Mathematical modelling and optimal control of constrained systems

This thesis is concerned with mathematical modelling and optimal control of constrained systems. Each of the systems under consideration is a system that can be controlled by one of the variables, and this control is subject to constraints. First, we consider middle-distance running where a runner's horizontal propulsive force is the control which is constrained to be within a given range. Middle-distance running is typically a strategy-intensive race as slipstreaming effects come into play since speeds are still relatively fast and runners can leave their starting lane. We formulate a two-runner coupled model and determine optimal strategies using optimal control theory. Second, we consider two applications of control systems with delay related to R&D; expenditure. The first of these applications relates to the defence industry. The second relates to the pharmaceutical industry. Both applications are characterised by a long delay between initial investment in R&D; and seeing the benefits of R&D; realised. We formulate models tailored to each application and use optimal control theory to determine the optimal proportion of available funds to invest in R&D; over a given time horizon. Third, we consider a mathematical model of urban burglary based on the Short model. We make some modifications to this model including the addition of deterrence due to police officer presence. Police officer density is the control variable, which is constrained due to a finite number of police officers. We look at different control strategies for the police and their effect on burglary hot-spot formation.