Increasing convergence rate using Optimal Control and Pontryagin Maximum Principle

Question

My question is in addition to Tuning the optimal control synthesized according to the Pontryagin maximum/minimum principle and choosing the cost function, but requires help from the mathematical side of view.

Given system of ODE:

$F=\begin{cases} \dot{x}=g \\ \dot{g}=-k_g \cdot g+k_g \cdot\frac{df}{dx}+u \\ \dot{h}=-k_h \cdot h+k_h \cdot\frac{d^2f}{d^2x} \end{cases}$

where:

$\boldsymbol{X}=(x,g,h)$ - vector of variables, which is moving towards some final state $\boldsymbol{X}_f$;

$f=e^{-(x-c)^2}$;

$c$ - unknown constant;

$k_g,k_h$ - parameters, can be selected from the range $3>k_g,k_h>1$;

$u$ - control input;

Feature: this system go to a state in which $g=0$ and $sign(h)=-1$.

Algorithm:

To ensure usual convergence, I chose the cost function in the form: $J= \int_{0}^{\infty} (g+u^2) dt$

1. Hamiltonian: $H=-(g+u^2)+\boldsymbol{\lambda} \cdot F$ where $\boldsymbol{\lambda}=(\lambda_1,\lambda_2,\lambda_3)$

2. Co-state equation: $\dot{\lambda}=\frac{dH}{d\boldsymbol{X}}$ where $\boldsymbol{X}=(x,g,h)$

3. Solve equation for control input $u$: $\frac{dH}{du}=0$

4.Write resulting system of equation:$\begin{cases} F=... \\ \dot{\boldsymbol{\lambda}}=... \end{cases}$

5. Solve numerically.

Problem: in what ways can the cost function be written to maximize convergence rate to the final state $\boldsymbol{X}_f$ for given parameters $k_g,k_h$.

$J= \int_{0}^{\infty} (???)dt$

I would be glad and grateful for any help.