Lev Semyonovich Pontryagin

Russian mathematician
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Also known as: Lev Semenovich Pontriagin, Lev Semyonovich Pontrjagin
Quick Facts
Also spelled:
Lev Semenovich Pontriagin
Or:
Pontrjagin
Born:
September 3, 1908, Moscow
Died:
May 3, 1988, Moscow (aged 79)

Lev Semyonovich Pontryagin (born September 3, 1908, Moscow—died May 3, 1988, Moscow) was a Russian mathematician, noted for contributions to topology, algebra, and dynamical systems.

Pontryagin lost his eyesight as the result of an explosion when he was about 14 years old. His mother became his tutor, describing mathematical symbols as they appeared to her, since she did not know their meaning or names. Entering Moscow State University in 1925, he soon became friends with Pavel Aleksandrov, who was developing point-set and combinatorial topology. Under Aleksandrov’s influence, Pontryagin spent most of the 1930s and ’40s investigating topology; his papers were collected and published as Topological Groups, which has been translated into several languages. In 1931 he was one of five signers of the Declaration on the Reorganization of the Moscow Mathematical Society, in which the signers pledged themselves to work to bring the organization in line with the policies of the Communist Party. He served for many years as a department chair at Moscow State University and as editor-in-chief of the prestigious journal Matematicheskii Sbornik (published in English as Sbornik: Mathematics).

In 1934 Pontryagin garnered international attention with his partial solution of one of David Hilbert’s famous set of 23 problems, which had challenged mathematicians since 1900. About this time he began studying control theory, work that led to his fundamental monograph, Theory of Optimal Processes (1961; English translation 1962). In later years he wrote several other expository works on mathematics.

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Pontryagin was showered with honours by the Soviet government and its Academy of Sciences, including four Orders of Lenin, the Order of the October Revolution, and the Lobachevsky Prize.

This article was most recently revised and updated by Encyclopaedia Britannica.

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. At that time, 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.

As long as human culture has existed, control has meant some kind of power over the environment. For example, cuneiform fragments suggest that the control of irrigation systems in Mesopotamia was a well-developed art at least by the 20th century bc. There were some ingenious control devices in the Greco-Roman culture, the details of which have been preserved. Methods for the automatic operation of windmills go back at least to the European Middle Ages. Large-scale implementation of the idea of control, however, was impossible without a high level of technological sophistication, and the principles of modern control started evolving only in the 19th century, concurrently with the Industrial Revolution. A serious scientific study of this field began only after World War II.

Although control is sometimes equated with the notion of feedback control (which involves the transmission and return of information)—an isolated engineering invention, not a scientific discipline—modern usage favours a wider meaning for the term. For instance, control theory would include the control and regulation of machines, muscular coordination and metabolism in biological organisms, and design of prosthetic devices, as well as broad aspects of coordinated activity in the social sphere such as optimization of business operations, control of economic activity by government policies, and even control of political decisions by democratic processes. If physics is the science of understanding the physical environment, then control theory may be viewed as the science of modifying that environment, in the physical, biological, or even social sense.

Much more than even physics, control is a mathematically oriented science. Control principles are always expressed in mathematical form and are potentially applicable to any concrete situation. At the same time, it must be emphasized that success in the use of the abstract principles of control depends in roughly equal measure on basic scientific knowledge in the specific field of application, be it engineering, physics, astronomy, biology, medicine, econometrics, or any of the social sciences.

Examples of modern control systems

To clarify the critical distinction between control principles and their embodiment in a real machine or system, the following common examples of control may be helpful.

Machines that cannot function without (feedback) control

Many basic devices must be manufactured in such a way that their behaviour can be modified by means of some external control. Generally, the same effect cannot be brought about (in practice and sometimes even in theory) by any intrinsic modification of the characteristics of the device. For example, transistor amplifiers introduce intolerable distortion in sound systems when used alone, but properly modified by a feedback control system they can achieve any desired degree of fidelity. Another example involves powered flight. Early pioneers failed, not because of their ignorance of the laws of aerodynamics but because they did not realize the need for control and were unaware of the basic principles of stabilizing an inherently unstable device by means of control. Jet aircraft cannot be operated without automatic control to aid the pilot, and control is equally critical for helicopters. The accuracy of inertial navigation equipment cannot be improved indefinitely because of basic mechanical limitations, but these limitations can be reduced by several orders of magnitude by computer-directed statistical filtering, which is a variant of feedback control.

Control of machines

In many cases, the operation of a machine to perform a task can be directed by a human (manual control), but it may be much more convenient to connect the machine directly to the measuring instrument (automatic control); e.g., a thermostat may be used to turn on or off a refrigerator, oven, air-conditioning unit, or heating system. The dimming of automobile headlights, the setting of the diaphragm of a camera, and the correct exposure for colour prints may be accomplished automatically by connecting a photocell directly to the machine in question. Related examples are the remote control of position (servomechanisms) and speed control of motors (governors). It is emphasized that in such cases a machine could function by itself, but a more useful system is obtained by letting the measuring device communicate with the machine in either a feedforward or feedback fashion.

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