Ocean: Difference between revisions

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

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Latest revision as of 10:50, 29 January 2026

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

$\begin{array}{lll} \min_{w} & - y (t_{f}) \\ subject to \\ \dot{y} (t) & = & \exp (- ρ \cdot t) \cdot (U (t) - A (t) - u_{1} (t) \cdot C (t) - D (t)), \\ \dot{S} (t) & = & u_{1} (t) - u_{2} (t) - γ \cdot (S (t) - ω \cdot D_{L} (t)), \\ \dot{R} (t) & = & - u_{1} (t) \\ y (0) & = & 0, \\ S (0) & = & 2 \cdot 1 0^{3}, \\ R (0) & = & 1 0^{4} \\ S (t), R (t) & \in & [0, 1 0^{5}], \\ u_{1} (t), u_{2} (t) & \in & [0, 40] \end{array}$

with auxiliary functions

$\begin{aligned} U (t) = b \cdot u_{1} (t) - μ \cdot u_{1} (t)^{2}, & D (t) = ν \cdot (0.3 \cdot S (t) - S_{preind})^{2}, \\ A (t) = a_{1} \cdot u_{2} (t) + a_{2} \cdot u_{2} (t)^{2}, & D_{L} (t) = D_{L, 0} + R_{0} + S_{0} - R (t) - S (t), \\ C (t) = c_{1} - c_{2} \cdot R (t) . \end{aligned}$

Parameters

Parameters
Symbol	Value
$t_{f}$	$400$
$ρ$	$0.03$
$γ$	$0.001$
$ω$	$0.1$
$b$	$50$
$μ$	$0.5$
$a_{1}$	$2$
$a_{2}$	$2$
$ν$	$1$
$c_{1}$	$50$
$c_{2}$	$0.004$
$S_{preind}$	$600$
$S_{0}$	$2000$
$R_{0}$	$1 0^{4}$
$D_{L, 0}$	$2.3 \cdot 1 0^{4}$

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

@@ Line 1: / Line 1: @@
 {{Dimensions
 |nd        = 1
-|nx        = 1
+|nx        = 3
-|nw        = 1
+|nw        = 2
 }}
-The '''Ocean problem''' describes fossil fuel consumption and sequestration into the ocean [169]. It is a
+The '''Ocean problem''' describes fossil fuel consumption and sequestration into the ocean [[#PersonalComm | [1]]]. It is a
 two box model where <math>S</math> describes the carbon stock in the atmosphere and upper layer ocean,
 <math>R</math> describes the carbon stock in fossil reserve and <math>D_L</math> the carbon stock in the deeper layer. The dynamics are given by an [[:Category:ODE model|ODE model]].
@@ Line 15: / Line 15: @@
 <math>
   \begin{array}{lll}
-  \displaystyle \min_{w} && y(t_f) \\
+  \displaystyle \min_{w} && -y(t_f) \\
   \text{subject to} \\
 \quad \dot{y}(t) & = &  \exp(-\rho \cdot t) \cdot (U(t)- A(t) - u_1(t) \cdot C(t) - D(t)),\\
@@ Line 22: / Line 22: @@
 \quad y(0) &=& 0, \\
 \quad S(0) &=& 2 \cdot 10^3, \\
-\quad R(0) &=& 10^4
+\quad R(0) &=& 10^4 \\
+\quad S(t), R(t) & \in & [0,10^5], \\
+\quad u_1(t), u_2(t) & \in & [0,40] \\
    \end{array}
 </math>
@@ Line 33: / Line 35: @@
   \begin{align}
    &U(t) = b \cdot u_1(t) - \mu \cdot u_1(t)^2, \quad \quad
-   & D(t) = \nu \cdot (0.3 \cdot S(t) - S_{\text{preind}})^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, \quad \quad
-   & D_L(t) = D_{L, 0} + R_0 + S_0 - R(t) - S(t), \\
+   && D_L(t) = D_{L, 0} + R_0 + S_0 - R(t) - S(t), \\
    &C(t) = c_1 - c_2 \cdot R(t). &
   \end{align}
 </math>
 </p>
+== Parameters ==
+{| class="wikitable"
+|+Parameters
+|-
+|Symbol
+|Value
+|-
+|<math>t_f</math>
+|<math>400</math>
+|-
+|<math>\rho</math>
+|<math>0.03</math>
+|-
+|<math>\gamma</math>
+|<math>0.001</math>
+|-
+|<math>\omega</math>
+|<math>0.1</math>
+|-
+|<math>b</math>
+|<math>50</math>
+|-
+|<math>\mu</math>
+|<math>0.5</math>
+|-
+|<math>a_1</math>
+|<math>2</math>
+|-
+|<math>a_2</math>
+|<math>2</math>
+|-
+|<math>\nu</math>
+|<math>1</math>
+|-
+|<math>c_1</math>
+|<math>50</math>
+|-
+|<math>c_2</math>
+|<math>0.004</math>
+|-
+|<math>S_{\text{preind}}</math>
+|<math>600</math>
+|-
+|<math>S_0</math>
+|<math>2000</math>
+|-
+|<math>R_0</math>
+|<math>10^4</math>
+|-
+|<math>D_{L,0}</math>
+|<math>2.3 \cdot 10^4</math>
+|}
 == Reference Solutions ==
@@ Line 45: / Line 101: @@
 Here is one local solution to the above control problem.
-<gallery caption="Reference solution plots" widths="180px" heights="140px" perrow="1">
+<gallery caption="Reference solution plots" widths="500px" heights="300px" perrow="1">
-  Image:Rao_Mease.png| States and discretized control for a local optimum.
+  Image:Ocean.png| States and discretized control for a local optimum. Due to the explicit time dependence the time <math>t</math> was added as an additional state.
 </gallery>
 == Miscellaneous and Further Reading ==
-The problem description and further references can be found in the PhD thesis of Michael Ernst Geiger [[#GeigerPhD|[1]]].
+The problem description and further references can be found in the PhD thesis of Dennis Janka [[#JankaPhD|[2]]].
 == References ==
-<span id="GeigerPhD">[1]</span> "Adaptive Multiple Shooting for Boundary Value Problems and Constrained Parabolic Optimization Problems" by M. E. Geiger  <br>
+<span id="PersonalComm">[1]</span> W. Rickels and S. Sager. Personal communication. 2015. <br>
+<span id="JankaPhD">[2]</span> 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 <br>
 [[Category:MIOCP]]
 [[Category:Sensitivity-seeking arcs]]