Some conservation laws for a class of Hamilton-Jacobi-Bellman equations

Abstract

The idea of a conservation law has its origin in mechanics and physics. Since a large number of physical theories, including some of the ‘laws of nature’, are usually expressed as systems of nonlinear differential equations, it follows that conservation laws are useful in both general theory and the analysis of concrete systems. In [3] one of the present authors has introduced the concept of weak selfadjoint equations. This definition generalizes the concept of self-adjoint and quasi self-adjoint equations that were introduced by Ibragimov in [8]. Recently [4] we found a class of weak self-adjoint Hamilton-Jacobi-Bellman equations which are neither self-adjoint nor quasi self-adjoint. In this paper, by using a general theorem on conservation laws proved in [7] and the new concept of weak self-adjointness [3] we find conservation laws for some of these partial differential equations.