# Particle steering problem

Particle steering problem
State dimension: 1
Differential states: 2
Discrete control functions: 1
Path constraints: 2
Interior point equalities: 7

The Particle steering problem minimizes "the time taken for a particle, acted upon by a thrust of constant magnitude, to achieve a given altitude and terminal velocity." (Cite and problem taken from the COPS library)

## Mathematical formulation

The problem is given by

$\begin{array}{llcl} \displaystyle \min_{x, u, t_f} & t_f \\[1.5ex] \mbox{s.t.} & \ddot{x}_1 & = & a \cos (u), \\ & \ddot{x}_2 & = & a \sin (u), \\ & x(0) &=& (0, 0)^T, \\ & \dot{x}(0) &=& (0, 0)^T, \\ & x_2 (t_f) &=& 5, \\ & \dot{x}(t_f) &=& (45, 0)^T, \\ & u(t) &\in& [-\frac{\pi}{2},\frac{\pi}{2}]. \end{array}$

where $(x_1, x_2)$ is the position of the particle, $u$ is the control angle and $a$ is the constant magnitude of thrust.

## Source Code

Model descriptions are available in