Lotka Volterra (terminal constraint violation)

Lotka Volterra (terminal constraint violation)
State dimension:	1
Differential states:	2
Discrete control functions:	1
Interior point inequalities:	1
Interior point equalities:	2

This site describes a Lotka Volterra variant where a terminal inequality constraint on the differential states is added. A violation of this constraint is penalized as part of the Mayer objective.

Mathematical formulation

The mixed-integer optimal control problem is given by

$\begin{array}{llclr} \displaystyle \min_{x, w, s} & x_2(t_f) + 10s \\[1.5ex] \mbox{s.t.} & \dot{x}_0 & = & x_0 - x_0 x_1 - \; c_0 x_0 \; w, \\ & \dot{x}_1 & = & - x_1 + x_0 x_1 - \; c_1 x_1 \; w, \\ & \dot{x}_2 & = & (x_0 - 1)^2 + (x_1 - 1)^2, \\[1.5ex] & x_0 & \geq & 1.1 - s, \\ & x(0) &=& (0.5, 0.7, 0)^T, \\ & w(t) &\in& \{0, 1\}, \\ & s & \geq & 0. \end{array}$

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 penalizes a biomass x(0) that is below 1.1 at the end of the time horizon.

Parameters

These fixed values are used within the model.

$\begin{array}{rcl} [t_0, t_f] &=& [0, 12],\\ (c_{0}, c_{1}) &=& (0.4, 0.2), \end{array}$

Reference Solutions

If the problem is relaxed, i.e., we demand that $w(t)$ is in the continuous interval $[0, 1]$ rather than being binary, the optimal solution can be determined by means of direct optimal control.

The optimal objective value of the relaxed problem with $n_t=12000, \, n_u=200$ is $x_2(t_f) =1.36548113$. The objective value of the solution with binary controls obtained by Combinatorial Integral Approximation (CIA) is $x_2(t_f) =1.38756111$.

• Reference solution plots
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  Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and $n_t=12000, \, n_u=200$.

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  Differential states and binary cotnrol determined by an direct approach (Radau collocation) with ampl_mintoc and $n_t=12000, \, n_u=200$. The relaxed controls were approximated by Combinatorial Integral Approximation.