Named after V.F. Krotov

Vadim F. Krotov, Founder and first Head of Laboratory No. 45

The Laboratory’s main research area is optimal control processes with application to different-nature objects (technical, physical, biological, and economic). In particular, this area concerns the mathematical apparatus and methods of optimal control, including algorithms for the design and optimization of control strategies for complex-structure systems based on heterogeneous objects described by differential and difference equations under random factors, uncertainty, and counteraction.

The Laboratory was established in 1982. Vadim F. Krotov, Dr. Sci., Prof., Honored Worker of Science of the Russian Federation, a mathematician, a mechanical engineer, and an expert in mathematical control theory, headed the Laboratory from the very beginning until passing away in March 2015. Even in the first series of works, which made him widely recognized, Krotov elaborated a theory to study discontinuous solutions of variational calculus problems. Moreover, a new class of solutions (optimal sliding modes) was discovered within the theory.

In his further research, Krotov endeavored to solve completely the well-known optimal control problem posed by Lagrange: reduce constrained optimization to unconstrained optimization. And he succeeded! Krotov formulated the general extension principle as an abstract modification of the Lagrange principle. He adopted the former principle to derive very general sufficient conditions for the global optimality of controlled dynamic systems, a fundamental contribution to optimal control theory. The research area based on these conditions organically covers almost all aspects of optimal control theory: analytical methods for studying optimal processes, including the proof of their global optimality; numerical methods for finding optimal processes, including nonlocal methods; construction of minimizing (maximizing) sequences when the optimal process does not exist; investigation of abnormal and degenerate processes; construction of simple suboptimal solutions with the estimation of their degree of optimality; the design of optimal feedback control strategies and simple suboptimal ones with the estimation of their degree of optimality. The proposed approaches differ from other approaches known in the literature: they yield the absolute optimum, contain advanced techniques for finding analytical solutions, and have algorithmic simplicity. (In particular, there is no need to solve boundary value problems.)

Krotov’s interests were not limited to optimal control. He proposed a theory extending the Poincaré–Einstein general theory of relativity, the so-called general field theory. It combines the theory of gravitation with electromagnetic and quantum-mechanical phenomena.

In the development of Krotov’s ideas and methods, a lot of the credit must go to Dr. Sci. (Eng.), Prof. Vladimir I. Gurman. He worked in the Laboratory from 2014 to 2016 and studied the extension principle proposed by Krotov and its generalizations as well as degenerate solutions of optimal control problems (in more detail). Gurman obtained exact and approximate solutions (with the estimation of their degree of optimality) for many essentially nonlinear and important applied problems arising in aerospace engineering, robotics, physics, and economic ecology, confirming the high efficiency of Krotov’s methods.

Academician Stanislav N. Vassilyev worked in the Laboratory from 2007 to 2017 and significantly contributed to its scientific bundle. He proposed efficient algorithms for the reduction method without a priori assumptions on inter-model links; investigated multi-mode systems, including those with the switching of nonlinear modes; established conditions for dynamic stabilizability-type properties in complex combinations with state-space constraints and transient quality. In collaboration with other institutes and universities, he led the work on stability and control in heterogeneous and some other models of dynamic and intelligent systems. Abductive-deductive inference methods were developed to intellectualize computer systems for research, design, and control automation.

The studies of Mark A. Krasnoselsky and his student, Nikolay A. Bobylev, adjoin Laboratory’ research. Note that Krasnoselsky organized the Laboratory of Mathematical Methods for Complex Systems Analysis at the Institute. Krasnoselsky and Bobylev obtained several important results in different fields of nonlinear analysis, optimization, and control theory. For example, their kinship theorems became an effective tool for investigating nonlinear problems in oscillation theory. These theorems interconnect the topological characteristics of the zeros of different vector fields arising in a particular problem. The results have many applications: the convergence of approximate methods for constructing periodic oscillations of automatic control systems, the periodic oscillations of time-delayed systems, and estimation of the number of oscillating modes.

In 2015–2018, the Laboratory was headed by Mikhail M. Khrustalev, Dr. Sci. (Phys.–Math.), Prof., and continuator of Krotov’s research. He proposed and rigorously justified sufficient and necessary conditions for the global optimality of systems described by ordinary differential equations. They extend Krotov’s conditions, being applicable to problems with state-space constraints. Khrustalev adopted the apparatus of Krotov-type functions to establish necessary and sufficient conditions for the weak (terminal) invariance problem formulated by L.I. Rozonoer.

Since 2019, the Laboratory has been headed by Dr. Sci. (Phys.–Math.), Prof. Aram V. Arutyunov. His research interests include optimal control theory, theory of optimum problems, and nonlinear analysis, particularly abnormal optimum problems, optimal control problems with state-space mixed constraints, and problems with time delays. Arutyunov developed a mathematical apparatus for investigating such problems, including theorems on inverse functions at abnormal points and new results on covering maps. This apparatus proved to be an effective tool in optimal control theory and economic-mathematical modeling.

Many Arutyunov’s students are working in Laboratory No. 45. Sergey E. Zhukovsky, a young Doctor of Physics and Mathematics and a leading researcher, is an expert in functional analysis applications, including optimal control. His research interests include inverse problems and implicit functions.

Natalya G. Pavlova, a senior researcher and Cand. Sci. (Phys.–Math.), deals with the theory of impulse control and its applications to mathematical economics. She obtained results in modeling market processes. Her postgraduates Alexander M. Kotyukov and Stanislav O. Nikanorov are also involved in this research.

Elena P. Ushakova, a leading researcher and Dr. Sci. (Phys.–Math.), significantly contributes to the work of the Laboratory. Together with Vladimir D. Stepanov, she develops the mathematical apparatus for investigating optimal control problems. In particular, she studies the properties of functional spaces and the properties of integral operators in the spaces of multivariate functions.

Kirill A. Tsarkov, a senior researcher and Cand. Sci. (Phys.–Math.), develops the theory of control of stochastic dynamic systems. He established results on the terminal invariance of deterministic and stochastic systems.

The scientific results and methods obtained by Laboratory’s employees were included in monographs and textbooks on mathematics and engineering as well as in university courses. They are used in research institutes and design bureaus when studying applied problems and developing specific products. The investigation methods of optimal processes are employed to optimize the trajectories of moving objects, analyze and design their control systems, model and analyze the development of a multisectoral economy, etc.

An international pool of scientists, including dozens of candidates and doctors of science (and holders of foreign academic degrees), has formed around Laboratory’s research.

Current research areas of the Laboratory (traditional and novel):

  • The basic Krotov’s principles (the extension principle): further development and transfer to new classes of optimal control problems.
  • Methods to design control strategies for swarms of unmanned aerial vehicles based on classical mechanics and potential velocity fields.
  • Theoretical foundations of control automation for mechatronic moving objects based on the situational methodology and artificial intelligence technologies using ellipsoidal and polyhedral optimization of control processes under resource constraints in standard, conflict, and critical situations.
  • Design of optimal control strategies for systems admitting extended linearization.
  • Nonlinear analysis.
  • Applied research on the optimal control of aerospace, marine, and robotic objects, macroeconomic processes, biological and medical processes, etc.
  • Economic-mathematical modeling and applications.