Showing papers presented at "Constructive Nonsmooth Analysis and Related Topics in 2017"

Quality estimation of the geopolitical actor development strategy

Abstract: A problem of geopolitical actor strategy quality estimation (socio-economic, military-political etc.) with a view to increasing or decreasing the assets market value is formalized and solved in the paper. Aiming to solve this problem the random process of assets market value changes is introduced as random function X(t). The geopolitical actor assets market value changes are represented as a random process and analyzed with the help of this random process's characteristics. Making use of these characteristics the quality of the geopolitical actor strategy is estimated. An illustrative example is given.

Integrability of nonsmooth one-variable functions

Abstract: In the article the authors state and prove necessary and sufficient conditions for primitive one-variable functions to exist and the ways to find these functions.

Optimal impulse control of bi-virus SIR epidemics with application to heterogeneous Internet of Things

Abstract: With the emerging Internet of Things (IoT) technologies, malware spreading over increasingly connected networks becomes a new security concern. To capture the heterogeneous nature of the IoT networks, we propose a continuous-time Susceptible-Infected-Recovered (SIR) epidemic model with two types of malware for heterogeneous populations over a large network of devices. The malware control mechanism is to patch an optimal fraction of the infected nodes at discrete points in time, which leads to an impulse controller. We use the Pontryagin's minimum principle for impulsive systems to obtain an optimal structure of the controller and use numerical experiments to demonstrate the computation of the optimal control and the controlled dynamics.

An irrational behavior proof condition for multistage multicriteria games

Abstract: We use the payment schedule based approach to ensure stable cooperation in multistage games with vector payoffs. On the example of the Shapley value in multicriteria game it is shown that the irrational behavior proof condition and the balance condition may be incompatible. We design a recurrent payment schedule that satisfies such advantageous properties as the efficiency condition, non-negativity and irrational behavior proofness.

Differential game of oil market with moving informational horizon and non-transferable utility

Abstract: Non-transferable utility game of oil market is considered. Special approach for defining solution is used. This approach enables to construct a real time models of conflicting processes. Connection between the solution in the game with moving information horizon and solutions on the truncated time intervals is shown.

Properties of solutions of nonlinear equations of mechanics control systems

Abstract: Particularities of the application of the equations of motion in gravitational fields in nonlinear problems to problems, when equations or solutions contain nonsmooth functions, are discussed. Classical mechanics is engaged in the study of properties, approximation and prediction of motion for problems with regular functions. Solving nonlinear equations of dynamics uses additional transformations to eliminate singularities of equations for complex systems that lead to a linear or simplified form. This gives the possibility of solving with regard to the stages of successive approximation. The application of such transformations is considered for solving the problems of controlled motion of space vehicles in the gravitational field, taking into account active and reactive forces. After reduction to the canonical form or regular elements of nonlinear equations of dynamics we obtain the initial approximation. The control by the relay view determines the switching times at corner points based on the Pontryagin maximum principle. It is required to join successive sections of the trajectory.

Multi-objective optimization in ship weather routing

Abstract: Ship weather routing problem has a crucial importance especially for transoceanic voyages, where voyage time reaches several days and the distance is thousands of nautical miles. This paper presents an optimization method for multi-objective ship weather routing problem. The voyage time and voyage risk are both taking into account and the multi-objective evolutionary algorithm NSGA-II (Non-Dominated Sorting Genetic Algorithm II) is applied to obtain the Pareto optimal routes set. At the same time the users can select route bi assigning a weight coefficient to meet their requirements in the obtained routes set. This paper focuses on the mathematical model of multi-objective ship weather routing problem and how to use NSGA-II to solve it. Furthermore, the effectiveness of the algorithm has been proved by carrying out the numerical simulation experiment.

Metric problems for algebraic manifolds: Analytical approach

Abstract: The problem of distance evaluation from a point to an algebraic manifold in Rn is treated in the framework of elimination of variables procedure applied for the algebraic equation system generated by Lagrange's method. The resulting univariate distance equation possesses a zero set coinciding with that of critical values of the squared distance function. We discuss also the problem of nearest point coordinates evaluation. For the case of a quadric manifold, the distance approximation formulas are suggested in the form of analytical expressions with respect to the coefficients of the manifold.

Combined control synthesis algorithm

Abstract: The objects of the research are homogeneous bilinear control systems. The control synthesis problem of moving a bilinear system to the origin of coordinates in a finite time is considered. To solve this problem it is suggested to use an auxiliary linear system. It is known that the program control for the linear system that moves it to the origin can be represented in an equivalent form of positional control. In this case the closed system has a structure similar to the bilinear control system. Based on this property the combined control synthesis method in homogeneous bilinear systems is proposed.

Optimal impulsive control problems with hysteresis

Abstract: This paper deals with an optimal control problem for measure-driven differential equations with hysteresis. The hysteresis is modelled as a scalar rate independent variational inequality. First, we propose a concept of solution to the impulsive control system of our measure-driven differential equations with hysteresis. In general, these solutions belong to the class of functions of bounded variation. Then, we present some approximate results for the impulsive control system with hysteresis. Finally, we establish the existence of a solution to the optimal control problem we consider.

On the choosing problem of PID controller parameters for a quadrocopter

Abstract: At present, quadrocopters are widely used. Therefore, the problems of controlling quadrocopters in real operating conditions, including emergency modes, are of particular interest. This is due to various weather conditions, restrictions on the landing place (the presence of water surface and other obstacles). So the stabilization problem is considered in this paper. To stabilize the apparatus in three planes, it is proposed to use three PID controllers. The parameters choice in this case has features related to the basic technical and geometric characteristics of the quadrocopter. The modeling results of the real time stabilization are presented. Some recommendations of simplification of the empirical choosing of PID controller parameters depending on the possibilities of the flight controller are given.

Stochastic model of the universe matter

Abstract: We discuss the mathematical model of the structure and properties of matter in the Universe. The basis of all processes is the movement. The start can with the basic principles and laws of physics, statistical and thermodynamic equilibrium. We consider the collective movement of quasiparticle as wave function which satisfies the Schrodinger equation of quantum mechanics. The solution of this equation is in the form of a spherical wave function of the longitudinal wave, which can be represented in a specific form of soliton solutions. Weak interaction transition for proton into a neutron, a positron and a neutrino, one can imagine that it is happening under the influence of the Coulomb interaction constraints caused by the interposition of electrical helicity neutrinos. The strong interaction may be due to rearrangement of the electron by combining atoms. The substantiation of the basic laws of physics-based motion protoparticle small size and low weight. We obtain the probability parameters of the basic characteristics.

Approximation of convex fuzzy sets

Abstract: In this paper we introduce the concepts of fuzzy space, the distances between points and sets of this space. In addition, a concept of a polyhedral fuzzy set, a convex hull, and a fuzzy direction are proposed. It is proved that a polyhedral fuzzy set is a convex hull of a finite number of fuzzy points and directions. In accordance with the main result of this paper any convex closed fuzzy set with a continuous membership function can be approximated with any degree of accuracy by a fuzzy polytop.

Adaptive stochastic mirror descent for constrained optimization

Abstract: Mirror Descent (MD) is a well-known method of solving non-smooth convex optimization problems. This paper analyzes the stochastic variant of MD with adaptive stepsizes. Its convergence on average is shown to be faster than with the fixed stepsizes and optimal in terms of lower bounds.

On the coupled orbit-attitude control motion of a celestial body in the neighborhood of the collinear libration point L 1 of the Sun-Earth system

Abstract: This paper considers the motion of a celestial body (as a rigid body) within the restricted three-body problem of the Sun-Earth system. The equations of controlled coupled attitude-orbit motion in the neighborhood of collinear libration point L 1 are investigated. The translational orbital motion of a celestial body is described using Hill's equations of circular restricted three-body problem of the Sun-Earth system. Rotational orbital motion is described using Euler's dynamic equations and quaternion kinematic equation. The problems of celestial body motion stability in relative equilibrium positions and stabilization of a celestial body translational motion with proposed control law in collinear libration point L 1 are investigated. Results of numerical integration are presented graphically.

Irregular H ∞ -optimization of control laws for marine autopilots

Abstract: The problem of external disturbances rejection is considered for control systems of marine ships steered by autopilot under the action of sea wave disturbances and bias terms. The essence of the problem is to find mathematical models of adjustable corrective items for control law with a special structure, to provide desirable values of correspondent functionals, which characterize accuracy and intensity of a ship's rudders action. It is shown that minimax representation, supplied by H ∞ -approach, is quite suitable to reflect multipurpose orientation of control system tuning. Specialized methods of control laws design are proposed based on the original spectral approach to SISO LTI-optimization problem. Its applicability and effectiveness are illustrated by the practical example of autopilot synthesis.

Evolutionary behavior of taxpayers in the model of information dissemination

Abstract: We consider an impact of spreading information about future tax audits on strategies of a population of taxpayers. We suppose that each agent selects the best method of behavior, which depends on the behavior of other agents. Its assumed that all taxpayers pay taxes in accordance with their income, if they know that the probability of an audit is high. Though one share of agents can hide their true income and then such behavior provokes the tax audit. However total audit is very expensive, hence fiscal system should choose new instruments to force the tax collection. We formulate an evolutionary model with network structure which describes the changes in the population of taxpayers under the impact of information about tax audit.

Existence of strong nash equilibrium in repeated and multistage games

Abstract: In the paper two types of games are considered: the infinity repeated games and multistage games with prescribed duration. The notion of Partially Strong Nash Equilibrium (PSNE) is introduced and the existence of (PSNE) proved for the version of the game with specially defined payoffs along cooperative trajectory.

Solvability of nonlinear initial boundary value problem with distributed parameters in a netlike domain

Abstract: In this paper, we study the problem of the existence a weak solution nonlinear differential systems with distributed parameters in a netlike connected bounded domain. The results are the basis of the analysis the optimal control problems for differential equations with distributed parameters in a netlike domain with an interesting analogy with the multiphase fluid dynamics problems.

On the bijectivity of controls pairs and pairs of heat and mass transfer local parameters in the hypersonic flow stagnation point

Abstract: The hypersonic aircraft permeable surfaces (side surface of circular cylinder and spherical nozzle surface) effective heat protection mathematical modeling problems are considered. The possibility of unique controls restoration is established and proved (in the stagnation point). For one-dimensional case: the conditions of heat and mass transfer local parameters and boundary layer parameters monotonous dependence on controls are given. For two-dimensional case: the diffeomorphness conditions for controls pairs and pairs of heat and mass transfer local parameters are established. The computational experiments results are presented: the domains of allowed values “heat-friction” are obtained.

Binary cut-and-branch method for solving mixed integer programming problems

Abstract: The report is dedicated to the description of the algorithm for one of the methods for solving mixed-integer linear programming problems, which is based on binary cuttings. One of its algorithms, a hybrid algorithm based on binary cuts and branches, which combines the idea of the branch-and-bound method with the construction of cutting planes, is extended to milp problems.

Polyhedral methodology for conflict control of competing objects in pursuit conditions

Abstract: A polyhedral methodology for optimization of discrete time control processes is exploited. A problem of polyhedral discrete time dynamic game of pursuit is defined. A polyhedral strategy of pursuit based on the concept of a guaranteed blunder is proposed.

Demyanov-Rubinov subdifferentials of real-valued functions

Abstract: A new notion of subdifferentiability of real-valued functions called the Demyanov-Rubinov subdifferential is introduced. For convex functions this notion coincides with the classical subdifferential in the sense of convex analysis while for nonconvex functions it contains the Frechet subdifferential as a (possibly empty) subset. In comparison with the latter the Demyanov-Rubinov subdifferential is non-trivial for a much broader class of functions.

The one-dimensional and two-dimensional inverse problems of heat and mass transfer on hypersonic aircraft permeable surfaces

Abstract: The hypersonic aircraft permeable cylindrical and spherical surfaces effective heat protection mathematical modeling problems are considered. The statements of one- and two-dimensional inverse problems of heat and mass transfer are given. The interpolation and approximation statements of the inverse problems are defined. The computational experiments results are discussed: two types of blowing laws restoration by heat and mass transfer parameters (local heat flow and local friction) for the inverse problem interpolation statement are presented as the examples. The different character of sensitivity of controllable parameters respectively to step changing of control (the blowing) is studied.

Algorithms of the interior point method

Abstract: The results of development, theoretical justification and experimental research of interior point algorithms for solving linear programming problems are presented in the article.

Dynamic Shapley value in the game with spanning forest

Abstract: The dynamic Shapley Value for N-customers and M-suppliers in two-stage minimum cost spanning forest game is considered. Define the players' cooperative behavior. Selecting strategies, players build a minimum cost spanning forest at each stage. At second stage, there is a particular player x ∊ N may drop out of the game. The probability p of the player's leaving depends solely upon all players' behavior in first stage. Along the cooperative trajectory, compute characteristic function value of any coalition. Define the Shapley Value for two-stage and one stage games. According to definition of imputation distribution procedure (IDP), construct the dynamic Shapley Value in the game. An example is proposed.

On boundary conditions at infinity for infinite horizon control problem

Abstract: In this paper we consider necessary conditions of optimality for infinite-horizon optimal control problems with overtaking optimality and with weakly overtaking optimality as optimality criterions. We propose a boundary conditions on the co-state arc that are necessary for the optimality. We also show that, under additional assumptions on the payoff function's asymptotic behavior, the Pontryagin Maximum Principle with these conditions becomes a complete system of relations, and this boundary condition points out the unique co-state arc through a Cauchy-type formula. Examples are given to clarify the application of this formula as an explicit expression of the co-state arc.

On the superadditivity of a characteristic function in cooperative differential games with negative externalities

Abstract: In this paper, a 3-player cooperative differential game with negative externalities is considered. We assume constant constraints on controls. For this game, a delta-characteristic function is constructed. Numerical example of such delta-characteristic function is presented.

Non-smooth resource allocation problem

Abstract: This paper deals with one of the canonical problems in networked systems — the resource allocation problem, that is based on user equilibrium principle. The user equilibrium principle was formulated in the middle of XX-th century by John Wardrop. Resource allocation problem has widespread applications. Urban road networks, financial networks, power grids and other systems could be modeled as network equilibrium problems with convex performance functions. The present paper is focused on the resource allocation problem with convex non-smooth piecewise linear nondecreasing performance function. The constructive tool to cope with such problem was developed.

Construction of nonsmooth solutions in one class of velocity problems

Abstract: Algorithms for constructing the optimal result function are proposed for a planar velocity problem with a circular velocity vectogram and a nonconvex target set with smooth boundary. The algorithms work with the case where the solution of the problem has a complicated (segmented) structure of the singular set. Differentiable dependence is detected for smooth segments of the singular set, which makes it possible to consider and construct these segments as arcs of integral curves. An example of the velocity problem is considered, for which the optimal result function and its singular set are calculated numerically. A visualization of the results is implemented.