t L(“(s), s, –(s))ds + g(“(T)): “(t) = x, ˙ “(s) = f (“(s), s, –(s))’s œ (t, T) < , (1) and the corresponding Hamilton-Jacobi (HJ) PDE Y ] [ ˆÏ(x, t) ˆt + sup –œRm {≠Èf (x, T ≠ t, –), Òx Ï(x, t)Í ≠ L(x, T ≠ t, –)} = 0, x œ , t œ [0, T], Ï(x, 0) = g(x), x œ . (2) Grid based method (e.g., Lax–Friedrichs, Engquist-Osher scheme) needs to satisfy the CFL condition in discretization. In this project, we provide an optimization method for solving certain HJ PDEs with a saddle point formulation and implicit time discretization. This allows us to choose a larger time stepsize and avoid CFL condition. In the literature, there are some optimization algorithms for solving HJ PDE [Darbon, Dower, Osher, Yegerov, . . . ]. Compared to these methods, the optimization algorithm proposed in this project can handle a more general Hamiltonian that depends on (x, t). Partial theoretical guarantee is provided. Meng, Liu, Li, Osher PDHG control August 4, 2024 2 / 5