Summary: The first author [J. Phys. A, Math. Theor. 44, No. 26, Article ID 262001, 6 p. (2011; Zbl 1223.35203
)] introduced the concept of weak self-adjoint equations. This definition generalizes the concept of self-adjoint and quasi self-adjoint equations that were introduced by N. H. Ibragimov
[J. Math. Anal. Appl. 318, No. 2, 742–757 (2006; Zbl 1102.34002
)]. In this paper we find a class of weak self-adjoint Hamilton-Jacobi-Bellman equations which are neither self-adjoint nor quasi self-adjoint. By using a general theorem on conservation laws proved in [N. H. Ibragimov
, J. Math. Anal. Appl. 333, 329–346 (2007; Zbl 1117.83127
)] and the new concept of weak self-adjointness we find conservation laws for some of these partial differential equations.