Spring 2023 6.8210 Lecture 9: Computing Lyapunov Functions II

The lecture provides a thorough and rigorous introduction to sum-of-squares (SOS) optimization for computing Lyapunov functions. The instructor’s pedagogical approach is effective, building on previous material and clearly explaining the mathematical concepts. The transition from sample-based verification to global certification is well-articulated, highlighting the power of convex optimization. The use of the ‘six-hump camel’ function as an example effectively demonstrates the application of SOS to global optimization, showing how a seemingly non-convex problem can be solved via lifting. The lecture also addresses important theoretical aspects, such as the difference between positivity and SOS in multivariate polynomials, and the implications for Lyapunov analysis. The technical level is appropriate for a graduate course, with detailed derivations and explanations. However, the lecture lacks explicit citations to external sources, which is common in lectures but limits the ability to verify claims independently. The instructor’s expertise is evident, and the content aligns with established literature in the field. The lecture is well-structured and provides a solid foundation for further study. The adéquation between title and content is excellent, as the lecture indeed focuses on computing Lyapunov functions using SOS. Overall, this is a high-quality educational resource, though its value is primarily for those with a background in control theory and optimization.

The title accurately reflects the content: the lecture focuses on computing Lyapunov functions using sum-of-squares methods, building on previous material.

8/10

The lecture is part of an MIT graduate course, presented by an expert in the field. It provides a rigorous introduction to sum-of-squares optimization for Lyapunov analysis, with clear explanations and examples. The content is technically sound and well-structured, though it does not include formal citations or references to external sources.

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

• 00:00Introduction and recap of previous lecture on sum-of-squares optimization.
• 02:00Discussion on moving from sample points to global certification using convex optimization.
• 05:00Explanation of semidefinite programming for linear systems and extension to nonlinear systems.
• 10:00Introduction to sum-of-squares decomposition and its role in Lyapunov analysis.
• 15:00Example: global optimization of the six-hump camel function using SOS.
• 20:00Discussion on decision variables and the lifting to higher-dimensional space.
• 25:00Q&A on multivariate polynomials and the gap between positivity and SOS.
• 30:00Further examples and explanation of coefficient matching.
• 35:00Discussion on the limitations and extensions to regions of attraction.