Optimal control problem with boundaries depending on control

Question

I have a relatively standard optimal quadratic control problem on infinite horizon : $\int_0^\infty (R-u)^\top (R-u) + C(t)^\top u~ dt $ subject to $\dot R = kR - k u $ but with one specific. The differential equation for $R$ above is artificially constructed from integral $$ R = \int_x^\infty ke^{k(x-t)} u(t) dt $$ where $k$ is positively definite matrix. Here $R,u$ are $n$-dimensional vectors. That means that $R(0)=\int_0^\infty k e^{-kt} u(t)dt$, $R(\infty)$ is somehow $u(\infty)$ if $u$ has a limit. So my boundary conditions for $R$ are determined by $u$. $u$ is in interval $[-1,1]^n$. Let us setup Hamiltonian $$ H = R^\top R + 2 R^\top u + u^\top u + C(t)^\top u + \lambda^\top (kR-ku) $$ and optimal $u$ in terms of $R$ and $\lambda$ from $$ \min_{|u|_\infty<1} ( R^\top R + 2 R^\top u + u^\top u + C^\top u +\lambda^\top (kR-ku)) $$ will be $$ u = \min(1,\max(-1,(-C(t)+k\lambda - 2R))) $$ The equations will be : $$ \dot R = k R - k u = \frac{\partial H}{\partial \lambda} $$ $$ \dot \lambda = -2R + 2u - C(t) - k \lambda= -\frac{\partial H}{\partial R} $$ Everything seems fine except one thing. I don't understand how to setup boundary conditions for $\lambda(t)$. Bounds for $R$ are not frozen and not completely free for $R$ since $u$ is not arbitrary. I would greatly appreciate any help here. Let's omit for now the problem of existence the integral converging etc.