The mathematical modeling of plant growth and applications to robotics

In the thesis, a novel modelling approach to study plant growth is proposed. Motivated by the ability of plants to survive and develop in stressful environments, the analysis here proposed focuses on the efficiency of the plant's growing strategies. Many mathematical models have been proposed in the literature to study plants. Here, most of them are critically reviewed. Among them, it still misses a quantitative definition of what the plant is optimising during the growth. In the thesis, the plant is thought as an active agent able to adapt its growing strategy to optimise something. Both the metabolism and the movements of a plant can be assumed as functions to optimise. Firstly, a mechanistic model of the main signals driving the growth has been developed and a novel definition of the plant efficiency arises based on the metabolic needs of the plant. Then, the optimal control theory has been applied to investigate the efficiency of plant motions. This thesis is the first application of optimal control theory to investigate plant motions while taking care of the quantitative and qualitative validation of the model with biological evidence and laboratory experiments. Such an approach is crucial to characterise the plant behaviour and to improve agricultural and ecological studies in those conditions in which laboratory experiments are difficult or expensive to perform. Furthermore, engineers are developing a new technology of robots able to grow and move like plants. Optimal control theory is a suitable tool to translate the efficiency of plant behaviours in features to design plant-inspired robots. In this thesis, the focus is on a specific movement performed by roots during the soil penetration: the circumnutation. The results presented in the thesis help to better study the root-soil interactions (where data lacks due to the soil complexity) and to support the design of autonomous devices able to explore the soil in the most efficient manner. In particular, the soil-root dynamics leads to a new class of optimal control problems describing phenomena in which a mechanical constrained is concerned. In this case, the investigation of the optimal trajectory does not follow in the standard manner. Then, the thesis concerns the characterisation of the optimal control for this new family of problems.