  Optimal control problems are crucial in various domains, including path
planning, robotics, and humanoid control, demonstrating their broad
applicability. The connection between optimal control and Hamilton-Jacobi (HJ)
partial differential equations (PDEs) underscores the need for solving HJ PDEs
to address these control problems effectively. While numerous numerical methods
exist for tackling HJ PDEs across different dimensions, this paper introduces
an innovative optimization-based approach that reformulates optimal control
problems and HJ PDEs into a saddle point problem using a Lagrange multiplier.
Our method, based on the preconditioned primal-dual hybrid gradient (PDHG)
method, offers a solution to HJ PDEs with first-order accuracy and numerical
unconditional stability, enabling larger time steps and avoiding the
limitations of explicit time discretization methods. Our approach has ability
to handle a wide variety of Hamiltonian functions, including those that are
non-smooth and dependent on time and space, through a simplified saddle point
formulation that facilitates easy and parallelizable updates. Furthermore, our
framework extends to viscous HJ PDEs and stochastic optimal control problems,
showcasing its versatility. Through a series of numerical examples, we
demonstrate the method's effectiveness in managing diverse Hamiltonians and
achieving efficient parallel computation, highlighting its potential for
wide-ranging applications in optimal control and beyond.
