Introduction to the Calculus of Variations and Optimal Control Problems: Theory, Methods, and Applications

• Grant number: 25/22520-3
• Support Opportunities: Scholarships in Brazil - Scientific Initiation
• Start date: February 01, 2026
• End date: January 31, 2027
• Field of knowledge: Physical Sciences and Mathematics - Mathematics - Analysis
• Principal Investigator: João Vitor da Silva
• Grantee: Paulo César Campos de Pinho
• Host Institution: Instituto de Matemática, Estatística e Computação Científica (IMECC). Universidade Estadual de Campinas (UNICAMP). Campinas , SP, Brazil

Abstract

This project will focus on guiding undergraduate student Paulo César Campos de Pinho through an introductory study of Variational Calculus, in order to understand and solve problems that, since ancient times, have aroused the interest of humanity and figures in the history of mathematics. Essentially, this Scientific Initiation project is divided into two parts: In the first,we introduce the classic variational problems-the Brachistochrone Problem and Didó's Problem.Then, we address the fundamental results of Variational Calculus, especiallythe Euler-Lagrange equations and the conditions for the existence of extrema: the Jacobi Condition andthe Weierstrass and Legendre Conditions. Finally, we conclude with Noether's Theorem andsome of its applications to Modern Physics. Brachistochrone Problem: The Brachistochrone Problem was proposed by Johann Bernoulli in 1696 and consists of finding a curve that a particle must describe when movingfrom point A to point B located in the same vertical plane, in the shortest possible time,under the action of gravity alone. In this case, point A is assumed to be above point B, but noton the same vertical line. If A and B are on the same vertical line, the solution is a straight line.We study the Brachistochrone Problem in two different ways: through the theory of Variational Calculusfor problems with fixed boundaries and also through the considerations made byJohann Bernoulli, using concepts from Optics and Geometry.Dido's Problem: Didó's Problem is probably the oldest known maximum and minimum problem.In short, Didó's Problem consists offinding, among all the curves of fixed length, the one that bounds the largest area.The final part of this project focuses on some applications of Variational Calculus, highlighting the resolutions of previously proposed problems and obtaining the Bliss Function and the best constant in the Hardy-Sobolev Inequality, using the concepts discussed in the first part of this proposal. We will also seek to provide a numerical approach to some of the problems addressed in the first part of the proposal.We will also study some classic problems of Variational Calculus:¿ Minimum Surface of Revolution;The minimum surface problem consists of finding a curve with fixed boundaries, whose rotation around the x-axis generates a surface of minimum area.¿ Geodesic or Arc Length Problem;The arc length problem consists of finding the equation of a class C1 curve with minimum length (geodesic) that connects two distinct (fixed) points. We will explore this problem in the context of the plane and in non-Euclidean environments, such as the sphere andthe cylinder, to name a few examples.Prerequisites (desirable but not mandatory) include knowledge of multivariable calculus, ODEs, basic physics, and Euclidean and analytic geometry. (AU)