Vadim F. Krotov

Vadim F. Krotov


Sunday, February 14, 1932

Passed away: 

Wednesday, March 4, 2015

Vadim Krotov, in full Vadim Fedorovich Krotov, was an outstanding Soviet and Russian scientist, expert in optimal control theory and its applications, and Honored Man of Science of the Russian Federation.


Krotov was born on February 14, 1932, in Khabarovsk.

In 1956, he graduated from Bauman Moscow State Technical University (BMSTU); from 1956 to 1958, worked as a design engineer at the Central Research Institute of Heavy Machine Building. In 1958—1961, Krotov was a postgraduate at BMSTU and started studying the theory of optimal control. His first research paper (1960) was devoted to discontinuous solutions of variational problems. At that period, Krotov obtained sufficient optimality conditions for optimal control problems.

From 1961 to 1969, Krotov taught at Moscow Aviation Institute (the Department of Flight Dynamics and Control, headed by Professor I.V. Ostoslavsky). In 1967, he received a professorship there.

In 1962, Krotov defended his candidate’s dissertation in physics and mathematics, entitled New Methods of Variational Calculus and Some Applications, at Steklov Mathematical Institute (the USSR Academy of Sciences). In 1963, he defended his doctoral dissertation in engineering, entitled Some New Methods of Variational Calculus and Their Applications to Flight Dynamics, at Moscow Aviation Institute.

From 1968 to 1972, Krotov headed the Department of Higher Mathematics at Moscow Aviation Technological Institute (MATI).

In 1969, he published the monograph entitled Novye metody variatsionnogo ischisleniya v dinamike poleta (New Variational Methods in Flight Dynamics) with coauthors V.I. Gurman and V.Z. Bukreev. The book presented a new method of optimal control theory based on Krotov’s sufficient optimality conditions with applications to aircraft motion optimization.

At that time, Krotov organized regular inter-institutional seminars on optimal control at the Department of Higher Mathematics (MATI). Distinguished experts in mathematics and novice mathematicians, who became famous in subsequent years, made presentations at the seminars. The foundations of the theory of degenerate problems for unbounded differential inclusions and optimal control for discrete-continuous systems were laid (V.I. Gurman); new computational methods were developed (V.F. Krotov and V.I. Gurman); necessary and sufficient conditions for the weak invariance of control systems and sufficient conditions for absolute invariance (with respect to perturbations and initial conditions) were established (M.M. Khrustalev). Based on those theoretical results, major applied R&D works were carried out: optimization of spacecraft orientation maneuvers (V.I. Gurman, A.M. Nikulin, and I.V. Ioslovich), optimization of helicopter takeoffs through reducing the takeoff distance by 40—50% (V.I. Gurman and B.T. Chuklov), the analytical solution of the spacecraft landing problem in a turbulent atmosphere (M.M. Khrustalev), and others. Note that many foreign researchers studied this topic.

Krotov was Professor (1972—1996) and Head (1974—1982) of the Department of Economic Cybernetics at Moscow Economics and Statistics Institute (MESI).

Working jointly with economists (particularly from Central Economics and Mathematics Institute (TsEMI) and the All-Union Research Institute of Systems Study (VNIISI)), Krotov applied the theory of optimal control to the nonlinear models of multi-sectoral economic development based on V.V. Leontief’s inter-sectoral balance. Moreover, he obtained sufficient conditions for the optimality of macroeconomic processes described by a system of nonlinear input-output models. The corresponding results were published in Automation and Remote Control in 1982 and 1983. Under Krotov’s guidance, many monographs and textbooks were written and several projects were implemented in the field of macroeconomic processes optimization and simulation. Among them, note the development of the System of Interactive Optimization (SIO) and the Nature-Economy Simulation System (NESSY).

From 1982 until the end of his life, Krotov headed the Laboratory of Optimal Controlled Systems (Laboratory No. 45) at Trapeznikov Institute of Control Sciences (the Russian Academy of Sciences). Here and in other research teams, Krotov’s scientific results were further developed in different directions with applications to the problems of flight dynamics, automatic control, ecology, and physics.

For example, M.M. Khrustalev extended Krotov’s sufficient optimality conditions and obtained several new results. In particular, he proposed analogs of Krotov’s conditions for stochastic diffusion systems, including those operating under information constraints. Also, Khrustalev established necessary and sufficient optimality conditions, in the Bellman dynamic programming form and the Krotov form, for general optimal control problems, including those with state-space constraints. The problems of orbit stabilization and satellite orientation under multiplicative control perturbations and the motion of unmanned aircraft in a turbulent atmosphere were solved.

A.I. Moskalenko developed the theory of joint optimality, where the nonlinear transformations of differential equations act as Krotov functions. With these transformations, minimizing control sequences are found by solving the analogous problem for a simplified model. For example, several problems for distributed parameter systems were reduced to models with concentrated parameters. S.N. Vassilyev showed that sufficient conditions of joint optimality can be obtained by developing the Matrosov comparison method. Methodologically, the relationship between Moskalenko transformations and vector Lyapunov functions (VLF) is similar to that between Krotov functions and traditional Lyapunov ones.

Krotov’s outstanding role in science and education was marked by high government awards. He was entitled Honored Man of Science of the Russian Federation.

Krotov’s unique feature as a researcher was the ability to see the simple in the complex. No matter what field of science he studied, he saw the deep essence of the problem and obtained fundamental results.

Main research results

Krotov made major contributions to variational calculus and optimal control, their applications to flight dynamics, automatic control, applied physics, and computational methods for optimizing control processes. His sufficient optimality conditions and iterative computational method (known as the global improvement method) have gained well-deserved fame in the theory of optimal control.

Krotov functions, used in the sufficient optimality conditions, have the properties of Lyapunov functions and, generally, are nonlocal Lagrange multipliers for the complete elimination of constraints.

Besides, Krotov obtained several important results in the relativistic mechanics of elastic medium and the observational theory of dynamical systems in the problems of quantum mechanics.

Variational calculus and optimal control theory

In a series of papers (1960—1965), Krotov formalized the concept of a discontinuous solution of a variational calculus problem; within the proposed approach, he investigated discontinuous optimal sliding modes.

At that period, Krotov also formulated sufficient conditions for the optimality of controlled dynamic systems. Subsequently, the conditions were employed by him and other researchers to develop analytical and numerical design methods for optimal program and feedback control laws. The corresponding results were included in monographs and textbooks on mathematical and engineering disciplines as well as university courses.

Theory and methods for calculating aircraft control systems and trajectories

Krotov’s mathematical results were used for studying many applied scientific and technical problems (e.g., optimization of moving object trajectories and analysis and design of control systems for such objects). Among these problems, note optimal control of aircraft maneuvers in the Earth’s atmosphere based on program adjustments of the thrust and the angle of attack.

Theoretical physics

Krotov’s research interests also covered the correlation between the foundations of fundamental physical disciplines and their minimum general mathematical description. He proposed and justified the equations of the relativistic theory of elasticity.

His version of the general relativistic theory of the world (a generalization of the Poincaré—Einstein theory) combines relativistic mechanics, the gravitation theory, and the Maxwell electromagnetic field theory.

In several publications devoted to quantum mechanics, Krotov explored the range of problems from its statistical, dynamical, and geometrical foundations to mathematical control methods for the quantum state of matter.

The global improvement method in quantum mechanics

The applications-relevant design and optimization of control for the quantum state of matter are especially interesting. At present, new physical technologies based on the quantum state control of matter through an electromagnetic field impact are rapidly developing. Such nanotechnologies allow synthesizing new materials by physical means (instead of chemical ones) and separating isotopes as well as have applications to photochemistry and others. Mathematical algorithms for finding appropriate controls are an essential part of these technologies.

According to many physicists, optimal control theory methods are an adequate apparatus for developing such algorithms. The corresponding problems are described by the systems of nonlinear differential equations with orders of several thousand. Their solutions were investigated using Krotov’s sequential global improvement methods. After publication in the 1990s, these methods attracted the interest of physicists.

Krotov’s main monographs are as follows:

  1. Global Methods in Optimal Control Theory, New York: Marcel Dekker, 1996. — 408 p.;
  2. Osnovy teorii optimal’nogo upravleniya (Foundations of Optimal Control Theory), Moscow: Vysshaya Shkola, 1990. — 430 p. (coauthors B.A. Lagosha, S.M. Lobanov, et al.);
  3. Metody i zadachi optimal’nogo upravleniya (Methods and Problems of Optimal Control), Moscow: Nauka, 1973. — 448 p. (coauthor V.I. Gurman);
  4. Novye metody variatsionnogo ischisleniya v dinamike poleta (New Variational Calculus Methods in Flight Dynamics), Moscow: Mashinostroenie, 1969. — 287 p. (coauthors V.Z. Bukreev and V.I. Gurman).

They are presented in the Institute’s database:Кротов

Many Krotov’s papers can be found at Math-Net.Ru:

In addition, his publications are available at:

Browse the Institute’s database as well:Кротов+В.Ф.

Articles about V.F. Krotov

1. M. M. Khrustalev, On the 90th Anniversary of V.F. Krotov’s BirthAvtomat. i Telemekh., 2022, no. 5, 164—168.
2. The 80th Birthday of V. F. KrotovAvtomat. i Telemekh., 2012, no. 4, 162—163.

Also, see the Wikipedia page devoted to Krotov:Кротов,_Вадим_Фёдорович

Scopus Author ID: 7006670499