Stanford AA203 Optimal and Learning-Based Control | Spring 2026 | Lecture 4: Optimal Control

This lecture provides a solid and rigorous introduction to indirect methods for optimal control, specifically focusing on the calculus of variations and the Euler-Lagrange equation. The instructor, Dr. Daniele Gammelli, demonstrates a deep understanding of the subject and presents the material in a clear, logical manner. The lecture begins with a recap of the fundamental theorem of calculus of variations, which is essential for understanding the necessary conditions for optimality. The derivation of the Euler-Lagrange equation is thorough, and the instructor takes care to explain the intuition behind each step, making the material accessible to students with a background in optimization and differential equations. The example of finding the shortest path between two points is well-chosen and effectively illustrates the application of the Euler-Lagrange equation. The instructor also addresses the issue of boundary conditions, explaining how to handle cases where the final time or final state is free, which is a common challenge in practical optimal control problems. The lecture is well-structured, with a clear roadmap and transitions between topics. The use of the whiteboard for derivations is effective, though the video quality may make it difficult to read some equations. The content is highly technical and assumes prior knowledge of calculus, linear algebra, and basic optimization. The lecture does not include any public engagement or discussion, but this is typical for a university lecture. Overall, this is an excellent lecture that provides a strong foundation for understanding indirect methods in optimal control. The mathematical rigor and clarity of presentation make it a valuable resource for students and practitioners in the field.

The title accurately reflects the content: a lecture on optimal control, specifically focusing on indirect methods.

9/10

Lecture by Dr. Daniele Gammelli, a researcher at Stanford and AI4I, based on the AA203 course materials. The content is mathematically rigorous, with derivations and references to a companion textbook. The source is institutional (Stanford Online) and the presentation is clear and well-structured.

Markers derived by PSI from the transcript: the creator did not define chapters.

• 00:00Introduction and roadmap of the course, positioning indirect methods within the broader context.
• 02:00Recap of the fundamental theorem of calculus of variations and the Euler-Lagrange equation.
• 05:00Example: shortest path problem, deriving the functional and applying the Euler-Lagrange equation.
• 10:00Derivation of the Euler-Lagrange equation for the shortest path problem, leading to the solution of a straight line.
• 15:00Extension to optimal control problems, relaxing fixed boundary conditions.
• 20:00Introduction of transversality conditions for free final time and free final state.
• 30:00Discussion of the necessary conditions for optimality in the presence of control inputs.
• 40:00Preview of direct methods and closed-loop control for the next lectures.

• AA203 Optimal and Learning-Based Control course page — Course information and enrollment details.
• Principles of Robot Autonomy (companion textbook) — Free online textbook referenced as companion material for the course.
• AA203 course schedule and syllabus — Course schedule and syllabus for Spring 2025-2026.
• Lecture 4 slides — Slides used in this lecture.
• Full playlist of AA203 lectures — Playlist containing all lectures of the course.

• Principles of Robot Autonomy — The companion textbook likely covers similar material on optimal control.
• AA203 course materials — Course syllabus and slides align with the lecture content.