Update of value function in continuous time - HJB
You iterate towards a fixed point, so you want to reach a situation where plugging in your current iterated value produces itself. Now using your notation, we are told that we should calculate

$$V_{n+1}(a) = V_n(a) + \Delta$$

where

$$\Delta = \ u(c(a^*)) + \dfrac{\partial V_n(a)}{\partial a}da_t(a^*) - \rho V_n(a)$$

Insert the second into the first to see what the iteration rule is:

$$V_{n+1}(a) = V_n(a) + \ u(c(a^*)) + \dfrac{\partial V_n(a)}{\partial a}da_t(a^*) - \rho V_n(a)$$

When you reach a point where

$$V_{n+1}(a) = V_n(a) $$

(or $\epsilon$-so)

It will mean

$$\rho V_n(a)= \ u(c(a^*)) + \dfrac{\partial V_n(a)}{\partial a}da_t(a^*) $$

which is what you have to satisfy.

Some stars etc may have to be adjusted in the above, for a fully consistent notation.