CONTROL THEORY

field of applied mathematics that is relevant to the control of certain physical processes and systems. Although control theory has deep connections with classical areas of mathematics, such as the calculus of variations and the theory of differential equations, it did not become a field in its own right until the late 1950s and early 1960s. After World War II, problems arising in engineering and economics were recognized as variants of problems in differential equations and in the calculus of variations, though they were not covered by existing theories. At first, special modifications of classical techniques and theories were devised to solve individual problems. It was then recognized that these seemingly diverse problems all had the same mathematical structure, and control theory emerged. The systems, or processes, to which control theory is applied have the following structure. The state of the system at each instant of time t can be described by n quantities, which are labeled x1(t), x2(t), . . . , xn(t). For example, the system may be a mixture of n chemical substances undergoing a reaction. The quantities x1(t), . . . , xn(t) would represent the concentrations of the n substances at time t. At each instant of time t, the rates of change of the quantities x1(t), . . . , xn(t) depend upon the quantities x1(t), . . . , xn(t) themselves and upon the values of k so-called control variables, u1(t), . . . , uk(t), according to a known law. The values of the control variables are chosen to achieve some objective. The nature of the physical system usually imposes limitations on the allowable values of the control variables. In the chemical-reaction example, the kinetic equations furnish the law governing the rate of change of the concentrations, and the control variables could be pressure and temperature, which must lie between fixed maximum and minimum values at each time t. Systems such as those just described are called control systems. The principal problems associated with control systems are those of controllability, observability, stabilizability, and optimal control. The problem of controllability is the following. Given that the system is initially in state a1, a2, . . . , an, can the controls u1(t), . . . , uk(t) be chosen so that the system will reach a preassigned state b1, . . . , bn in finite time? The observability problem is to obtain information about the state of the system at some time t when one cannot measure the state itself, but only a function of the state. The stabilizability problem is to choose control variables u1(t), . . . , uk(t) at each instant of time t so that the state x1(t), . . . , xn(t) of the system gets closer and closer to a preassigned state as the time of operation of the system gets larger and larger. Probably the most prominent problem of control theory is that of optimal control. Here, the problem is to choose the control variables so that the system attains a desired state and does so in a way that is optimal in the following sense. A numerical measure of performance is assigned to the operation of the system, and the control variables u1(t), . . . , uk(t) are to be chosen so that the desired state is attained and the value of the performance measure is made as small as possible. To illustrate what is meant, consider the chemical-reaction example as representing an industrial process that must produce specified concentrations c1 and c2 of the first two substances. Assume that this occurs at some time T, at which time the reaction is stopped. At time T, the other substances, which are by-products of the reaction, have concentrations x3(T), x4(T), . . . , xn(T). Some of these substances can be sold to produce revenue, while others must be disposed of at some cost. Thus the concentrations x3(T), . . . , xn(T) of the remaining substances contribute a cost to the system, which is the cost of disposal minus the revenue. This cost can be taken to be the measure of performance. The control problem in this special case is to choose the temperature and pressure at each instant of time so that the final concentrations c1 and c2 of the first two substances are attained at minimal cost. The control problem discussed here is often called deterministic, in contrast to stochastic control problems, in which the state of the system is influenced by random disturbances. The system, however, is to be controlled with objectives similar to those of deterministic systems.