Approximation of an optimal value of a Bolza functional

Abstract

In this article, we show that under reasonable assumptions every Lipschitz-continuous solution to a Hamilton–Jacobi inequality approximates with a priori known error the optimal value of a respective Bolza functional and that such approximation is stable. The solutions of Hamilton–Jacobi variational inequalities can be easily obtained by well-known numerical methods as approximate solutions of Hamilton–Jacobi equations resulting from related Bolza functionals. The main strength of this approach lies in the fact that both precise solution to the Hamilton–Jacobi PDE and the distance between that solution and its numerical approximation need not be known in order to solve the original Bolza problem.

Keywords:

• Bolza problem
• Hamilton–Jacobi equation
• Hamilton–Jacobi inequality
• optimal control
• numerical approximation

AMS Subject Classifications::

• 49L20
• 34H05
• 93C15
• 65K10