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Mathematical modelling, simulation, and optimisation are core methodologies for future
developments in engineering, natural, and life sciences. This work aims at applying these
mathematical techniques in the field of biological processes with a focus on the wine
fermentation process that is chosen as a representative model.
In the literature, basic models for the wine fermentation process consist of a system of
ordinary differential equations. They model the evolution of the yeast population number
as well as the concentrations of assimilable nitrogen, sugar, and ethanol. In this thesis,
the concentration of molecular oxygen is also included in order to model the change of
the metabolism of the yeast from an aerobic to an anaerobic one. Further, a more sophisticated
toxicity function is used. It provides simulation results that match experimental
measurements better than a linear toxicity model. Moreover, a further equation for the
temperature plays a crucial role in this work as it opens a way to influence the fermentation
process in a desired way by changing the temperature of the system via a cooling
mechanism. From the view of the wine industry, it is necessary to cope with large scale
fermentation vessels, where spatial inhomogeneities of concentrations and temperature
are likely to arise. Therefore, a system of reaction-diffusion equations is formulated in
this work, which acts as an approximation for a model including computationally very
expensive fluid dynamics.
In addition to the modelling issues, an optimal control problem for the proposed
reaction-diffusion fermentation model with temperature boundary control is presented
and analysed. Variational methods are used to prove the existence of unique weak solutions
to this non-linear problem. In this framework, it is possible to exploit the Hilbert
space structure of state and control spaces to prove the existence of optimal controls.
Additionally, first-order necessary optimality conditions are presented. They characterise
controls that minimise an objective functional with the purpose to minimise the final
sugar concentration. A numerical experiment shows that the final concentration of sugar
can be reduced by a suitably chosen temperature control.
The second part of this thesis deals with the identification of an unknown function
that participates in a dynamical model. For models with ordinary differential equations,
where parts of the dynamic cannot be deduced due to the complexity of the underlying
phenomena, a minimisation problem is formulated. By minimising the deviations of simulation
results and measurements the best possible function from a trial function space
is found. The analysis of this function identification problem covers the proof of the
differentiability of the function–to–state operator, the existence of minimisers, and the
sensitivity analysis by means of the data–to–function mapping. Moreover, the presented
function identification method is extended to stochastic differential equations. Here, the
objective functional consists of the difference of measured values and the statistical expected
value of the stochastic process solving the stochastic differential equation. Using a
Fokker-Planck equation that governs the probability density function of the process, the
probabilistic problem of simulating a stochastic process is cast to a deterministic partial
differential equation. Proofs of unique solvability of the forward equation, the existence of
minimisers, and first-order necessary optimality conditions are presented. The application
of the function identification framework to the wine fermentation model aims at finding
the shape of the toxicity function and is carried out for the deterministic as well as the
stochastic case.