14 - Time-Optimal Control in the Phase Plane

Summary

Many physical systems that are amenable to mathematical modelling do not exist in isolation from human intervention. A good example is the British economy, for which the Treasury has a complicated mathematical model. The state of the system (the economy) is given by values of the dependent variables (for example, unemployment, foreign exchange rates, growth, consumer spending and inflation), and the government attempts to control the state of the system to a target state (low inflation, high employment, high growth) by varying several control parameters (most notably taxes and government spending). There is also a cost associated with any particular action, which the government tries to minimize (some function of, for example, government borrowing and, one would hope, the environmental cost of any government action or inaction). The optimal control leads to the economy reaching the target state with the smallest possible cost.

Another system, for which we have studied a simple mathematical model, consists of two populations of different species coexisting on an isolated island. For the case of two herbivorous species, which we studied in Chapter 9, we saw that one species or the other will eventually die out. If the island is under human management, this may well be undesirable, and we would like to know how to intervene to maintain the island close to a state of equilibrium, which we know, if left uncontrolled, is unstable. We could choose between either continually culling the more successful species, continually introducing animals of the less successful species or some combination of these two methods of control. Each of these actions has a cost associated with it.