# Increasing convergence rate using Optimal Control and Pontryagin Maximum Principle

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:$$\begin{cases} F=... \\ \dot{\boldsymbol{\lambda}}=... \end{cases}$$

1. 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.