Fixed end points optimal control problem

Given two points $(t_0,x(t_0)=x^{0})$ and $(t_1,x(t_1)=x^{1})$ in the $(t,x)$ plane, the objective is to find an optimal trajectory $x^{*}(t)$ such that the cost function

begin{equation}label{eq:1}
J(x) = int_{t_0} ^ {t_1} g(x,dot{x},t) dt tag{1}
end{equation}

has a relative extremum with $g$ being a function with continuous first and second partial derivatives with respect to all its arguments. In this case where the end points are fixed, the necessary condition for minimizing the functional $J(x)$ is obtained by means of Euler-Lagrange equation

begin{equation}label{eq:2}
dfrac{partial{g(x^{*},dot{x}^{*},t)}}{partial{x}} – dfrac{d}{dt} dfrac{partial{g}(x^{*},dot{x}^{*},t)}{partial{dot{x}}} = 0.tag{2}
end{equation}

Next, as I was studying the same problem (from Modern Control System Theory by M. Gopal) with the modified constraint – terminal time $t_1$ fixed but $x(t_1)$ free, I encountered a bit of difficulty (highlighted in bold texts) in understanding the flow of arguments. It goes as follows.

Given $x$ be any curve in the admissible class $Omega$ and $delta{x}$ represents the variation in $x$ which is defined as an infinitesimal arbitrary change in $x$ for a fixed value of variable $t$.

The first variation in $J$ is given as

begin{equation}label{eq:3}
delta{J}(x,delta{x}) = dfrac{partial{g}(x,dot{x},t)}{partial{dot{x}}}delta{x(t)}|_{t_0} ^ {t_1}+int_{t_0}^{t_1}Bigg{dfrac{partial{g(x,dot{x},t)}}{partial{x}} – dfrac{d}{dt} dfrac{partial{g}(x,dot{x},t)}{partial{dot{x}}}Bigg} delta{x} dt. tag{3}
end{equation}

For an extremal (minimal or maximal) $x^{*}(t)$, we know that $delta{J}(x,delta{x})$ must be zero. In the following we show that the integral in eqref{eq:2} must be zero on an extremal.

Suppose that a curve $x^{*}(t)$ is an extremal for the problem under consideration; $t_1$ specified and $x(t_1)$ free. The value of $x^{*}(t)$ at $t_1$ is say $x^{*}(t_1)=x^{1}$. Now consider the fixed end-points problem with the functional $J$ in eqref{eq:1} and the end-points $(t_0,x^{0})$ and $(t_1,x^{*}(t_1))$. The curve $x^{*}(t)$ must be an extremal for this fixed end-points problem and therefore must be a solution of the Euler-Lagrange eqn. eqref{eq:2}. The integral term in eqref{eq:3} must be zero on an extremal and

begin{equation}label{eq:4}
dfrac{partial{g}(x^{*},dot{x}^{*},t)}{partial{dot{x}}}|_{t_1}delta{x(t_1)}=0.tag{4}
end{equation}

Since $x(t_1)$ is free, $delta{x(t_1)}$ is arbitrary; therefore it is necessary that

begin{equation}label{eq:5}
dfrac{partial{g}(x^{*},dot{x}^{*},t)}{partial{dot{x}}}|_{t_1} = 0. tag{5}
end{equation}

Equation eqref{eq:5} provides the second required boundary condition for the solution of the second-order Euler-Lagrange equation.

---

Now my question is- why should one consider a fixed-end points problem for this case? I am stuck at this problem for quite some time now and therefore sincerely appreciate any thoughts, or suggestions.