Optimal Control Parameter Values: Converge/Fail?

[kalish1]

Motivation:

I am working with a code that minimizes the objective functional value in an optimal control problem. It takes $A_1,A_2,A_3,A_4$ (the balancing factors for various components of the objective functional) as inputs, and then outputs the values of the state variables, control functions, and adjoint functions as solutions.

Method:

The algorithm can be summarized as follows:

Here, $\vec{x} = (x_1, \ldots, x_8)$ and $\vec{\kappa} = (\kappa_1, \ldots, \kappa_8)$ denote the vector approximations for the states and adjoints.

1. Make initial guesses for $u_1,u_2$ over the interval $[0,t_f].$ Store the guesses as $u_1,u_2.$
2. Using the initial condition $x_1 = x(0) = a$ and the stored values for $u_1,u_2,$ solve $\vec{x}$ forward in time (using a $4$th-order Runge-Kutta scheme) according to its system of differential equations in the optimality system.
3. Using the transversality condition $\kappa_8 = \kappa(t_f) = 0$ and the stored values for $u_1,u_2,\vec{x},$ solve $\vec{\kappa}$ backward in time (using the same scheme as in 2) according to its system of differential equations in the optimality system.
4. Update $u_1,u_2$ by entering the new $\vec{x}$ and $\vec{\kappa}$ values into the characterization of the optimal controls.
5. Check convergence. If either the relative error between all state variables, the adjoint functions, and the control functions is less than a fixed $\delta$ or the number of iterations of the procedure exceeds a fixed $k^*,$ output the current values of the state variables and adjoint/control functions as solutions. Otherwise return to 2.----------
Question:

Is there a robust procedure to find reasonable ranges of $A_1,A_2,A_3,A_4$ such that the code converges? And values for which the code fails to converge?

I could brute-force the casework, but that is not ideal.

Thanks in advance for any help. Please let me know if excerpts of the code are necessary.

 

[jvicens]

Thank you for sharing your approach and code with us. I can understand your concern about finding a robust procedure to determine the appropriate ranges for the $A_1,A_2,A_3,A_4$ balancing factors in order to ensure convergence of the code. Brute-force casework may not be the most efficient method, especially if the number of iterations increases with the complexity of the problem.

One approach that you could consider is to perform a sensitivity analysis on these balancing factors. This involves systematically varying each factor individually while keeping the others constant and observing the effect on the convergence of the code. This will give you an idea of the range of values for each factor that leads to successful convergence. Additionally, you could also vary multiple factors simultaneously to see if there are any interactions between them that affect the convergence.

Another option is to use optimization techniques, such as gradient descent or genetic algorithms, to find the optimal values for these balancing factors that minimize the objective functional value and ensure convergence. These techniques can be computationally expensive but can be more efficient than brute-force casework.

In terms of values for which the code fails to converge, it may be helpful to analyze the behavior of the code for extreme or unrealistic values of the balancing factors. This can give you insight into the sensitivity of the code to these factors and help identify any potential issues or limitations.

I hope this helps. Good luck with your research!