The lectures will be an overview of various tools from optimization and control theory and applications of these ideas to studying properties of nonlinear dynamical systems. A list of topics to be covered includes:
1) Quadratically Constrained Quadratic Programs (QCQPs), a class of optimization problems that are known to be hard in general. We will cover various special cases under which QCQPs are known to be efficiently solvable, and describe the relationship of these results to various "hidden convexity" results.
2) Modern tools from nonlinear control theory: Integral Quadratic Constraints (IQCs), Contraction and Differential Positivity.
3) Applications of these ideas to analyzing properties of nonlinear dynamical systems, including some applications to studying properties of numerical algorithms.
1) Pólik, Imre, and Tamás Terlaky. "A survey of the S-lemma." SIAM review 49.3 (2007): 371-418.
2) Lessard, Laurent, Benjamin Recht, and Andrew Packard. "Analysis and design of optimization algorithms via integral quadratic constraints." arXiv preprint arXiv:1408.3595 (2014).
3) Forni, Fulvio, and Rodolphe Sepulchre. "Differentially positive systems." arXiv preprint arXiv:1405.6298 (2014).