Optimal stopping problem with restricted stopping times

Question

Consider that you have some ergodic, time-homogenous Markov process $(X_n)_{n \ge 0}$ taking values in a finite space $S$. To solve $$ g(N, x) = \sup_{\tau \le N} E \left[ f(X_\tau) \mid X_0 = x \right] $$ one may use the value equations $$ v(n, x) = \max \left\{ f(x), E \left[ v(n+1, X_1) \mid X_0 = x \right] \right\}, n = 1, \ldots, N-1, \, v(N,x) = f(x) $$ and find an optimal solution by the relationship $v(0,x) = g(N, x)$. In the infinite horizon case $$ g(\infty, x) = \sup_{\tau} E \left[ f(X_\tau) \mid X_0 = x \right] $$ I know that you similarly deploy the value function. But what I have been wondering is if it possible to restrict the stopping times to being in a subset of values $A \subset S$ and then find the optimal stopping time amongst these? That is to consider the problem $$ \sup_{\tau \in C} E \left[ f(X_\tau) \mid X_0 = x \right] $$ where $$ \tau \in C_A \iff \tau \stackrel{a.s.}{<} \infty, X_\tau \stackrel{a.s.}{\in} A $$ Any papers or links are much appreciated!