The semigroup property of value functions in Lagrange problems

Wolenski, Peter R.

doi:10.1090/S0002-9947-1993-1156301-6

The semigroup property of value functions in Lagrange problems
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Trans. Amer. Math. Soc. 335 (1993), 131-154 Request permission

Abstract:

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 $t = 0$. 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 $L$ is in addition finite-valued, the results are given in terms of pointwise convergence.

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