Particle steering problem

Particle steering problem
State dimension:	1
Differential states:	2
Discrete control functions:	1
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

$\begin{array}{llcl} \min_{x, u, t_{f}} & t_{f} \\ s.t. & {\ddot{x}}_{1} & = & a \cos (u), \\ {\ddot{x}}_{2} & = & a \sin (u), \\ x (t_{0}) & = & (0, 0)^{T}, \\ \dot{x} (0) & = & (0, 0)^{T}, \\ x_{2} (t_{f}) & = & 5, \\ \dot{x} (t_{f}) & = & (45, 0)^{T}, \\ u (t) & \in & [- \frac{π}{2}, \frac{π}{2}] . \end{array}$

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

AMPL/TACO code at Particle steering problem (TACO)