The authors propose a complex mathematical model for determining the functional reliability of flight crew members. The components of the model are the model of control of oxygen modes in the human body, the model of transport and mass exchange of respiratory gases in the human body, and the model of self-organization of the respiratory system and adaptation of the human body to extreme disturbances. The model is shown to allow (given a corresponding data array) one to determine if a particular individual can adapt to work under extreme conditions of professional activity.

The problem of approach of controlled objects with different inertia in dynamic game problems is considered. For such controlled objects, it is characteristic that the Pontryagin condition is not satisfied on a time interval. To solve the problem, a special multivalued mapping and a matrix function are introduced to equalize the players’ control resources, and then the additional resource is compensated by the body component of the cylindrical terminal set. With the use of the lower resolving function for objects with different inertia, two modified schemes of the first direct Pontryagin method are proposed, which guarantee the successful completion of the conflict-controlled process in the class of countercontrols. The upper resolving function is introduced, and the corresponding modified schemes of the method of resolving functions are presented for controlled objects with different inertia in the class of quasi-strategies and counter-controls. New theoretical results are illustrated by a model example.

A differential pursuit game in a stochastic descriptor linear system is analyzed. The dynamic of the system is described by Ito’s stochastic differential algebraic equation. Solutions of the equation are presented by the stochastic formula of the variation of constants in terms of the initial data and control unit. Constraints on the support functionals of two sets defined by the behaviors of the pursuer and evader are used to obtain the game completion conditions. The method of resolving functions is applied to construct pursuer’s control bringing the dynamic vector of the system to the terminal set. The results are illustrated by an example of a stochastic descriptor system that describes transients in a radio engineering filter with random disturbances in the form of white noise.

The problem of a guaranteed result in game problems of approach of controlled objects is considered. A method for solving such problems is proposed. It involves constructing some scalar functions that qualitatively characterize the course of approach of controlled objects and the efficiency of decisions. Such functions are called resolving functions. In contrast to the main scheme of the method, the case is considered where the classical Pontryagin condition does not hold. In this situation, instead of the Pontryagin selector, which does not exist, some shift functions are considered and special multivalued mappings are introduced with their help. They generate upper and lower resolving functions, which are used to formulate the sufficient conditions for the game completion in a certain guaranteed time. An example is given to illustrate the approach of controlled objects with a simple motion, in order to obtain upper and lower resolving functions in explicit form, which allows making a conclusion about the possibility of ending the game when the Pontryagin condition does not hold.

The problem of the approach of controlled objects with different inertia in dynamic game problems is considered. Modified sufficient conditions for ending the game in a finite guaranteed time when Pontryagin’s condition is not satisfied are formulated. Some shift functions are considered instead of the Pontryagin selector, and special multi-valued mappings are introduced with their help. They generate the upper and lower resolving functions of a special type, and based on them, two types of modified schemes are proposed: the scheme of Pontryagin’s first method and the method of resolving functions. This ensures the completion of the conflict-controlled process for objects with different inertia in the class of quasi-strategies and counter-controls. New theoretical results are illustrated by a model example.

The authors propose a new approach to generating control strategies in the problem of approach of conflict-controlled objects. Modifications of the first direct method are developed for the stroboscopic strategy in the class of counter-controls where the classical Pontryagin’s condition is not satisfied. The lower resolving function is considered, which plays a key role in formulating the results and can be determined in the general case using the Minkowski functional of a multivalued mapping. An upper resolving function is introduced, and a modified scheme of the method of resolving functions is proposed, which guarantees the termination of the conflict-controlled process in the class of quasi-strategies and counter-controls when the classical Pontryagin’s condition is not satisfied. The guaranteed times are compared for different schemes of the considered methods. The theoretical results are illustrated on a second-order model example with a special non-convex control domain of the pursuer.

This chapter is devoted to quasilinear conflict-controlled processes with a cylindrical terminal set. A specific feature is that, instead of a dynamical system, we start with representation of a solution in a form that allows one to include an additive term with the initial data and a control unit. This makes it possible to consider a broad spectrum of dynamic processes in a unified scheme. Our study is based on the method of resolving functions. We obtain sufficient conditions for the solvability of the pursuit problem at a certain quaranteed time in the class of strategies that use information on the behavior of the opponent in the past as well as in the class of stroboscopic strategies. We also find conditions under which information on the prehistory of the evader does not matter.

The authors propose a method for solving the problem of approach of controlled objects in dynamic game problems with a terminal payoff function. The method is reduced to the systematic use of the Fenchel–Moreau ideas on the general scheme of the method of resolving functions. The essence of the method is that the resolving function can be expressed in terms of the function conjugate to the payoff function and, using the involutivity of the connection operator for a convex closed function, it is possible to obtain a guaranteed estimate of the terminal value of the payoff function represented by the payoff value at the initial instant of time and integral of the resolving function. A feature of the method is the cumulative principle used in the current summation of the resolving function to assess the quality of the game before reaching a certain threshold. The notion of the upper and lower resolving functions of two types is introduced and sufficient conditions of a guaranteed result in the differential game with the terminal payoff function are obtained in the case where Pontryagin’s principle does not hold. Two schemes of the method of resolving functions with extremum strategies of approach of controlled objects are constructed and the guaranteed times are compared.

Introduction. The application of binary-reflected (mirror, reflexive) Gray codes for solving combinatorial problems with pseudo-Boolean functions (polynomials from Boolean variables) is considered. A recursive Ehrlich algorithm is given for generating a sequence of lines n-bit Gray codes, in which each subsequent line differs from the previous one by only one digit (bit). As an example of the effectiveness of the use of these codes, the solution of two combinatorial problems with Boolean variables with a complete enumeration of solutions is considered, and it is shown how these codes can be used to efficiently calculate the values of the objective function and constraints. The results of an experimental study are presented, which show that Gray codes can be practically applied in branching schemes, for example, in the branch and bound method, when the number of variables in the branching nodes of the decision algorithm does not exceed 35. Purpose. The purpose of the article is to show the developers of algorithms and programs how to apply Gray codes in various branching schemes of the decision algorithm, for example, in the branch and bound method, when the number of binary (Boolean) variables at the nodes of the tree is small (less than 35). The technique. The research methodology is based on a computational experiment for solving the 0-1 knapsack problem with the proposed algorithm of exhaustive search the solution with partial and full recalculation of the values of objective function and constraint of the problem. During the experiment, the accuracy of solving the problem by a “greedy” heuristic algorithm with time complexity O(n2) was also checked. Results. As a result of the experiment, it was found that the algorithm with a partial recalculation of the objective function and restrictions can be used for practical calculations in branching schemes, when the number of variables in the nodes of the branching tree does not exceed 35. The algorithm with partial recalculation is faster than the algorithm with full recalculation on average by 7 times. The heuristic "greedy" algorithm can be applied in practice to solve the 0-1 problem of a knapsack of large dimension (more than 10,000 items), when need to obtain an approximate value of the objective function at the limited computing resources. Scientific novelty and practical significance. The novelty of the work lies in the proposed approach to solving combinatorial optimization problems with pseudo-Boolean functions using Gray codes. The efficiency of the proposed algorithm with a partial recalculation of the values of the objective function and constraints is shown, and its can be applied in practice in various branching schemes of the decision algorithm. Keywords: Gray codes, combinatorial optimization problems, problem solving time.

We study the game problem of approach for a system whose dynamics is described by a stochastic differential equation in a Hilbert space. The main assumption on the equation is that the operator multiplying the system state generates a strongly continuous semigroup (a semigroup of class C-0). Solutions of the equation are represented by a stochastic variation of constants formula. Using constraints on the support functionals of sets defined by the behavior of the pursuer and the evader, we obtain conditions for the approach of the system state to a cylindrical terminal set. The results are illustrated with a model example of a simple motion in a Hilbert space with random perturbations. Applications to distributed systems described by stochastic partial differential equations are considered. By taking into account a random external influence, we consider the heat propagation process with controlled distributed heat sources and sinks.

We study the optimal control problem for a descriptor system whose evolution is described by Ito’s differential-algebraic equation. The quadratic cost functional is considered. The main constraint is that the characteristic matrix pencil corresponding to the equation is regular. We establish the conditions for the existence and uniqueness of the optimal control and the corresponding optimal state. The results are illustrated on an example of a descriptor system that describes transient states in a radio engineering filter with random perturbations in the form of white noise.

We establish the conditions to decompose a complex descriptor control system into simpler subsystems. The system state and input are described by equations not solved with respect to the derivative of the state. We consider two types of decompositions: sequential and parallel ones. The decomposition conditions are formulated in terms of the existence of invariant pairs of subspaces for operator pencils consisting of system coefficients. The results are illustrated by the example of a descriptor system that describes transient states in a radio-engineering filter. We perform the cascade–parallel decomposition of forth-order filter into the simplest first-order filters, each containing one inertial element.

A conflict-controlled process of the approach of a trajectory to a cylindrical terminal set is studied. The problem statement encompasses a wide range of quasilinear functional-differential systems. We use the technique of set-valued mappings and their selections to derive sufficient conditions for the game termination in a finite time. The methodology used is close to the scheme that involves the time of the first absorption. By way of illustration, quasilinear integro-differential games are examined. For this purpose, their solutions are presented in the form of an analog of the Cauchy formula. The calculations are performed for the case of a system with a simple matrix; the control sets of the players are balls centered at the origin and the terminal set is a linear subspace. Depending on the relations between the initial state of the system and the parameters of the process, sufficient conditions for the game termination are derived. An explicit form of the guaranteed time is found in one specific case.

This chapter suggests that a general scheme for investigation of conflict-controlled processes, illustrates on various types of functional-differential systems. Employment of the Extremal Targetting Rule, developed in, allows for the game termination in the "first absorption" time in the regular and regularized cases. Similar result was obtained by B. N. Pshenichnyi with the use of the convex analysis technique. On the one hand, this method is the result of extension of the Pontryagin Maximum Principle to the game problems. Positional conflict control by the systems of integral and integro-differential equations was studied in the papers of V. L. Pasikov, G.Ts. Chikrii and K. Volyanskij. Investigations of M. S. Gabrielyan and A. V. Kryazhimskii are devoted to the study of positional conflict counteraction of controlled objects groups. It should be noted that, the Extremal Targetting Rule, as applied to linear systems, is based on using the apparatus of support functions, the notion of Aumann integral of set-valued mapping and the Lyapunov theorem on vector measures.

Изучаются квазилинейные конфликтно-управляемые процессы общего вида на предмет сближения траекторий с заданным цилиндрическим множеством. В основу исследований положен метод верхних и нижних решающих функций. Основное внимание уделено ситуации, когда нет места условию Понтрягина, к тому же телесная часть терминального множества не является выпуклой. Предложена схема метода, которая позволяет в случае невыпуклости телесной части зафиксировать некоторую точку в ней, точку прицеливания, и реализовать процесс сближения. Получены достаточные условия для решения задачи сближения для разных классов стратегий. При этом использованы стробоскопические стратегии Хайека, определяющие управление М.М. Красовским. Процесс сближения состоит из двух этапов: активного и пассивного. На активном этапе накапливается верхняя разрешающая функция первого типа, а после момента переключения используется нижняя разрешающая функция второго типа. Эти функции дают возможность построить измерительное управление первого игрока на основе теорем об измеримом выборе, в частности теоремы Филиппова-Кастена. Полученные результаты для обобщенных квазилинейных процессов позволяют охватить широкий круг функционально-дифференциальных систем, систем с дробными и частными производными. Указаны возможности для развития предложенной методики.

April this year marks the 80-th birth anniversary of prominent Ukrainian scientist, specialist in the field of applied mathematics and cybernetics, Academician of NAS of Ukraine Boris Nikolaevich Pshenichnyi. B.N. Pshenichnyi graduated from mechanical and mathematical faculty of Ivan Franko Lvov University, for most part of his life he worked as the head of department at V.M. Glushkov Institute of Cybernetics and for his last years he worked in educational-scientific complex “Institute of applied systems analysis of NTUU “KPI” MON and NAS of Ukraine. It is worth noting that Kiev school of extreme problems is known worldwide for more than half a century, and one of its most powerful directions is connected with scientific activity of B.N. Pshenichnyi. Great are his contributions to the domestic science: they manifested in development of fundamental methods, proof of subtle mathematical results, creating the scientific school, organization of some investigation structures, training scientific personnel. The circle of scientist’s scientific interests is unusually wide. It includes problems of designing networks and graph theory, numerical optimization methods and mathematical theory of optimal control, convex analysis and necessary extremum conditions, theory of multivalued mappings and differential inclusions, methods of differential games and problems of seeking moving objects, models of economic dynamics, methods for constructing invariant sets of dynamic systems, minimax estimation of parameters, solution of variational inequalities, methods of laying geometric figures.