I am faced with an optimal control problem in continuous time which includes a path constraint which involves controls at two distinct points in time.
I do not know how to approach this problem. I do not even know what to search for, since I am not sure about the correct terminology regarding the constraint. I would be happy about any type of input regarding how to approach this problem, as well as for references or terminology. Here is more detail:
Let $\mathbf{x}(t)\in\mathbb{R}^{n}$ be the state and $\mathbf{u}(t)\in\mathbb{R}^{m}$ the control, $t_{0}$ the initial time and $t_{f}$ the final time. The goal is to minimize $$ \int_{t_{0}}^{t_{f}}f\left(\mathbf{x}(t)\right)dt $$ subject to laws of motion $$ \dot{\mathbf{x}}(t)=g(\mathbf{x}(t),\mathbf{u}(t)) $$ an integral constraint $$ \int_{t_{0}}^{t_{f}}h\left(\mathbf{x}(t),\mathbf{u}(t)\right)dt=0 $$ path constraints $$ k(\mathbf{x}(t),\mathbf{u}(t))=0\ \forall t\in[t_{0},t_{f}], $$ and a path constraint of the type $$ u_{1}(t)-u_{2}(\tau\cdot t)=0\ \forall t\in\left\{ t:t_{0}\leq\tau\cdot t\leq t_{f}\right\} , $$ with $\tau>0$ a scalar constant, and $u_{1}(t)$ and $u_{2}(t)$ are elements of $\mathbf{u}(t)$. It is this final path constraint which troubles me.