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

$\begin{array}{llclr} \min_{x, w, s} & x_{2} (t_{f}) + 10 s \\ s.t. & {\dot{x}}_{0} & = & x_{0} - x_{0} x_{1} - c_{0} x_{0} w, \\ {\dot{x}}_{1} & = & - x_{1} + x_{0} x_{1} - c_{1} x_{1} w, \\ {\dot{x}}_{2} & = & (x_{0} - 1)^{2} + (x_{1} - 1)^{2}, \\ x_{0} & \geq & 1.1 - s, \\ x (0) & = & (0.5, 0.7, 0)^{T}, \\ w (t) & \in & {0, 1}, \\ s & \geq & 0 . \end{array}$

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
Optimal relaxed controls and states determined by an direct approach with ampl_mintoc (Radau collocation) and $n_{t} = 12000, n_{u} = 200$ .
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.