Lotka Volterra multi-arcs problem

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Lotka Volterra multi-arcs problem
State dimension: 1
Differential states: 3
Discrete control functions: 1
Interior point equalities: 3

This site describes a Lotka Volterra variant with three singular arcs instead of only one in the standard variant.

Mathematical formulation

The mixed-integer optimal control problem is given by


\begin{array}{llclr}
 \displaystyle \min_{x, w} & x_2(t_f)   \\[1.5ex]
 \mbox{s.t.} 
 & \dot{x}_0 & = &  x_0 - x_0 x_1 - \;  c_0\; x_0 \; w, \\
 & \dot{x}_1 & = & - x_1 + x_0 x_1 - \; c_1\; x_1 \; w,  \\
 & \dot{x}_2 & = & (x_0 - p_0(t))^2 + (x_1 - p_1(t))^2,  \\[1.5ex]
 & x(0) &=& (0.5, 0.7, 0)^T, \\
 & w(t) &\in&  \{0, 1\}, 
\end{array}

Here the differential states (x_0, x_1) describe the biomasses of prey and predator, respectively. The third differential state is used here to transform the objective, an integrated deviation, into the Mayer formulation \min \; x_2(t_f). This problem variant allows to choose between three different fishing options.

Parameters

These fixed values are used within the model.


\begin{array}{rcl}
[t_0, t_f] &=& [0, 18],\\
(c_0, c_1,) &=& (0.4, 0.2),\\
p_{0}(t) &=& \left\{ 	\begin{array}{ll}
1 \quad \quad &\text{for } \ t\in [0,7.2],\\
0.9 \quad \quad &\text{for } \ t\in [7.2,12],\\
1 \quad \quad &\text{for } \ t\in [12,18].
\end{array}	\right. \\
p_1(t) &=& p_0(t).
\end{array}

Reference Solutions

If the problem is relaxed, i.e., we demand that w(t) be in the continuous interval [0, 1] instead of the binary choice \{0,1\}, the optimal solution can be determined by means of direct optimal control.

The optimal objective value of the relaxed problem with  n_t=18000, \, n_u=300  is x_2(t_f) =1.41842699. The objective value of the binary controls obtained by Combinatorial Integral Approimation (CIA) is x_2(t_f) =1.4237735 .