Hanging chain problem: Difference between revisions

Hanging chain problem
State dimension:	1
Differential states:	2
Discrete control functions:	1
Interior point equalities:	2

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Revision as of 17:42, 5 May 2016

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

$\begin{array}{llcl} \min_{x, u} & x_{2} (t_{f}) \\ s.t. & {\dot{x}}_{1} & = & u, \\ {\dot{x}}_{2} & = & x_{1} (1 + u^{2})^{1 / 2}, \\ {\dot{x}}_{3} & = & (1 + u^{2})^{1 / 2}, \\ x (t_{0}) & = & (a, 0, 0)^{T}, \\ x_{1} (t_{f}) & = & b, \\ x_{3} (t_{f}) & = & L p, \\ x (t) & \in & [0, 10], \\ u (t) & \in & [- 10, 20] . \end{array}$

Parameters

In this model the parameters used are

$\begin{array}{rcl} [t_{0}, t_{f}] & = & [0, 1], \\ (a, b) & = & (1, 3), \\ L p & = & 4 . \end{array}$

Source Code

Model descriptions are available in

AMPL/TACO code at Hanging chain problem (TACO)

@@ Line 37: / Line 37: @@
 \begin{array}{rcl}
 [t_0, t_f] &=& [0, 1],\\
-(a,b) &=& (0.4, 0.2),\\
+(a,b) &=& (1,3),\\
 Lp &=& 4.
 \end{array}