Optimal Control

There are two examples realized with GEKKO:

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Simple one: Mixed Integer Non Linear Programming problem:

• There is no time-horizon
• State and control variables are the same

$$ \begin{gather*} \min\limits_{x_1, x_2, x_3, x_4} x_1 x_4 (x_1 + x_2 + x_3) + x_3 \\ x_1 x_2 x_3 x_4 \geq 25 \\ x^2_1 + x^2_2 + x^2_3 + x^2_4 = 40 \\ 1 \geq x_1, x_2, x_3, x_4 \geq 5 \\ x_0 = (1, 5, 5, 1) \end{gather*} $$

Current results:

• The algorithm (Gauss-Newton) has been implemented entirely. Everything works correctly.

Current bottlenecks:

• The GN algorithm doesn't work better than the existing method from GEKKO (Branch and Bound based algorithm).

What I plan to do:

• Find more different examples of the problem and try the method.

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Complicated one: Mixed Integer Non Linear Optimal Control problem:

• There is a time-horizon

$$ \begin{split} F(\textbf{x}, \textbf{i}) = \frac{1}{2} & \int\limits^{T}_{t_0} (x(t) - x_{ref})^2 dt \rightarrow \min \\ \text{such that:} \quad & x(t_0) = x_0 \\ & \dot{x}(t) = x^3(t) - i(t) \\ & \textbf{i} \in P \cap \mathbb{Z}^{N} \\ & N = 30, \ x_0 = 0.8, \ x_{ref} = 0.7 \end{split} $$

Current results:

• The first step (S1) of the tree-step GN algorithm has been implemented.
• [UPD] The second and the third methods have been implemented.
• Completed steps are substantiated theoretically. All the necessary calculations are attached.

Current bottlenecks:

• The GN algorithm doesn't work better than the existing method from GEKKO.