In this course we will present a unified perspective on recent developments in optimization and control theory. We will focus on convex methods, and their applications in control and dynamical systems, estimation, system identification, and machine learning. We will make particular emphasis on computational methods, duality as suboptimality or infeasibility certificates, and focus on the exciting developments that have occurred in the last few years, including convex relaxations of combinatorial optimization problems, algebraic methods such as sum-of-squares, and applications to sparsity and rank minimization problems. The following topics will be covered: Introduction to dynamical systems. First-order methods for optimization. Semidefinite programming and sum of squares techniques. Solvers for semidefinite programs. Lyapunov functions, KYP lemma, analysis via integral quadratic constraints (IQCs). Sparse approximation approaches to estimation, system identification, and machine learning