Hanging chain problem

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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}{llcl}
 \displaystyle \min_{x, u} & x_2(t_f)   \\[1.5ex]
 \mbox{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