Carathéodory Equivalence, Noether Theorems, and Tonelli Full-Regularity in the Calculus of Variations and Optimal Control

Torres, Delfim F. M.

doi:10.1023/b:joth.0000013565.78376.fb

Cited by 21 publications

(16 citation statements)

References 32 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…2]. Here we just recall that (3) is one of the cornerstone results of the calculus of variations and optimal control [35]: it has been used, for example, to prove existence, regularity of minimizers, conservation laws, and to explain the Lavrentiev phenomena. Main result of the paper gives an extension of (3) to an arbitrary time scale (cf.…”

Section: Introductionmentioning

confidence: 99%

The Second Euler-Lagrange Equation of Variational Calculus on Time Scales

Bartosiewicz

Martins

Torres

2011

European Journal of Control

View full text Add to dashboard Cite

The fundamental problem of the calculus of variations on time scales concerns the minimization of a deltaintegral over all trajectories satisfying given boundary conditions. In this paper we prove the second Euler-Lagrange necessary optimality condition for optimal trajectories of variational problems on time scales. As an example of application of the main result, we give an alternative and simpler proof to the Noether theorem on time scales recently obtained in [J. Math.

show abstract

Section: Introductionmentioning

confidence: 99%

The Second Euler-Lagrange Equation of Variational Calculus on Time Scales

Bartosiewicz

Martins

Torres

2011

European Journal of Control

View full text Add to dashboard Cite

show abstract

“…In optimal control theory and in its many applications is standard to consider objective functionals with integrands that are convex with respect to the control variables [17]. Such convexity easily ensures the existence and the regularity of solution to the problem [31] as well as good performance of numerical methods [8]. In our case, we considered a quadratic expression of the control in order to indicate nonlinear costs potentially arising at high treatment levels, as proposed in [22].…”

Section: Optimal Control Problemmentioning

confidence: 99%

An epidemic model for cholera with optimal control treatment

Lemos-Paião

Silva

Torres

2017

Journal of Computational and Applied Mathematics

View full text Add to dashboard Cite

We propose a mathematical model for cholera with treatment through quarantine. The model is shown to be both epidemiologically and mathematically well posed. In particular, we prove that all solutions of the model are positive and bounded; and that every solution with initial conditions in a certain meaningful set remains in that set for all time. The existence of unique disease-free and endemic equilibrium points is proved and the basic reproduction number is computed. Then, we study the local asymptotic stability of these equilibrium points. An optimal control problem is proposed and analyzed, whose goal is to obtain a successful treatment through quarantine. We provide the optimal quarantine strategy for the minimization of the number of infectious individuals and bacteria concentration, as well as the costs associated with the quarantine. Finally, a numerical simulation of the cholera outbreak in the Department of Artibonite (Haiti), in 2010, is carried out, illustrating the usefulness of the model and its analysis.Comment: This is a preprint of a paper whose final and definite form is with 'Journal of Computational and Applied Mathematics', ISSN 0377-0427, available at [http://dx.doi.org/10.1016/j.cam.2016.11.002]. Submitted 19-June-2016; Revised 14-Sept-2016; Accepted 04-Nov-2016. This version: some typos detected while reading the proofs were correcte

show abstract

“…Noether's theorem has become a fundamental tool of modern theoretical physics [1], the calculus of variations [10,11], and optimal control [7,8,9]. It states that when an optimal control problem is invariant under a one parameter family of transformations, then there exists a corresponding conservation law: an expression that is conserved along all the Pontryagin extremals of the problem [7,8,9,12].…”

Section: Preliminariesmentioning

confidence: 99%