The semigroup property of value functions in Lagrange problems
Peter R. Wolenski · Transactions of the American Mathematical Society · 1993
The Lagrange problem in the calculus of variations exhibits the principle of optimality in a particularly simple form. The binary operation of inf-composition applied to the value functions of a Lagrange problem equates the principle of optimality with a semigroup property. This paper finds the infinitesimal generator of the semigroup by differentiating at t = 0 t = 0 . The type of limit is epigraphical convergence in a uniform sense. Moreover, the extent to which a semigroup is uniquely determined by its infinitesimal generator is addressed. The main results provide a new approach to existence and uniqueness questions in Hamilton-Jacobi theory. When L L is in addition finite-valued, the results are given in terms of pointwise convergence.