Department of Differential Equations

Department of Differential Equations (Department of Ordinary Differential Equations before 2002) was created in 1959 by L. S. Pontryagin, who headed the department until his death in 1988.

Up to the early 1950s, L. S. Pontryagin was mainly focused on topology. However, later he decided to change his field of research and turned to ordinary differential equations and related problems (including those connected with control theory, which was called—and represented—automatic control theory at that time). To discuss these problems, in fall 1952, Pontryagin organized a seminar at the Steklov Institute (Pontryagin led this seminar together with his younger colleagues V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko). Later, the seminar gave rise to several other seminars while continuing as the main seminar of the department.

In the new area, L. S. Pontryagin had four fields of interest at different times:

• singular perturbation theory,
• stability of equilibrium states of relay systems,
• mathematical theory of optimal processes,
• differential games.

Of these investigations, the most renowned and recognized are Pontryagin's studies on optimal modes in control processes. Everything started with several mathematical models of specific problems that were presented by engineers at Pontryagin's seminar. Up to 1955, Pontryagin found a necessary optimality condition, now known as the Pontryagin maximum principle. For a wide class of linear systems, the maximum principle was proved by R. V. Gamkrelidze in 1957. In 1958, V. G. Boltyanskii proved it in the general case. And already in 1961, Pontryagin, Boltyanskii, Gamkrelidze, and Mishchenko wrote the monograph Mathematical Theory of Optimal Processes, for which the entire group received the country's highest scientific award, the Lenin Prize. Since then optimal control theory has grown into a separate field of research in theoretical and applied mathematics.

Among outstanding results obtained in the department, one should point out the following:

• L. S. Pontryagin and E. F. Mishchenko's studies on the theory of relaxation oscillations and differential games;
• Yu. S. Osipov and A. V. Kryazhimskiy's studies on differential games, inverse problems in dynamics, and optimization theory;
• D. V. Anosov's studies on the properties of geodesic flows on closed Riemann surfaces of negative curvature, on averaging in systems of differential equation with rapidly oscillating solutions, and on the behavior at infinity of the lifts (to the universal covering surfaces) of trajectories of flows on closed surfaces;
• Negative solution of the Riemann–Hilbert problem (of Hilbert's 21st problem for linear Fuchsian systems), which was obtained by A. A. Bolibruch in 1989, as well as Bolibruch's studies on isomonodromic deformations of Fuchsian systems;
• A. A. Agrachev and R. V. Gamkrelidze's studies on geometric theory of optimal control and related problems of differential geometry, calculus of variations, and Hamiltonian dynamics;
• S. M. Aseev's results on regularization of nonsmooth dynamic optimization problems, multivalued analysis, and mathematical theory of optimal economic growth problems.

In 1988–1997 the department was headed by R. V. Gamkrelidze, and in 1997–2014, by D. V. Anosov.

Since February 2014, the department is headed by S. M. Asseev.

At present the department is developing the following fields of research:

• dynamical systems, ergodicity, structural stability, asymptotics (D. V. Anosov, A. I. Bufetov, A. V. Klimenko);
• Differential and geometric methods in the theory of extremal problems (A. A. Agrachev, R. V.  Gamkrelidze);
• optimal economic growth problems (S. M. Aseev, K. O. Besov, A. V. Kryazhimskiy, M. S. Nikol'skii);
• optimal control, differential games, inverse problems of the dynamics of control systems (A. V. Kryazhimskiy, Yu. S. Osipov, M. S. Nikol'skii).

In 1962, the group headed by L. S. Pontryagin and consisting of V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko was awarded the Lenin Prize for the "Cycle of studies on ordinary differential equations and their applications to optimal control theory and oscillation theory in 1956–1961".

D. V. Anosov:

Moscow Mathematical Society award — 1965,

USSR State Prize — 1976,

Humboldt Foundation Research Award ­­­­— 1999,

A. M. Lyapunov Prize of the Russian Academy of Sciences — 2001

Yu. S. Osipov:

Lenin Prize — 1976,

State Prize of the Russian Federation — 1993,

L. Euler Gold Medal — 1997,

The Order For Merit to the Fatherland — 1999

A. A. Bolibruch:

A. M. Lyapunov Prize of the Russian Academy of Sciences — 1995,

State Prize of the Russian Federation — 2001

R. I. Grigorchuk:

Moscow Mathematical Society award — 1980

A. S. Mishchenko:

State Prize of the Russian Federation — 1996 (jointly with A. T. Fomenko),

Lomonosov Prize — 2001

K. O. Besov:

MAIK "Nauka/Interperiodika" Prize for the cycle of papers on the blow-up of solutions to quasilinear parabolic systems — 2005