Optimal control theory


A set of mathematical techniques and theorems concerned with finding optimal time-paths of particular systems. The state of the system at any point in time is completely described by the numerical values of a set of variables called state variables. The values of these variables, and so the state of the system, change over time according to some dynamic relationships which must be specified mathematically. There is a second set of variables, called the control variables, whose values at each point in time are to be chosen by the decision-taker and which then determine the time-paths of the state variables. The decision-taker will have a preference ordering over alternative time-paths of the system, and this must also be specified mathematically. Optimal control theory then studies the problem of choosing those time-paths of the control variables out of the set of paths which are feasible, which leads to the preferred time-path of the entire system.
The main application of optimal control theory in economics has been in the theory of optimal economic growth. The system concerned is an aggregate macroeconomic model of the economy. The problem is to find the time-path of investment which meets some objective such as maximizing consumption per head of the population. However, many other applications of the theory exist and it has been a vital tool in the development of dynamic analysis in all areas of economics.

Reference: The Penguin Dictionary of Economics, 3rd edt.