# Rendezvous problem

## 1 Problem

We have two bodies in discrete time: the first is described by its coordinate x_i and its speed v_i, the second has coordinate z_i and speed u_i. Each body has its own dynamics, which we denote as linear systems with matrices A, B, C, D:

\begin{align*} x_{i+1} = Ax_i + Bu_i \\ z_{i+1} = Cz_i + Dv_i \end{align*}

We want these bodies to meet in future at some point T in such a way, that preserve minimum energy through the path. We will consider only kinetic energy, which is proportional to the squared speed at each point of time, thatâ€™s why optimization problem takes the following form:

\begin{align*} & \min \sum_{i=1}^T \|u_i\|_2^2 + \|v_i\|_2^2 \\ \text{s.t. } & x_{t+1} = Ax_t + Bu_t, \; t = 1,\ldots,T-1\\ & z_{t+1} = Cz_t + Dv_t, \; t = 1,\ldots,T-1\\ & x_T = z_T \end{align*}

Problem of this type arise in space engineering - just imagine, that the first body is the spaceship, while the second, say, Mars.

## 2 Code

Open In Colab{: .btn }

## 3 References

- Jupyter notebook by A. Katrutsa