Hamilton–Jacobi–Bellman equation under exponential discounting

Question

If we want to choose $\mathbf{u}(t)$ in order to minimize $$ J(\mathbf{x}(t),t) = \int_{t}^{T} g(\mathbf{x}(s),\mathbf{u}(s),s) \, \textrm{d}s + h(\mathbf{x}(T),T), $$ subject to $ \dot{\mathbf{x}}(t) = \mathbf{a}(\mathbf{x}(t),\mathbf{u}(t),t), $ then the Hamilton–Jacobi–Bellman (HJB) equation tells us that the optimal $J^{*}(x(t),t)$ satisfies: $$ 0 = \frac{\partial J^{*}}{\partial t}(\mathbf{x}(t),t) + \min_{\mathbf{u}(t)} \Big\{ \, g(\mathbf{x}(t),\mathbf{u}(t),t) + \frac{\partial J^{*}}{\partial \mathbf{x}}(\mathbf{x}(t),t) \cdot \mathbf{a}(\mathbf{x}(t),\mathbf{u}(t),t) \, \Big\}. $$ Sometimes in economics papers, where $g(\mathbf{x}(t),\mathbf{u}(t),t) = \textrm{e}^{-\rho t} \cdot \tilde{g}(\mathbf{x}(t),\mathbf{u}(t),t)$, and when $T=\infty$, I see the HJB equation written as: $$ \frac{\partial J^{*}}{\partial t}(\mathbf{x}(t),t) = -\rho J^{*}(\mathbf{x}(t),t). $$ When is this okay to do?