Nonlinear controllability via the initial state : with application to the spread of rabies
There are many problems in medicine and biology involving some kind of spatial spread. Often the aim in such problems is to control the spread. A large proportion of medical and biological systems distinguish themselves from the types of system found in engineering by the way the control acts. This is illustrated by considering the specific example of the spread of rabies among foxes. A brief description of a model for the spatial spread of rabies among foxes, developed by Murray et al. (1986), is given. This model is then extended to include the control mechanism. The problem is to prevent the spread of the rabies virus by vaccinating or culling foxes via the distribution of bait in a region around an observed outbreak. The extended model can be formulated as a nonlinear time-varying control system described by partial differential equations. In contrast to most engineering type control problems the control does not continuously affect the system but only acts through the initial distributions. A general theory is developed for dealing with such nonlinear systems by the use of a fixed point theorem. In a similar way to Pritchard and Salamon (1987) and Hinrichsen and Pritchard (1994) the dynamics are considered on a triple of Banach spaces Z C Z c Z to allow for the possible unboundedness of the nonlinearity. Thus the nonlinearity is considered as a map from Z into Z. A mild form of the time-varying system is introduced to allow for a wider class of nonlinearities. Assumptions are introduced so that the mild form of system equation is well-defined and has a fixed point that, at least partially, solves the control problem. An adaptive scheme is introduced that constructs the control that gives rise to the fixed-point but is easier to implement computationally. This scheme is less intuitive than that provided by the fixed point theorem. However the method exploits the existence of the fixed point while only requiring the final states (and not the states on the whole time interval) to be stored at each step. By assuming that the linear part of the system is a time-varying perturbation of a time-invariant operator it is shown how a mild form for the system equation can be derived from the original dynamics. Moreover suppose that the time-invariant operator is the generator of a strongly continuous semigroup. Then the conditions for the mild form of the system to be well-defined and have a fixed point can be reduced to conditions on the semi group and perturbation. Existence theorems are provided for solutions of semilinear systems with unbounded nonlinearities. The theory is applied to the rabies model. The problem and the theory are illustrated by some numerical simulations.