American Institute of Mathematical Sciences

The Hamilton-Jacobi theory is a classical subject that was extensively developed in the last two centuries. The Hamilton-Jacobi equation is particularly useful in identifying conserved quantities for mechanical systems, which may be possible even when the mechanical problem itself cannot be solved completely. Indeed, the powerful of this method is that, in spite of the difficulties to solve a partial differential equation instead of an ordinary differential one, in many cases it works, being an extremely useful tool, usually more than Hamilton’s equations. Indeed, in these cases the method provides an immediate way to integrate the equations of motion. The modern interpretation relating the Hamilton-Jacobi procedure with the theory of lagrangian submanifolds is an important source of new results and insights.
   In addition, the Hamilton-Jacobi-Bellman equation is a partial differential equation which is central to optimal control theory. The equation is a result of the theory of dynamic programming which was pioneered in the 1950s by Richard Bellman and coworkers. The corresponding discrete-time equation is usually referred to as the Bellman equation. In continuous time, the result can be seen as an extension of earlier work in classical physics on the Hamilton-Jacobi equation by Hamilton and Jacobi.
   This special issue on Hamilton-Jacobi theory wants to bring specialists coming from different areas of research and show how the Hamilton-Jacobi theory is so useful in their domains: completely integrable systems, nonholonomic mechanics, Schrödinger equation, optimal control theory, and, in particular, applications in engineering and economics.