Oil Shale Pyrolysis

Oil Shale Pyrolysis
State dimension:	1
Differential states:	2
Continuous control functions:	1
Discrete control functions:	0
Interior point equalities:	2

The following problem is an example from the global optimal control literature and was introduced in <bibref>Wen1977</bibref>.

Mathematical formulation

$\begin{array}{ll}\underset{u}{\min }& -x_1(t_N{)}^2\\ \text{s.t.}& {\dot{x}}_0(t)=-k_0{x}_0(t)-(k_2+k_3+k_4)x_0(t)x_1(t)\\ & {\dot{x}}_1(t)=k_0{x}_0(t)-k_1{x}_1(t)+k_2{x}_0(t)x_1(t)\\ & k_i=a_i{e}^{-u(t)\frac{b_i}{R}},\mathrm{\forall }i\in \{1,\dots ,5\}\\ [1.5ex]& t\in [t_0,t_N]\\ & u(t)\in [698.15/748.15,1]\\ & x(t_0)=(1,0{)}^T\\ \end{array}$

where this is the normalized form with

$u(t)=\frac{1}{u_{temp}}$, with

$u_{temp}\in [698.15,748.15]$

Parameters

State variables

Symbol	Initial value ($t_0$)
$x_0(t)$	$1$
$x_1(t)$	$0$

Parameters

Symbol	Value
$a_1$	$8.86$
$a_2$	$24.25$
$a_3$	$23.67$
$a_4$	$18.75$
$a_5$	$20.7$
$b_1$	$20.3$
$b_2$	$37.4$
$b_3$	$33.8$
$b_4$	$28.2$
$b_5$	$31.0$

Control variable

Symbol	Interval
$u(t)$	[698.15/748.15,1]

Measurement grid

Reference solution

Coming soon.

References

<bibreferences/>