Given system of equation:
\begin{cases} \frac{dx_1(t)}{dt}= \frac{df}{dx_1} \\ \frac{dx_2(t)}{dt}+x_2(t)=k(t) \cdot \frac{d^2f}{d^2x_1} \\ \frac{dk(t)}{dt}=??? \end{cases}
where $x_1(t),x_2(t),k(t)$ - variables, $f$ - any unimodal function, for experiment $f=\operatorname{sech}(x_1)$.
Variable $k(t)$ has a special purpose. It acts as a "gain/attenuation" factor.
Task: construct dynamic of the third equation ($???$) in such a way that $\nabla_{x_1}^2f \rightarrow -1$, regardless of the function $f$.
Let me rewrite your system a bit more compactly yet explicitly,
$$ \begin{aligned} \dot{x}_1(t) &= f'(x_1(t)), \\ \dot{x}_2(t) &= -x_2(t) + k(t) f''(x_1(t)), \end{aligned} $$
where we'll take $k: \mathbb{R}\to \mathbb{R}$ to be any control law. Unfortunately, there exists initial conditions $x_1(0)$ and functions $f$ where no such $k$ exists to make $f''(x_1)\to -1.$ The state $x_1$ isn't controllable. We can see that the $x_1(t)$ dynamics is entirely determined by $x_1(t).$ It is an independent subsystem whose dynamics can be resolved independent of $k(t).$
So, for example, consider the (unimodal) bump function $$ f(x) = \left\{ \begin{array}{lll} \mathrm{exp}\left(-\frac{1}{1 - x^2}\right), & \quad & \mathrm{if}\; x \in (-1,1)\\ 0, & \quad & \mathrm{otherwise} \end{array} \right. $$ Although the dynamics are going to be locally asymptotically stable around the equilibrium $x_1 = 0$ where $f$ attains it maximum, they will not be in the region where $f'(x_1) = 0.$ If you are looking at minimums, just flip $f$ and make the same observation.