Hanging chain problem

Hanging chain problem
State dimension:	1
Differential states:	3
Continuous control functions:	1
Path constraints:	4
Interior point equalities:	5

The Hanging chain problem is concerned with finding a chain (of uniform density) of length $L$ suspendend between two points $a,b$ with minimal potential energy. (Problem taken from the COPS library)

Mathematical formulation

The problem is given by

$\begin{array}{ll}\underset{x,u}{\min }& x_2(t_f)\\ \text{s.t.}& {\dot{x}}_1& =& u,\\ & {\dot{x}}_2& =& x_1(1+u^2{)}^{1/2},\\ & {\dot{x}}_3& =& (1+u^2{)}^{1/2},\\ & x(t_0)& =& (a,0,0{)}^T,\\ & x_1(t_f)& =& b,\\ & x_3(t_f)& =& Lp,\\ & x(t)& \in & [0,10],\\ & u(t)& \in & [-10,20].\end{array}$

Parameters

In this model the parameters used are

$\begin{array}{rcl}[t_0,t_f]& =& [0,1],\\ (a,b)& =& (1,3),\\ Lp& =& 4.\end{array}$

Source Code

Model descriptions are available in

• AMPL/TACO code at Hanging chain problem (TACO)
• GEKKO Python code at Hanging chain problem (GEKKO)