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30 pages, 495 KiB

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Geometric Numerical Methods for Lie Systems and Their Application in Optimal Control

by Luis Blanco Díaz, Cristina Sardón, Fernando Jiménez Alburquerque and Javier de Lucas

Symmetry 2023, 15(6), 1285; https://doi.org/10.3390/sym15061285 - 19 Jun 2023

Cited by 1 | Viewed by 2604

A Lie system is a nonautonomous system of first-order ordinary differential equations whose general solution can be written via an autonomous function, the so-called (nonlinear) superposition rule of a finite number of particular solutions and some parameters to be related to initial conditions. This superposition rule can be obtained using the geometric features of the Lie system, its symmetries, and the symmetric properties of certain morphisms involved. Even if a superposition rule for a Lie system is known, the explicit analytic expression of its solutions frequently is not. This is why this article focuses on a novel geometric attempt to integrate Lie systems analytically and numerically. We focus on two families of methods based on Magnus expansions and on Runge–Kutta–Munthe–Kaas methods, which are here adapted, in a geometric manner, to Lie systems. To illustrate the accuracy of our techniques we analyze Lie systems related to Lie groups of the form SL

(n, R)

, which play a very relevant role in mechanics. In particular, we depict an optimal control problem for a vehicle with quadratic cost function. Particular numerical solutions of the studied examples are given. Full article

(This article belongs to the Special Issue Symmetry and Applications of Differential Geometry to the Differential Equations of Mathematical Physics)

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23 pages, 508 KiB

Open AccessArticle

Optimal Intervention Strategies for a SEIR Control Model of Ebola Epidemics

by Ellina V. Grigorieva and Evgenii N. Khailov

Mathematics 2015, 3(4), 961-983; https://doi.org/10.3390/math3040961 - 21 Oct 2015

Cited by 12 | Viewed by 5571

Abstract

A SEIR control model describing the Ebola epidemic in a population of a constant size is considered over a given time interval. It contains two intervention control functions reflecting efforts to protect susceptible individuals from infected and exposed individuals. For this model, the problem of minimizing the weighted sum of total fractions of infected and exposed individuals and total costs of intervention control constraints at a given time interval is stated. For the analysis of the corresponding optimal controls, the Pontryagin maximum principle is used. According to it, these controls are bang-bang, and are determined using the same switching function. A linear non-autonomous system of differential equations, to which this function satisfies together with its corresponding auxiliary functions, is found. In order to estimate the number of zeroes of the switching function, the matrix of the linear non-autonomous system is transformed to an upper triangular form on the entire time interval and the generalized Rolle’s theorem is applied to the converted system of differential equations. It is found that the optimal controls of the original problem have at most two switchings. This fact allows the reduction of the original complex optimal control problem to the solution of a much simpler problem of conditional minimization of a function of two variables. Results of the numerical solution to this problem and their detailed analysis are provided. Full article

(This article belongs to the Special Issue Optimal Control and Management of Infectious Diseases)

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