Pontryagin Maximum Principle with terminal and initial conditions

Question

Consider a control problem with Lagragian $L(t,x,u)$ (where $u$ is the control, $x \in \mathbb{R}^d$ the state) and dynamics $\dot{x}=f(x,u,t)$. I have mostly seen problems in which the dynamical system is coupled with fixed initial conditions $x_{i}(0)$ for $i=1,\dots,d.$ Is there any analogous result in the case with hybrid terminal and initial conditions? I.e., say $x_i(0)$ fixed for some $i \in \mathcal{I}$ and $x_i(T)$ fixed for $i \in \mathcal{I}^c.$