## Applied Math Colloquium: Christian Parkinson

**Title: **The Hamilton-Jacobi Formulation of Optimal Path Planning for Autonomous Vehicles

**Speaker: **Christian Parkinson (University of Arizona)

**Abstract: **We present a partial-differential-equation-based optimal path planning framework for simple self-driving cars. This formulation relies on optimal control theory, dynamic programming, and a Hamilton-Jacobi-Bellman equation, and thus provides an interpretable alternative to black-box machine learning algorithms. We design grid-based numerical methods used to resolve the solution to the Hamilton-Jacobi-Bellman equation and generate optimal trajectories. We then describe how efficient and scalable algorithms for solutions of high dimensional Hamilton-Jacobi equations can be developed to solve similar problems in higher dimensions and in nearly real-time. We demonstrate our methods with several examples.