Non-uniqueness of solutions of the Hamilton–Jacobi–Bellman equation for time-average control

In control of diffusion processes a very useful instrument is the equation for optimal strategy and cost. For the version of infinite time horizon with time averaging this equation is much more complicated than for the version of finite time horizon, and even than for the version of infinite time horizon with discounting. In particular, the equation solution may be non-unique. This problem of non-uniqueness is researched in book of A. Arapostathis et al., 2012, for special models – near-monotone. The result received in the book is extended in the article to an important general case—models with restrictions in control which guarantee ergodicity of the process. Besides we correct the proofs from the book.