Condition to apply the verification theorem

Question

I am solving a optimal control problem, the SDE is this $$ dX(t) = [X(t)(r-\beta(t))+ \theta(t)+k_1e^{\int_t^T (r-\beta(s))ds}]dt + k_2e^{\int_t^T (r-\beta(s))ds}dW(t) $$ with $ X(0)=x_0$, where $\beta(t)$ and $\theta(t)$ are deterministic functions and $r$,$k_1$ and $k_2$ are constants. We call $\Lambda(t)=x(r-\beta(t))+ \theta(t)+k_1e^{\int_t^T (r-\beta(s))ds}$ where $X(t)=x$. One of the conditions for the application of Verification Theorem is that $$|\Lambda_t| + |\Lambda_x| \leq C $$ It follows that $\Lambda_t = -x \beta'(t)+ \theta'(t) +k_1e^{\int_t^T(r-\beta(s))ds}(\beta(t)-r)$ and $\Lambda_x=r-\beta(t)$. When I applied HJB, I obtained the condition $x \times u(t) = k_3e^{\int_t^T(r-\beta(s))ds} > 0$ so I conclude that $x,u(t) \neq 0$ and there is no possibility for an indeterminate value and $x < \infty$. This approach is right? if not how can face this problem? There is some way to use the solution of the SDE? $$X(t)=e^{\int_t^T (r-\beta(s))ds}\left[x_0 + \int_0^t e^{-\int_0^s (r-\beta(v))dv}\left(\theta(s)+k_1e^{\int_s^T (r-\beta(v))dv}\right)ds \right. $$ $$\left. + \int_0^t k_2 e^{-\int_0^s (r-\beta(v))dv}e^{\int_s^T (r-\beta(v))dv}dWs\right] $$