Hamilton-Jacobi Equations

This thesis presents the theory of Hamilton-Jacobi equations. It is first shown how the equation is derived from the Lagrangian mechanics, then the traditional methods for searching for the solution are presented, where the Hopf-Lax formula along with the appropriate notion of the weak solution is defined. Later the flaws of this approach are remarked and the new notion of viscosity solutions is introduced in connection with Hamilton-Jacobi equation. The important properties of the viscosity solution, such as consistency with the classical solution and the stability are proved. The introduction into the control theory is presented, in which the Hamilton-Jacobi-Bellman equation is introduced along with the existence theorem. Finally multiple numerical methods are introduced and aligned with the theory of viscosity solutions. The knowledge of the theory of partial differential analysis, calculus and real analysis will be helpful