroo, Fariba
Fonte: IEEE Publicador: IEEE
Tipo: Conference Paper
Relevância na Pesquisa
75.79%
The article of record as published may be located at http://ieeexplore.ieee.org; Approved for public display, distribution unlimited; IEEE Transactions on Automatic Control, vol. 49, no. 8, August 2004 (Journal Article); This note presents some preliminary results on combining two new ideas from nonlinear control theory and dynamic optimization. We show that the computational framework facilitated by pseudospectral methods applies quite naturally and easily to Fliess implicit state variable representation of dynamical systems. The optimal motion planning problem for differentially flat systems is equivalent to a classic Bolza problem of the calculus of variations. In this note, we exploit the notion that derivatives of flat outputs given in terms of Lagrange polynomials at Legendre'Gauss'Lobatto points can be quickly computed using pseudospectral differentiation matrices. Additionally, the Legendre pseudospectral method approximates integrals by Gauss-type quadrature rules. The application of this method to the two-dimensional crane model reveals how differential flatness may be readily exploited.

Automatic computation of conservation laws in the calculus of variations and optimal control

Gouveia, Paulo D.F.; Torres, Delfim F.M.
Fonte: Instituto Politécnico de Bragança Publicador: Instituto Politécnico de Bragança
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
75.82%
We present analytical computational tools that permit us to identify, in an automatic way, conservation laws in optimal control. The central result we use is the famous Noether's theorem, a classical theory developed by Emmy Noether in 1918, in the context of the calculus of variations and mathematical physics, which was extended recently to the more general context of optimal control. We show how a Computer Algebra System can be very helpful in ¯nding the symmetries and corresponding conservation laws in optimal control theory, thus making useful in practice the theoretical results recently obtained in the literature. A Maple implementation is provided and several illustrative examples are given.

Automatic computation of conservation laws in the calculus of variations and optimal control

Gouveia, Paulo D.F.; Torres, Delfim F.M.
Fonte: Maplesoft Publicador: Maplesoft
Tipo: Trabalho em Andamento
ENG
Relevância na Pesquisa
75.82%
Computer Application; We present analytic computational tools that permit us to identify, in an automatic way, conservation laws in optimal control. The central result we use is the famous Noether's theorem, a classical theory developed by Emmy Noether in 1918, in the context of the calculus of variations and mathematical physics, and which was extended recently to the more general context of optimal control. We show how a Computer Algebra System can be very helpful in finding the symmetries and corresponding conservation laws in optimal control theory, thus making useful in practice the theoretical results recently obtained in the literature. A Maple implementation is provided and several illustrative examples given.

Dynamics of Dengue epidemics using optimal control

Rodrigues, Helena Sofia; Monteiro, M. Teresa T.; Torres, Delfim F. M.
Fonte: Elsevier Publicador: Elsevier
Tipo: Artigo de Revista Científica
Publicado em /11/2010 ENG
Relevância na Pesquisa
75.79%
We present an application of optimal control theory to Dengue epidemics. This epidemiologic disease is important in tropical countries due to the growing number of infected individuals. The dynamic model is described by a set of nonlinear ordinary differential equations, that depend on the dynamics of the Dengue mosquito, the number of infected individuals, and people's motivation to combat the mosquito. The cost functional depends not only on the costs of medical treatment of the infected people but also on the costs related to educational and sanitation campaigns. Two approaches for solving the problem are considered: one using optimal control theory, the other carried out by first discretizing the problem and then solving it with nonlinear programming. The results obtained with OC-ODE and IPOPT solvers are given and discussed. We observe that with current computational tools it is easy to obtain, in an efficient way, better solutions to Dengue problems, leading to a decrease in the number of infected mosquitoes and individuals in less time and with lower costs.; Fundação para a Ciência e a Tecnologia (FCT)

Dynamics of Dengue epidemics when using optimal control

Rodrigues, H.S.; Monteiro, M.T.T.; Torres, D.F.M.
Fonte: Elsevier Publicador: Elsevier
Tipo: Artigo de Revista Científica
ENG
Relevância na Pesquisa
75.79%
We present an application of optimal control theory to Dengue epidemics. This epidemiologic disease is important in tropical countries due to the growing number of infected individuals. The dynamic model is described by a set of nonlinear ordinary differential equations, that depend on the dynamics of the Dengue mosquito, the number of infected individuals, and people's motivation to combat the mosquito. The cost functional depends not only on the costs of medical treatment of the infected people but also on the costs related to educational and sanitation campaigns. Two approaches for solving the problem are considered: one using optimal control theory, the other carried out by first discretizing the problem and then solving it with nonlinear programming. The results obtained with OC-ODE and IPOPT solvers are given and discussed. We observe that with current computational tools it is easy to obtain, in an efficient way, better solutions to Dengue problems, leading to a decrease in the number of infected mosquitoes and individuals in less time and with lower costs. © 2010 Elsevier Ltd.

Numerical solution of the variational PDEs arising in optimal control theory

Costanza,Vicente; Troparevsky,Maria I.; Rivadeneira,Pablo S.
Fonte: Sociedade Brasileira de Matemática Aplicada e Computacional Publicador: Sociedade Brasileira de Matemática Aplicada e Computacional
Tipo: Artigo de Revista Científica Formato: text/html
Publicado em 01/01/2012 EN
Relevância na Pesquisa
75.75%
An iterative method based on Picard's approach to ODEs' initial-value problems is proposed to solve first-order quasilinear PDEs with matrix-valued unknowns, in particular, the recently discovered variational PDEs for the missing boundary values in Hamilton equations of optimal control. As illustrations the iterative numerical solutions are checked against the analytical solutions to some examples arising from optimal control problems for nonlinear systems and regular Lagrangians in finite dimension, and against the numerical solution obtained through standard mathematical software. An application to the (n + 1)-dimensional variational PDEs associated with the n-dimensional finite-horizon time-variant linear-quadratic problem is discussed, due to the key role the LQR plays in two-degrees-of freedom control strategies for nonlinear systems with generalized costs. Mathematical subject classification: Primary: 35F30; Secondary: 93C10.

Optimal control theory with arbitrary superpositions of waveforms

Meister, Selina; Stockburger, Jürgen T.; Schmidt, Rebecca; Ankerhold, Joachim
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Relevância na Pesquisa
75.78%
Standard optimal control methods perform optimization in the time domain. However, many experimental settings demand the expression of the control signal as a superposition of given waveforms, a case that cannot easily be accommodated using time-local constraints. Previous approaches [1,2] have circumvented this difficulty by performing optimization in a parameter space, using the chain rule to make a connection to the time domain. In this paper, we present an extension to Optimal Control Theory which allows gradient-based optimization for superpositions of arbitrary waveforms directly in a time-domain subspace. Its key is the use of the Moore-Penrose pseudoinverse as an efficient means of transforming between a time-local and waveform-based descriptions. To illustrate this optimization technique, we study the parametrically driven harmonic oscillator as model system and reduce its energy, considering both Hamiltonian dynamics and stochastic dynamics under the influence of a thermal reservoir. We demonstrate the viability and efficiency of the method for these test cases and find significant advantages in the case of waveforms which do not form an orthogonal basis.; Comment: 16 pages, 6 figures

On Certain Hypotheses in Optimal Control Theory and the Relationship of the Maximum Principle with the Dynamic Programming Method Proposed by L. I. Rozonoer

Wu, Hanzhong
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 01/01/2008
Relevância na Pesquisa
75.78%
In this paper we will study three hypotheses proposed by L. I. Rozonoer (Automation and Remote Control, 2003, vol.64, no.8, pp.1237--1240) in optimal control theory in order to derive conditions for the existence of an optimal control under all initial conditions, and the relationships between Pontryagin maximum principle and the dynamic programming method.; Comment: 18pages

Optimal control theory : a method for the design of wind instruments

Vey, Georges Le
Fonte: Universidade Cornell Publicador: Universidade Cornell
Tipo: Artigo de Revista Científica
Publicado em 19/01/2010
Relevância na Pesquisa
75.73%
It has been asserted previously by the author that optimal control theory can be a valuable framework for theoretical studies about the shape that a wind instrument should have in order to satisfy some optimization criterion, inside a fairly general class. The purpose of the present work is to develop this new approach with a look at a specific criterion to be optimized. In this setting, the Webster horn equation is regarded as a controlled dynamical equation in the space variable. Pressure is the state, the control being made of two parts: one variable part, the inside diameter of the duct and one constant part, the weights of the elementary time-harmonic components of the velocity potential. Then one looks for a control that optimizes a criterion related to the definition of an {oscillation regime} as the cooperation of several natural modes of vibration with the excitation, the {playing frequency} being the one that maximizes the total generation of energy, as exposed by A.H. Benade, following H. Bouasse. At the same time the relevance of this criterion is questioned with the simulation results.; Comment: To appear in Acta Acustica united with Acustica, 2010