Efficient Numerical Methods for Hamilton–Jacobi Equations in Optimal Control

Abstract

The research presented here develops algorithms which facilitate efficient solutions to Hamilton-Jacobi (HJ) equations for motivating problems belonging to the trajectory optimization, pursuit-evasion, and flight controller synthesis subclasses of optimal control theory. The HJ equation provides both the necessary, and sufficient conditions for solutions to optimal control problems but practical solutions suffer from severe computational roadblocks due to exponential dimensional scaling properties. Three methods are proposed that inherit the optimality guarantees prescribed by the HJ equation but exploit algorithmic features leading to efficient solution methodologies. First, motivated by recent advancements regarding point wise solutions to the HJ equation, a nonconvex proximal splitting optimization algorithm is presented to solve the Hopf Formula for differential games. Applying this method demonstrates near real-time performance for high dimensional pursuit evasion problems. Next, a data-driven method for flight controller synthesis is proposed which leverages the HJ optimality conditions, enforced along an arbitrary trajectory of flight test data, to synthesis an optimal LQR controller. This novel flight testing paradigm is demonstrated by designing, and flight testing, the model-free LQR controller on a subscale aircraft. Last, a reachable set computation algorithm, capable of rapidly approximating the forward reachable set of a vehicle, is established. This procedure is used to compute the reachable set of low thrust spacecraft operating in highly nonlinear dynamical environments and can be enhanced for space domain awareness and reactive collision avoidance applications.