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

$\begin{array}{ll}\underset{x,u,t_f}{\min }& t_f\\ \text{s.t.}& {\ddot{x}}_1& =& a\cos (u),\\ & {\ddot{x}}_2& =& a\sin (u),\\ & x(0)& =& (0,0{)}^T,\\ & \dot{x}(0)& =& (0,0{)}^T,\\ & x_2(t_f)& =& 5,\\ & \dot{x}(t_f)& =& (45,0{)}^T,\\ & u(t)& \in & [-\frac{\pi }{2},\frac{\pi }{2}].\end{array}$

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

• AMPL/TACO code at Particle steering problem (TACO)