Numerical optimization and optimal control for dynamical systems [146442]

Lecture Notes

Author

Affiliation

Enrico Bertolazzi

University of Trento, Department of Industrial Engineering

Contents

These notes develop the mathematical and numerical tools used to model, analyse, and optimize dynamical systems. The mathematical material is organized in five parts: constrained optimization and Karush–Kuhn–Tucker conditions (KKT), ordinary differential equations (ODE), differential-algebraic equations (DAE), functional minimization, and optimal control problems (OCP). A sixth part collects the reproducible Python laboratories.

Exercises, worked examples and case studies are collected in a companion volume, Exercises and worked examples. What remains here are only the examples that carry the exposition forward. The algorithms, their order and stability analysis and the Python laboratories are in a third volume, Numerical Methods.

The presentation moves from differential models and their numerical integration to algebraic constraints and variational methods, providing the foundations required for the formulation and solution of optimal control problems.

• Constrained Minimization
  • Mathematical Preliminaries
  • Lagrange Multipliers
  • Sylvester’s Criterion
  • Inequality Constraints and KKT Conditions
• Ordinary Differential Equations
  • Introduction to ODE
  • ODE Theory and Properties
• Differential-Algebraic Equations
  • Linear DAEs
  • Index Reduction for Linear DAEs
  • The Differential Index
  • Semi-Explicit Form and Its Classification
  • General Index Reduction
  • Dummy Variables
  • DAEs from Lagrangian Mechanics
• Calculus of Variations
  • Variational Foundations
  • Fundamental Lemma
  • Euler–Lagrange Equations
  • Higher-Order Functionals
  • Isoperimetric Constraints
  • Path Constraints
  • Summary
• Optimal Control
  • Introduction
  • Optimal Control Problems
  • Transformations of Optimal Control Problems
  • Summary of Optimal Control Conditions
• References

Acknowledgement

License: CC BY 4.0

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