Subject to equations (14)-(16) where \(\eta=(x,y,M)\).
LPS method for the IHOC problem
One of the methods applied to numerically solve continuous-time problems
such as OC problems is the pseudo-spectral (PS) methods [18, 26,
27].
This method compared to finite element methods and finite-difference
methods has a high order of accuracy. By using this method the
continuous-time problems is reformed to an NLP problem. One type of
these methods is the LPS method. The Legendre-Gauss (LG) points defined
in the open interval \((-1,\ 1)\), the Legendre-Gauss-Radau (LGR)
points defined in the half-open interval \([-1,\ 1)\), the
Legendre-Gauss-Lobatto (LGL) points defined in the close interval\([-1,\ 1]\) are three commonly used sets of interpolating
points.
The IHOC problem (14)-(17) is defined on the interval\([0,\infty)\).
At first, the following change of variables is used to transform the
interval \([0,\infty)\) into the finite interval\([-1,\ 1)\):