Ocean

Ocean
State dimension:	1
Differential states:	3
Discrete control functions:	2

The Ocean problem describes fossil fuel consumption and sequestration into the ocean [1]. It is a two box model where $S$ describes the carbon stock in the atmosphere and upper layer ocean, $R$ describes the carbon stock in fossil reserve and $D_L$ the carbon stock in the deeper layer. The dynamics are given by an ODE model.

The optimal control function exhibits a singular arc.

Mathematical formulation

$\begin{array}{lll}\underset{w}{\min }& & -y(t_f)\\ \text{subject to}\\ \dot{y}(t)& =& \exp (-\rho \cdot t)\cdot (U(t)-A(t)-u_1(t)\cdot C(t)-D(t)),\\ \dot{S}(t)& =& u_1(t)-u_2(t)-\gamma \cdot (S(t)-\omega \cdot D_L(t)),\\ \dot{R}(t)& =& -u_1(t)\\ y(0)& =& 0,\\ S(0)& =& 2\cdot 10^3,\\ R(0)& =& 10^4\\ S(t),R(t)& \in & [0,10^5],\\ u_1(t),u_2(t)& \in & [0,40]\\ \end{array}$

with auxiliary functions

$\begin{array}{rlrl}& U(t)=b\cdot u_1(t)-\mu \cdot u_1(t{)}^2,& & D(t)=\nu \cdot (0.3\cdot S(t)-S_{\text{preind}}{)}^2,\\ & A(t)=a_1\cdot u_2(t)+a_2\cdot u_2(t{)}^2,& & D_L(t)=D_{L,0}+R_0+S_0-R(t)-S(t),\\ & C(t)=c_1-c_2\cdot R(t).& \end{array}$

Parameters

Parameters

Symbol	Value
$t_f$	$400$
$\rho$	$0.03$
$\gamma$	$0.001$
$\omega$	$0.1$
$b$	$50$
$\mu$	$0.5$
$a_1$	$2$
$a_2$	$2$
$\nu$	$1$
$c_1$	$50$
$c_2$	$0.004$
$S_{\text{preind}}$	$600$
$S_0$	$2000$
$R_0$	$10^4$
$D_{L,0}$	$2.3\cdot 10^4$

Reference Solutions

Here is one local solution to the above control problem.

• Reference solution plots
• 
  States and discretized control for a local optimum. Due to the explicit time dependence the time $t$ was added as an additional state.

Miscellaneous and Further Reading

The problem description and further references can be found in the PhD thesis of Dennis Janka [2].

References

[1] W. Rickels and S. Sager. Personal communication. 2015.
[2] Janka, D.: Sequential quadratic programming with indefinite Hessian approximations for nonlinear optimum experimental design for parameter estimation in differential-algebraic equations. Ph.D. thesis, Ruprecht-Karls-Universität Heidelberg (2015). URL https://mathopt.de/publications/Janka2015.pdf