I have the following version of time optimal control problem for a two dimensional system with terminal equality and state inequality constraint. \begin{align} \mathbb{J}(u)= (T-\int_{0}^{t_f}\mathrm{d}t)^2,\;\;\;\;\; T = \text{given}\\ \text{s.t.}\:\; \dot{V}= AV+bu\\ v_1(t_f)= \bar{V}, v_2(t) \leq V_{g},\;\; \forall t\\ 0<u(t)<\mathcal{U}\;\; \forall t, \;\;\; \mathcal{U}=\text{given} \\ v_1(0) = a,\;\; v_2(0) = b, \;\;\;\;\; a,b =\text{given} \end{align} Now I know how to solve this problem when the cost is separable in terms of running and terminal cost. I'd really appreciate any help.
Thanks
Add a new variable $V_{N+1}$ s.t. $\dot V_{N+1} = 1 $. Now your functional reads $$ J(u) = \big(T- V_{N+1}(t_f) \big)^2 $$ and you have a Mayer problem.