Canonicial Transformations and the Hamilton‐Jacobi Theorem in the Optimum Control Theory

This paper deals with the problem to establish a profound relationship between Pontryagin's maximum principle and Bellman 's dynamic programming method via the canonical transformations of the variables, as it is a case in classical mechanics. A rigorous form of the Hamilton‐Jacobi theorem is proved for optimal control systems. Further it is shown that the controlled systems may be treated by a partial differential equation of the Hamilton‐Jacobi type, which, however, is fundamentally different of the Bellman functional equation. The solution of this equation is a function of the generalized momenta, which appear in Pontryagin's theory, and the time. In some cases this equation can be advantageously used in comparison with the Bellman equation. An example is solved to illustrate the presented theory.