Theorem (Solution to free 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 and $x(t_{0})=x_{0}$. If $\exists K:[t_{0},t_{1}]\to M_{n}(\mathbb{R})$ with $K(t)=K^{\top}(t),\,\forall t\in[t_{0},t_{1}]$ which satisfies the Riccati Differential Equation: $$\begin{align*} \dot{K}(t)&= -A^{\top}(t)K(t)-K(t)A(t)-L(t)+K(t)B(t)B^{\top}(t)K(t)\\ K(t_{1})&= Q \end{align*}$$the solution to which we denote by $t\mapsto \Pi(t,Q,t_{1})$, then there exists an optimal control $u^{*}:[t_{0},t_{1}]\to \mathbb{R}^{m}$ for the Free endpoint problem with the cost $$J(u)=\int\limits _{t_{0}}^{t_{1}}\left(x^{\top}_{u}(t)L(t)x_{u}(t)+u^{\top}(t)u(t)\right) \, dt + x_{u}^{\top}(t_{1})Qx_{u}(t_{1})$$In particular, the optimal control is $$u^{*}(t)=-B^{\top}(t)\Pi(t,Q,t_{1})x(t),\quad t\in[t_{0},t_{1}]$$ a continuous function, and the optimal cost is $$J_{min}=x^{\top}_{0}\Pi(t_{0},Q,t_{1})x_{0}$$