# Lectures and Codes

• Week  1: Introduction and Course Overview (Slides)

Contents: Planning under uncertainty, Risk aware and robust approaches.

Contents: Optimality Conditions, Newton’s Method, Interior Point Method, Convex Optimization, Dual Optimization.

• Week 3: Sum of Squares (SOS) Based Nonlinear Optimization (Slides) , (Codes)

Contents: Semialgebraic sets, Nonnegative polynomials, Sum of squares (SOS) polynomials, Nonnegative polynomials for nonlinear optimization, Sum of squares relaxation for nonlinear optimization, Semidefinite programming (SDP) for nonlinear optimization.

• Week 4: Measure and Moments Based Nonlinear Optimization  (Slides) , (Codes)

Contents: Probability measures and moments, Probability measures for nonlinear optimization, Moments for nonlinear optimization, Moment based semidefinite programming for nonlinear optimization.

• Week 5: Duality of SOS and Moment based Semidefinite Programs (SDPs)  (Slides) , (Codes)

Contents: Conic duality, Duality of measures and continuous functions, Duality of moments of polynomials, Duality of moment-SDP and SOS-SDP for nonlinear optimization.

• Week 6: Modified SOS for Large Scale Nonlinear Optimization (Slides) , (Codes)

Contents: Scaled diagonally dominant SOS optimization (DSOS, SDSOS), Bounded degree SOS optimization (BSOS), Sparse SOS Optimization (SSOS), Sparse bounded degree SOS Optimization (SBSOS)

• Week 7: Nonlinear Chance Constrained and Chance Optimization (Slides) , (Codes)

Contents: Mathematical formulation of "chance optimization" and "chance constrained optimization", Geometrical interpretation, Challenges, Moment-SDP for chance optimization, Dual formulation for chance optimization, SOS-SDP for chance constrained optimization, Outer and inner approximations of chance constrained sets.

• Week 8: Nonlinear Robust and Distributionally Robust Chance Constrained Optimization (Slides) , (Codes)

Contents: Mathematical formulation of robust optimization, Challenges, Robust SOS Optimization, Robust sets for robust optimization (SOS based formulation), Robust games (SOS based formulation), Distributionally robust chance constrained optimization (SOS based formulation).

• Week 9: Algorithms for Large Scale Semidefinite Programs (SDPs)

• Week 10: Probabilistic Nonlinear Safety Verification (Slides) , (Codes)

Contents: Probabilistic safety verification along the trajectories of uncertain nonlinear systems, Uncertainty (moment) propagation through nonlinear uncertain dynamics, Risk estimation in presence of nonlinear safety constraints, Uncertainty set construction from the moment information, Probability density function construction from the moment information.

• Week 11: Risk Aware Planning and Control Of Probabilistic Nonlinear Dynamical Systems (Slides) , (Codes)

Contents: Polynomial representation of obstacles and dynamical systems, Risk bounded trajectory planning in uncertain environments, Nonlinear state feedback control, Receding horizon control, Flow-Tube based control of probabilistic nonlinear systems, Chance constrained backward reachability set, SOS optimization for continuous-time path planning.

• Week 12: Mean and Covariance Based Control and Safety Verification of Probabilistic Dynamical Systems (Slides)

Contents: Chane constrained trajectory optimization, Chane constrained RRT*, Chane constrained PRM, Chane constrained motion primitives, Continuous-time probabilistic safety guarantees, Distributionally robust chance constrained control, Chance constrained covariance control, SOS optimization for continuous-time probabilistic safety verification, Distributionally robust chance constrained set for moment ambiguity sets.

• Week 13: Occupation Measure Based Control of Continuous-Time Nonlinear Dynamical System (Slides)

Contents: Occupation measures, Liouville’s equation, Occupation-measure based SDP for trajectory optimization, Optimal control, Region of attraction set, Nonlinear feedback control, Backward reachable sets.

• Week 14: Sum-of-Squares Optimization Based Robust Planning for Uncertain Nonlinear Systems (Slides)

Contents: SOS optimization for robust control, Lyapunov stability and SOS optimization, Barrier function based safety, Region of attraction set estimation and design, Invariant set estimation and deign, Funnel based robust control, Reachable sets, Constrained volume optimization for robust planning.