Aleksei Fedorovich Filippov (; 29 September 1923 – 10 October 2006) was a Russian mathematician who worked on differential equations,

differential inclusion In mathematics, differential inclusions are a generalization of the concept of ordinary differential equation of the form :\frac(t)\in F(t,x(t)), where ''F'' is a multivalued map, i.e. ''F''(''t'', ''x'') is a ''set'' rather than a single point ...

s, diffraction theory and

numerical methods Numerical analysis is the study of algorithms that use numerical approximation (as opposed to symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics). It is the study of numerical methods t ...

. Born in

Moscow Moscow is the Capital city, capital and List of cities and towns in Russia by population, largest city of Russia, standing on the Moskva (river), Moskva River in Central Russia. It has a population estimated at over 13 million residents with ...

in 1923, Filippov served in the

Red Army The Workers' and Peasants' Red Army, often shortened to the Red Army, was the army and air force of the Russian Soviet Republic and, from 1922, the Soviet Union. The army was established in January 1918 by a decree of the Council of People ...

during the

Second World War World War II or the Second World War (1 September 1939 – 2 September 1945) was a World war, global conflict between two coalitions: the Allies of World War II, Allies and the Axis powers. World War II by country, Nearly all of the wo ...

, then attended

Moscow State University Moscow State University (MSU), officially M. V. Lomonosov Moscow State University,. is a public university, public research university in Moscow, Russia. The university includes 15 research institutes, 43 faculties, more than 300 departments, a ...

( Faculty of Mechanics and Mathematics). After graduating in 1950, he remained to work at the school. He got his Ph.D. under the supervision of I. G. Petrovsky, and became a professor in 1978. He taught until his death in 2006. Filippov showed interest in continuous loops in 1950 when he constructed a proof that they divide a plane into interior and exterior parts. Known as the

Jordan curve theorem In topology, the Jordan curve theorem (JCT), formulated by Camille Jordan in 1887, asserts that every ''Jordan curve'' (a plane simple closed curve) divides the plane into an "interior" region Boundary (topology), bounded by the curve (not to be ...

, it exemplifies a mathematical proposition easily stated but difficult to prove. In 1955 Filippov and V. S. Ryaben'kii became interested in

difference equation In mathematics, a recurrence relation is an equation according to which the nth term of a sequence of numbers is equal to some combination of the previous terms. Often, only k previous terms of the sequence appear in the equation, for a parameter ...

s and wrote ''On the Stability of Difference Equations''. The work was developed into a

textbook A textbook is a book containing a comprehensive compilation of content in a branch of study with the intention of explaining it. Textbooks are produced to meet the needs of educators, usually at educational institutions, but also of learners ( ...

in 1961 which was used in Moscow State University and many other Russian universities for several decades. In 1959 he published a paper containing a lemma about

implicit function In mathematics, an implicit equation is a relation of the form R(x_1, \dots, x_n) = 0, where is a function of several variables (often a polynomial). For example, the implicit equation of the unit circle is x^2 + y^2 - 1 = 0. An implicit func ...

s designed for use in

optimal control theory Mathematical optimization (alternatively spelled ''optimisation'') or mathematical programming is the selection of a best element, with regard to some criteria, from some set of available alternatives. It is generally divided into two subfiel ...

that is named after him (''Filippov's lemma''). Filippov made an important contribution in the theory of discontinuous ordinary differential equations with his monograph ''Differential Equations with Discontinuous Righthand Sides'' (1985).A.F. Filippov (1985
Differential Equations with Discontinuous Righthand Sides

Kluwer Academic Publishers Springer Science+Business Media, commonly known as Springer, is a German multinational publishing company of books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing. Originally founded in 1842 in ...

, link from

Google Books Google Books (previously known as Google Book Search, Google Print, and by its code-name Project Ocean) is a service from Google that searches the full text of books and magazines that Google has scanned, converted to text using optical charac ...

, review by J. Jarník: Such set-valued dynamical systems arise in

sliding mode control In control systems, sliding mode control (SMC) is a nonlinear control method that alters the dynamic system, dynamics of a nonlinear system by applying a discontinuous control signal (or more rigorously, a set-valued control signal) that forces th ...

, an important class of feedback control systems demonstrating

robust control In control theory, robust control is an approach to controller design that explicitly deals with uncertainty. Robust control methods are designed to function properly provided that uncertain parameters or disturbances are found within some (typical ...

. Such systems model some mechanical systems with Coulomb friction, and more recently genetic networks. A. F. Filippov was awarded the

's Lomonosov Award in 1993.

References

External links

*
All-Russian Mathematical Portal

Russian Encyclopedia: Filippov, Aleksei Fedorovich
(in Russian) * Izobov, N. A.; et al. (1998) "Alekseĭ Fedorovich Filippov" (on the occasion of his 75th anniversary). ''Differential Equations'' 34, no. 10, 1297–1302. {{DEFAULTSORT:Filippov, Aleksei Fedorovich 20th-century Russian mathematicians Soviet mathematicians Moscow State University alumni Soviet military personnel of World War II 1923 births 2006 deaths