The Hamilton-Jacobi-Bellman Equation with a Gradient Constraint.
Abstract
As the first order Hamilton Jacobi equation is related to a control problem associated with ordinary differential equations, the Hamilton Jacobi Bellman (HJB) equation arises from a control problem with random noise. In the stationary problem, the HJB equation has the form sup when alpha is an element of A of < (L superscript alpha)u - (f superscript and alpha> = 0 where L superscript alpha are second order linear elliptic operators with parameter alpha an element of A. In this paper, we are concerned with the HJB equation of the form max<L superscript 1 u - (f superscript 1,...,(L superscript m u)) - (f superscript m), (absolute value of Du) - g. We prove the existence of solutions which satisfy the equation almost everywhere. Using the notion of weak solution (so called viscosity solution) we prove the uniqueness of the solution in the class of continuously differentiable functions. The method of the uniqueness proof is also applicable to other obstacle problems. We prove uniqueness results in the class of continuous functions for two model problems. Keywords: Obstacle problem; Minimax equations; Variational inequalities; and Viscosity solutions.
Document Details
- Document Type
- Technical Report
- Publication Date
- Mar 01, 1986
- Accession Number
- ADA167532
Entities
People
- Naoki Yamada
Organizations
- University of Wisconsin–Madison