Lotka Volterra fishing problem: Difference between revisions

Lotka Volterra fishing problem
State dimension:	1
Differential states:	3
Algebraic states:	0
Continuous control functions:	0
Discrete control functions:	1
Continuous control values:	0
Discrete control values:	0
Path constraints:	0
Interior point inequalities:	0
Interior point equalities:	3

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Revision as of 20:47, 6 July 2008

This problem was set up as a small-scale benchmark problem. The optimal solution contains a singular arc, making the Lotka Volterra fishing problem an ideal candidate for benchmarking of algorithms.

In this problem the Lotka Volterra equations for a predator-prey system have been augmented by an additional linear term, relating to fishing by man.

Model dimensions and properties

The model has the following dimensions:

It is thus an ODE model with a single integer control function. The interior point equality conditions fix the initial values of the differential states.

Mathematical formulation

For $t \in [t_{0}, t_{f}]$ the mixed-integer optimal control problem is given by

$\begin{array}{llcl} \min_{x, w} & x_{2} (t_{f}) \\ s.t. & {\dot{x}}_{0} (t) & = & x_{0} (t) - x_{0} (t) x_{1} (t) - c_{0} x_{0} (t) w (t), \\ {\dot{x}}_{1} (t) & = & - x_{1} (t) + x_{0} (t) x_{1} (t) - c_{1} x_{1} (t) w (t), \\ {\dot{x}}_{2} (t) & = & (x_{0} (t) - 1)^{2} + (x_{1} (t) - 1)^{2}, \\ x (0) & = & x_{0}, \\ w (t) & \in & {0, 1} . \end{array}$

Initial values and parameters

These fixed values are used within the model.

$\begin{array}{rcl} t_{0} & = & 0 \\ t_{f} & = & 12 \\ c_{0} & = & 0.4 \\ c_{1} & = & 0.2 \\ x_{0} & = & (0.5, 0.7, 0)^{T} \end{array}$

Reference Solutions

The two differential states and corresponding adjoint variables in the indirect approach

Source Code

  double ref0 = 1, ref1 = 1;                 /* steady state with u == 0 */

  rhs[0] =   xd[0] - xd[0]*xd[1] - p[0]*u[0]*xd[0];
  rhs[1] = - xd[1] + xd[0]*xd[1] - p[1]*u[0]*xd[1];
  rhs[2] = (xd[0]-ref0)*(xd[0]-ref0) + (xd[1]-ref1)*(xd[1]-ref1);

Miscellaneous

The Lotka Volterra fishing problem was introduced by Sebastian Sager in a proceedings paper <bibref>Sager2006</bibref> and revisited in his PhD thesis <bibref>Sager2005</bibref>. These are also the references to look for more details.

References

@@ Line 8: / Line 8: @@
 == Model dimensions and properties ==
+The model has the following [[model dimensions|dimensions]]:
 {{Dimensions
-|image      = [[Image:lotkaindirektStates.png|thumb]]
+|nd        = 1
-|age        = 19
+|nx        = 3
-|gender     = Female
+|nz        = 0
-|sexuality  = Bisexual
+|nu        = 0
-|partner    = [[Celena]]
+|nw        = 1
-|rank       = [[Green riders|Green rider]]
+|np        = 0
-|location   = [[Main Page|Lakeside Weyr]]
+|nrho      = 0
-|parents    = [[Marney]] (mother)<br /> [[K'tol]] (father)
+|nc        = 0
-|siblings   = [[B'ran]]<br /> [[Armana]]
+|nri       = 0
-|dragon     = [[Tyrath]] (Tyr)
+|nre       = 3
-|wing       = Not assigned
 }}
-The model has the following [[model dimensions|dimensions]]:
-<math>
-\begin{array}{rcl}
-n_x &=& 3\\
-n_z &=& 0\\
-n_u &=& 0\\
-n_w &=& 1\\
-n_p &=& 0\\
-n_{\rho} &=& 0\\
-n_c &=& 0\\
-n_{r^\mathrm{i}} &=& 0\\
-n_{r^\mathrm{e}} &=& 3
-\end{array}
-</math>
 It is thus an [http://en.wikipedia.org/wiki/Ordinary_differential_equation ODE] model with a single integer control function. The interior point equality conditions fix the initial values of the differential states.