This chapter discusses the Hamilton–Jacobi–Bellman equation for two operators via variational inequalities. The results concern only the case where the family Aα consists of two operators. It is not clear the way to extend the proofs to the case of more than two operators. On the other hand, method has some advantages: (1) it is quite simple and constructive, (2) it leads to classical solutions u ∊ C2,α (Ω ), and (3) it is very flexible and can be adapted to the case where A1 and A2 are parabolic and A1 is elliptic and A2 is parabolic. The main results of this study are presented in the chapter.
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