SCIENTIFIC INTERESTS AND LINES OF RESEARCH
Convex and applied nonlinear analysis, theory of set-valued mappings, Aumann’s integral, measurable structures, extremal problems
Mathematical theory of optimal control, differential inclusions, complete conflict controllability
- Game problems for discrete systems, Pontryagin’s direct methods, semigroup operators of Pshenicnyi
- Theory of search for moving objects, stochastic dynamic games
- Differential games of pursuit and evasion, nonlinear problems on moving objects collision avoidance
- Evolutionary game problems
- Game problems for systems of integral, integral-differential and differential-difference equations
- Dynamic games with groups of participants. Group pursuit, pursuit in turn (dynamic “commercial traveler” problem), game problems under state constraints
- Systems with impulse control, integral and mixed constraints on control
- Game problems for systems with fractional derivatives of Riemann – Liouville, Dzharbasyan – Nersesyan – Caputo, Miller – Ross, Hilfer, Gryunvald – Letnikov, Hadamard
- Investigation of the method of resolving functions, upper and lower resolving functions, matrix resolving functions, choice of extremal selections
- Positional methods in game problems, generalization of the Krasovskii extremal targeting rule, Pontryagin’s maximum principle
- Dynamic systems of variable structure, hybrid systems, failure of controlling devices, descriptor systems
- Game problems for systems with distributed parameters
- Methods of the theory of making decision, artificial intelligence, operation research and system analysis
- Systems of control by spacecrafts and systems of search for sea objects
- Exploration of functional human possibilities in extreme conditions
- Modeling of sport single combating, competition of firms
- Aircraft control systems, soft landing
- Protection of specially importand objects in conflict situation
MOST IMPORTANT FUNDAMENTAL RESULTS
- General method to study differential games is developed – the
method of resolving functions (MRF), based on using the inverse Minkovski functionals and special set-valued mappings constructed in accordance to the parameters of conflict-controlled process. MRF makes it feasible from the unified position to study a wide range of problems in condition of conflict and uncertainty. These are the problems with many participants and under state constraints, problems of the pursuit in turn (dynamic “commercial traveller” problems) in condition of conflict, control of various inertia objects and various types of constraints on dynamics. This method justifies the classic rule of parallel pursuit, well known to engineers engaged in design of rocket and airspace technology.
- MRF makes it possible to study game problems for the processes evolving according to relationships more complicated than ordinary differential equations. In particular, also studied are differential-difference, integral-differential games, as well as game problems for the Volterra and Fredholm integral equations and systems with fractional derivatives. The method is extended to the case of matrix resolving functions and the resolving functionals, connected to the case of equations with distributed parameters. Upper and lower resolving functions of various types are introduced, with their help sufficient condition of the game termination in class of quasi and stroboscopic strategies are obtained.
- The positional method of pursuit, associated with the time of first absorption, is developed which justifies pursuit along the ‘line of sight’. The Krasovskii rule of extremal targeting is extended to the case of the group pursuit, the cases with exchange and without exchange of information in the group are examined, leading to different kinds of regularity conditions. The cases of integral and mixed constraints on controls, delay of information, systems of variable structure, impulse controls are explored, with account of possibility for state constraints.
- It is laid a foundation of the nonlinear theory of collision avoidance. The analog of the Taylor formula for a solution of the nonlinear dynamic system is established that plays a key role in elaborating the methods. The methods of evasion along a direction and of variable directions, method of invariant subspaces and recursive method are suggested. The conditions of first and higher order are studied. The μ-problem of L.S. Pontryagin is solved. Sufficient conditions for escape a group of pursuers as well as conditions of evasion under the moving objects groups’ counteraction are derived. Also obtained are sufficient conditions of collision avoidance in the class of ε-strategies and ε-counter-strategies in the minmax or maxmin form. Comparison with the workaround method of L.S. Ponntryagin is made.
- A cell markovian model is proposed to analyze game problems with incomplete information (so called search problems) in which the probability of detection or the mean time of detection plays the role of performance criterion. The process of search is described by a bilinear system with a stochastic matrix standing for the control block. Solving the problem is reduced to evaluating minmax or multiple minmax of certain polynomial function of many variables. The discrete Pontryagin maximum principle and the Bellman method of dynamic programming are used as tools for optimization. The search on a line, in the region and search on summon, group search with exchange and without exchange of information and secret search by homogeneous and heterogeneous forces are studied.
APPLIED ELABORATIONS AND PRACTICAL APPLICATIONS
A number of the problem-oriented computer systems, modeling complexes and simulators were worked out related with control of various nature moving objects in conditions of conflict and uncertainty. In particular:
- Search for moving objects. To solve game problems with incomplete information (when only probabilistic distribution of the initial position is known) the computer system of search and tracking for moving objects for need of navy was developed on the basis of the cell model of search. The work refers to special subject-matter and is implemented in corresponding institutions. Search for submarines is performed by heterogeneous forces (aircrafts, helicopters, airborne ships). Note that the game cell model of search is connected with the discretization of the process both in state and time. Search on a finite set of possible states is defined by the transition law of probability distribution of the players with the transition stochastic matrix depending on their controls. Such process appears as bilinear and markovian. To study it the technique of finite markovian chains is used. The discrete maximum principle and the method of dynamic programming are applied for optimization of detection probability of the target and the mean detection time.
The cases of search in the region, on summon, on the line, by a group of objects, and interaction of objects’ groups are encompassed in the computer system; dependence of detection radius on the motion velocity is taken into account. The technique can be applied for the search of crashed objects in difficult of access regions, for search of fish shoals and sunk ships.
- Cosmic Research. The method of resolving functions and the positional methods are especially efficient in the analysis and modeling of moving objects groups’ interaction. On their basis, in cooperation with the airspace institutions, the modeling complex for the “star wars” program was created for optimization of interaction of controlled cosmos based objects groups, moving along circular or elliptic orbits. The method of decomposition is used that makes it possible to reduce the process of optimization to several simpler problems of group and in turn pursuit. First of them is based on situation of encirclement by Pshenychnyi and efficiency of its solving crucially depends on the players’ disposition and control resources. Pursuit in turn is a combination of the ‘commercial traveller’ type problem and the control problem which should be solved together. Computer realization on specific examples allows reducing the time of variants’ rundown. In so doing, control is constructed on the basis of parallel pursuit, substantiated by the method of resolving functions.
Certain ideas of groups’ interaction are used in modeling of aerial battle of groups of aircrafts.
- Aviation. Safe aircraft take off and landing is a problem of paramount importance, especially in extreme conditions (lateral wind, rain, covering with ice of the take-off and landing strip etc.) The algorithms are created and on their basis simulators for training pilots in order to minimize risk. The work is performed in cooperation with the State Research Institute of Aviation.
By suggestion of American colleagues, the game problem on soft landing (coincidence of geometric coordinates and velocities) is solved. It is realized on computer for the dynamic systems of second order under friction and simulates the process of aircraft landing on aircraft carrier. The ocean surface stays for the state constraint (aircraft can not dive) that essentially complicates the problem.
Several ways of soft landing are proposed, on the basis of combination of classic methods of the dynamic games and mathematical methods of optimal control. Simulation package, realizing the soft landing is developed. The work is performed in cooperation with NIST (National Institute of Standards and Technology, Gaithersburg, USA). Joint book is published. Chinese institutes – the 28-th Institute (Nankin) and the Politechnical Institute (Harbin ) take an interest in this elaboration.
- Collision avoidance. The original problem of Pontryagin – Mishchenko on avoidance of moving objects collision from arbitrary initial positions on semi-infinite interval of time is taken up in planning safe movement in airports and seaports by dispatcher offices. Methods of collision avoidance for nonlinear controlled systems are elaborated. The counteraction of groups of controlled objects is studied. In airports the flights schedules are making up with account of the airliners dynamics and the dispatcher should be ready to interfere in precarious situation, which threatens to become an accident. Analogous circumstances arise in the places of great amount of floating fund. Knowledge of the potentials of controlled crafts and water area, on the basis of preliminary computations, makes it possible to avoid collisions.
- Control of particle beams. One of the applications of mathematical theory of control of the systems with distributed parameters is control of charged particles on the basis of Vlasov and Fokker –Planck –Kolmogorov equations. It was realized in the frames o the project STCU with Kharkiv and Kiev physicists in cooperation with the Brookhaven National Laboratory (USA). The related software is presented in the report materials and mathematical part is published in the joint papers in international journals.
- Interception of targets. Rules of pursuit along the line of sight, parallel pursuit and pursuit along a ray are traditional engineer methods of mobile targets interception in conflict condition. They are theoretically substantiated on the basis of ideology of the method of resolving functions and extremal targeting. In so doing, the situations of group approach and state constraints are encompassed that makes it feasible to solve a number of model examples from the classic book of R. Isaacs. Because the guaranteed controls in examples are found in explicit form, the computer realization allows the process visualization.
A complex of software is created for interception of mobile targets in various situation of conflict counteraction, which are applied in performing the tasks dealing with special subject-matter.
- Medicine, physiology, sport. One of the civil applications of the methods of making decision is analysis and mathematical modeling of the human organism systems, functioning in extreme conditions of vital functions, in particular, in condition of highland and underwater. It is created software to model the processes of control and the dynamics correction with the goal of optimization with respect to the given criteria of functioning efficiency and stability.
Models and software are developed for analysis of functional state, development of fighter tactic and strategy in sport single battles.
This work is performed in cooperation with the scientists from the Bogomolets Institute of Physiology of NAS of Ukraine, the National University of Physical Education and Sport of Ukraine and the Institute of Applied Problems of Physics and Biophysics of NAS of Ukraine
