Mini-Course: Calculus in Infinite Dimensions

Please REGISTER here if you are planning to join.

Description:

This course introduces differential calculus in infinite-dimensional spaces, with applications to the calculus of variations and optimal control. The goal is to clarify the basic concepts of infinite dimensional calculus and illustrate their use in concrete optimization problems. Topics include Fréchet differentiability, optimality conditions, Euler–Lagrange equations, and the Pontryagin Maximum Principle.

Schedule

Target Audiences: Mathematicians and Physicists 

Detailed Course Plan

Lecture 1 – From Vector Calculus to Infinite Dimensions

• Review of finite-dimensional calculus and optimization
• What changes in infinite dimensions?
• Motivating problems from calculus of variations and optimal control

Lecture 2 – Fréchet Differentiability and Extrema

• Banach spaces and functionals
• Gâteaux and Fréchet differentiability
• Chain rule and basic calculus
• Necessary and sufficient conditions for extrema

Lecture 3 – Calculus of Variations

• Variational problems and action functionals
• First variations and Fréchet derivatives
• Euler–Lagrange equations
• Convexity and minimizers
• The Dirichlet energy and the Laplace equation

Lecture 4 – Optimal Control and the Pontryagin Maximum Principle

• Optimal control problems and state equations
• Linearization and adjoint equations
• The Pontryagin Maximum Principle
• Switching functions and bang-bang controls
• A minimum-time problem with an explicit bang-bang solution

Lecturer: Dr. Denis Brazke Postdoctoral scholar at Analysis and Partial Differential Equations (APDE) Unit