$$\begin{cases}\min\limits_{{\bf u}(t)}&\displaystyle\int_0^{t_f}ϕ(\mathbf{u}(t))\,{\rm d}t\\&\dot{\bf x}(t)=A{\bf x}(t)+B{\bf u}(t)\\&C{\bf x}(t_f)=α\\&g({\bf x}(t_f))\le β\\&{\bf x}(0)=0\end{cases}$$ I am dealing with an optimal control problem of the above form, where $A$, $B$ are matrices and $C$ is a vector of appropriate size. $\phi$ and $g$ are scalar functions of the control and state respectively.
I would like to use the Pontryagin’s minimum principle to solve this problem but I am not sure how to deal with the final state inequality constraint $g(\mathbf{x}(t_f)) \le \beta$. Can anyone help me out and give me some ideas about how to set up the principle?