Theorem (Solution of fixed endpoint problem)

Consider a LTVC system $$\dot{x}(t)=A(t)x(t)+B(t)u(t)$$where $A,B$ are Continuous functions of time, $x(t_{0})=x_{0}$, and $x(t_{1})=x_{1}$.

1. If $u_{0}$ is any control input of the form $$u_{0}(t)=-B^{\top}(t)\Phi^{\top}(t_{0},t)\eta$$where $\eta$ satisfies $$W(t_{0},t_{1})\eta=x_{0}-\Phi(t_{0},t_{1})x_{1}$$and $W(t_{0},t_{1})$ is the Controllability Gramian, then the control $u_{0}$ drives the system from $x_{0}$ at time $t_{0}$ to $x_{1}$ at time $t_{1}$.
2. If $u_{1}$ is any other control input that steers the system from $x_{0}$ at time $t_{0}$ to $x_{1}$ at time $t_{1}$ then $$\int\limits _{t_{0}}^{t_{1}}u^{\top}_{1}(t)u_{1}(t) \, dt \ge \int\limits _{t_{0}}^{t_{1}}u^{\top}_{0}(t)u_{0}(t) \, dt$$Moreover, if $W(t_{0},t_{1})$ is nonSingular (i.e. has determinant not equal to zero $\iff$ full Rank $\iff$ system Controllable), then $$\int\limits _{t_{0}}^{t_{1}}u_{0}^{\top}(t)u_{0}(t) \, dt =(x_{0}-\Phi(t_{0},t_{1})x_{1})^{\top}W^{-1}(t_{0},t_{1})(x_{0}-\Phi(t_{0},t_{1})x_{1})$$