Let $(\lambda_k)$ be an unbounded and increasing sequence of real numbers, and $f:\mathbb{R}^n \ \times \mathbb{R}^m \rightarrow \mathbb{R}^n$ be sufficiently smooth such that $\dot{x}(t) = f(x(t),u(t))$ has an unique solution for each $x(0) \in \mathbb{R}^n$.
Suppose that the optimal control problem
$$ J_k = \min \int_{0}^{t_f} L(x(s),u(s))\, ds + \lambda_k \| x(t_f) - x_f\|^2 $$ subject to $$ \begin{align} \dot{x}(t) &= f(x(t),u(t)), \quad t\in[0, t_f],\\ x(0) &= x_0, \end{align} $$ has an unique solution for each $\lambda_k$.
If the optimal control problem $$ J = \min \int_{0}^{t_f} L(x(s),u(s))\, ds $$ subject to $$ \begin{align} \dot{x}(t) &= f(x(t),u(t)), \quad t\in[0, t_f],\\ x(0) &= x_0, \\ x(t_f) &= x_f \end{align} $$
has also an unique solution, it is true that $J_k \rightarrow J$?
