Lotka Volterra multi-arcs problem

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 $x_2(t_f) =1.4237735$.

• Reference solution plots
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  Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and $n_t=18000, \, n_u=300$.

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  Optimal binary controls and states determined by an direct approach (Radau collocation) with ampl_mintoc and $n_t=18000, \, n_u=300$. The relaxed controls were approximated by Combinatorial Integral Approximation.