# Lotka Volterra (terminal constraint violation)

Lotka Volterra (terminal constraint violation) | |
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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

Here the differential states 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 . 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.

## Reference Solutions

If the problem is relaxed, i.e., we demand that is in the continuous interval 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 is . The objective value of the solution with binary controls obtained by Combinatorial Integral Approximation (CIA) is .