Keywords: Differential Games
Abstract: The evasion problem is considered in which group of pursuers and group of evaders participate under the condition that the pursuers include participants whose capabilities do not yield place to the capabilities of evaders and participants with lesser capabilities. The goal of the group of pursuers is to "catch" all the evaders. The goal of the group of evaders is to prevent this from happening, i.e., to make it possible for at least one of the evaders to avoid an encounter. The pursuers and the evaders employ piecewise-program strategies. It is shown that if in a differential game avoidance of an encounter by at least one evader takes place over an innite time interval, then upon the addition of weak" pursuers avoidance will take place on any nite time interval.

Keywords: Differential Games
Abstract: The differential game under consideration belongs to the class of pursuit-evasion games in which pursuers are less than targets. Namely, the differential game of one pursuer against a coalition of two coherently dodging targets, one of which is false, is considered on the plane. The probabilities of targets classification are given. The pursuer has a simple motion. Each of the targets has a restriction on the minimum allowable turning radius. The main criterion is the mathematical expectation of the distance to the true target at a terminal point in time that is not fixed in advance and chosen by the pursuer during the pursuit process. The saddle point of the game in program and positional strategies was found. Illustrative examples are given.

Keywords: Optimal Control
Abstract: This paper concerns with optimal impulsive control problems with trajectories of bounded variation. Necessary optimality conditions based on weakly monotone solutions of the Hamilton--Jacobi inequality and feedback controls are discussed. A particular attention is paid to necessary optimality conditions with feedback controls, called Feedback minimum principle. The latter is generalized the corresponding principle for classical optimal control problems and is formulated in terms of Pontryagin Maximum Principle. An example illustrating these results is considered.

Keywords: Robust Control and Stabilization, Stochastic Optimization
Abstract: In this paper, a problem of robust anisotropy-based control with regional pole assignment for descriptor systems with norm-bounded parametric uncertainties is considered. The goal is to find a state-feedback control law, which guarantees desirable disturbance attenuation level from stochastic input with unknown covariance to controllable output of the closed-loop system, and ensures, that all finite eigenvalues of the closed-loop system belong to the given region inside the unit disk for all uncertainties from the given set.

Keywords: Optimal Control, Multi-Objective Control and Optimization, Real-Time Control Problems
Abstract: In this paper, we consider the differential inclusion with unbounded right-hand side and with local conditions called {it conditions of measurable pseudo-Lipschitz mappings}, which were proposed by the author. In these works, under the conditions of measurable pseudo-Lipschitz right-hand side of the differential inclusion many results were received, namely: existence theorem for solutions, theorem on the relaxation of differential inclusion, theorem on the differentiation of the initial data and other properties of trajectorys of differential inclusion with unbounded right-hand side. We consider a~differential inclusion with conditions that are very closed to that introduced by Clarke.

Necessary conditions incorporating the Euler--Lagrange inclusion are put forward. Our results weaken the hypotheses and strengthen the conclusions of the previously available works---they allow the admissible velocity sets to be unbounded and nonconvex under a certain pseudo-Lipschitz hypothesis.

The preset paper continues the study of the direct method, which was proposed by the author . This direct method is a development of the classical method of variations underlying the Pontryagin maximum principle for classical optimal control problems. The first steps of developing the Pontryagin's direct method to optimization problems involving differential inclusions were made in the works of B.Pshenichnyi (1980), V.Blagodatskikh (1986) and others. In this way we prove necessary optimality conditions incorporating the Euler--Lagrange inclusion with no use of the Clarke normal cone or the limiting normal cone.

Keywords: Multi-Objective Control and Optimization
Abstract: We propose an Aristotle's mechanics based method for synthesis of analytic control strategies for a flock of unmanned mobile robots. Synthesized strategies ensure bypass prohibited areas, non-collision of elements in the flock, preservation of the flock integrity, and fulfillment of the common goal.

Keywords: Multi-Objective Control and Optimization, Differential Games, Applications in Economics, Management and Environmental Science
Abstract: The new approaches to construct equilibria in dynamic multicriteria games with finite planning horizon are presented. We consider a dynamic, discrete-time game model where the players use a common resource and have different criteria to optimize. The multicriteria Nash equilibrium is obtained via the Nash bargaining design (Nash products), and the cooperative equilibrium is determined by the Nash bargaining procedure for the entire planning horizon. Furthermore, the suggested approach is applied to determine cooperative strategies and payoffs for any coalition to be formed. Dynamic multicriteria bioresource management problem with finite harvesting times is considered. The players' strategies and the size of the resource are compared under cooperative and noncooperative behavior.

Keywords: Optimal Control, Multi-Objective Control and Optimization, Robust Control and Stabilization
Abstract: In this paper, a novel optimal placement and parameter tuning approach for Power System Stabilizer (PSS) is proposed to damp both local and inter-area modes in mesh like power system. The design target is two-level PSS consisting of ΔP type local PSS and Δθ type Wide Area Damping Controller (WADC). For the PSS placement, modeshape with eigenvector sensitivity and coherency analysis are used to determine the appropriate placement of both local PSS and WADC. For the PSS parameter tuning, metaheuristics based approach via Mean Variance Mapping Optimization (MVMO) is employed in order to consider the time domain analysis. The proposed objective function designs at first the local PSS only to improve damping coefficient and damping ratio of eigenvalue. After that, it designs the WADC overlapping with already designed local PSS by time domain analysis, taking into account the transport delay in the remote signal between Phasor Measurement Unit (PMU) and PSS. The proposed method is applied in IEEE New England 39-Bus (NE 39-bus) system model. Then, modeshape and coherent grouping analysis give us understandable system aspect, the local PSS and the WADC optimized by using MVMO considering the transport delay improve both local and inter-area modes whereas a method ignoring the transport delay make system poorly damped.

Keywords: Stochastic Optimization, Applications in Economics, Management and Environmental Science, Robust Control and Stabilization
Abstract: A problem of the analysis and prevention of catastrophic shifts in stochastically forced ecosystems is considered. For the solution of this problem, a new mathematical approach based on the analysis and synthesis of the stochastic sensitivity of dynamic regimes in population models is suggested. Technical details of this approach are discussed for the conceptual stochastically forced Bazykin-Berezovskaya predator-prey model with the Allee effect. For this population model, a phenomenon of the noise-induced extinction is analysed by the method of confidence domains. By reducing these domains we provide a stabilization of the persistence regime for both interacting species.

Keywords: Stochastic Optimization, Applications in Economics, Management and Environmental Science, Robust Control and Stabilization
Abstract: We consider a model of thermochemical reactor proposed by Nowakowski. Stochastic effects in the bistability zone are studied. A parametric analysis of noise-induced transitions between coexisting equilibria is carried out on the basis of the stochastic sensitivity technique and confidence ellipses method. We solve the problem of stabilization of the equilibrium regime under incomplete information. The feedback regulator which reduces the stochastic sensitivity and stabilizes the randomly forced equilibrium is constructed.

Keywords: Optimization under Uncertainties Including the Theory of Noise Measurements, Stochastic Optimization
Abstract: In spite of the system identification, as (in accordance to L. Ljung (2010)) a science and art of constructing mathematical models via sample data, is a polyhedral process, selecting an identification criterion within an identification problem statement is a constituent part requiring both accounting its adequacy to the data available and practical suitability of implementation. The paper presents an approach to the identification of input/output mappings of stochastic systems in accordance to information-theoretic criteria that are derived by constructing a symmetric divergence measure based on Tsallis entropy of an arbitrary order. Meanwhile, a parameterized description of the system under study is utilized combined with a corresponding technique of estimation of the mutual information constructed by use of Tsallis entropy. This leads, finally, to a problem of the finite dimensional optimization to be solved by a suitable technique.

Keywords: Optimization under Uncertainties Including the Theory of Noise Measurements, Optimization Methods, Stochastic Optimization
Abstract: The solution of this problem is based on the combination of the possibilities of adaptive robust and guaranteeing techniques for signal processing. The proposed technique relies on the theory of multilevel optimization of stochastic systems on the basis of non-classical cost functions. This technique includes the following: the key parameter is formed for the first level of optimization, which characterizes the accuracy of estimation; the generalized parameter is formed, which characterizes the reliability of estimation; tolerances on the generalized parameter are specified; the likelihood function is formed, which includes the parameters of the first and second levels of optimization; using the maximum principle technique and the method of invariant imbedding, the two-point boundary-value problem is solved. The algorithms synthesized for adaptive robust data processing with guaranteeing tuning from the generalized parameter is shown.Results of the mathematical simulation are given.

Keywords: Large Scale Optimization Problems, Applications in Economics, Management and Environmental Science, Numerical Methods for Optimization
Abstract: We present a parallel parameter optimization algorithm for reproducing future projections of certain model outputs in dynamic general equilibrium models. The optimization problem is reduced to a nonlinear system of equations. The Jacobian matrix for a Newton-type solver in the problem is generated in parallel. The parameter optimization algorithm is implemented for parallel systems with distributed memory by using MPI. To achieve better performance of the parallel algorithm we use the parallel Fair-Taylor algorithm for computing an equilibrium path. Calculation of prices, input-output ratios and international trade for different time steps is carried out in parallel at each iteration of the method. The solution method is implemented for parallel systems with shared memory by using OpenMP. The effectiveness of the hybrid MPI+OpenMP parallel code for parameter optimization is demonstrated in the example of a global multi-sector energy economics model with scenarios that are used for studying climate change impacts on land use.

Keywords: Optimization under Uncertainties Including the Theory of Noise Measurements, Numerical Methods for Optimization, Optimization Methods
Abstract: This paper addresses the problem of numerically ecient computations for the auxiliary performance index (API) based linear time-variant stochastic models optimization. The main result of the paper is the new algorithm for numerically efficient (i.e. robust against round-off errors) array LD-computations for the API and its gradient.

Keywords: Real-Time Control Problems, Optimization under Uncertainties Including the Theory of Noise Measurements
Abstract: The problem of reconstructing unknown inputs in a quasi-linear stochastic system with diffusion depending on the phase state is investigated by means of the approach of the theory of dynamic inversion. The statement when the simultaneous reconstruction of disturbances in the deterministic and stochastic terms of the system is performed from the discrete information on a number of realizations of the stochastic process is considered. The problem is reduced to an inverse problem for ordinary differential equations describing the mathematical expectation and covariance matrix of the process. A finite-step software-oriented solving algorithm based on the method of auxiliary controlled models is proposed. The key result of the paper is an estimate for the convergence rate of the algorithm with respect to the number of measurable realizations.

Keywords: Optimization under Uncertainties Including the Theory of Noise Measurements, Optimal Control, Control Design for Hybrid Systems
Abstract: This article reviews the application of the least squares method for the processing of hybrid data. We formulated and solved the problem of fuzzy estimation of the parameters of a model with fuzzy basis functions, during the solution of which fully fuzzy systems of linear equations appear. In order to illustrate the solution of the problem by the inverse matrix method, two fuzzy basis functions are reviewed: fuzzy unit and fuzzy linear dependence.

Keywords: Differential Games, Applications in Economics, Management and Environmental Science, Multi-Objective Control and Optimization
Abstract: In this paper, we consider a coalitional partition of a set of players, N, in which each coalition S_isubset N , iin 1,...,m (S_icap S_i = emptyset), ineq j of players plays against the other coalitions in a non-zero sum cooperative differential game with prescribed duration and non-transferable payoffs. At the same time players within a coalition play a cooperative differential game with prescribed duration and transferrable utility. The solution concept for such type of differential games is proposed and its properties, namely time-consistency or dynamic stability, investigated.

Keywords: Robust Control and Stabilization, Control Design for Hybrid Systems, Multi-Objective Control and Optimization
Abstract: In this work, a new adaptive control scheme is presented for a class of bilateral teleoperation system with system uncertainties, jittering time delays, and state-constraint problem. A fixed-time control strategy with a novel proposed exponential-type Barrier Lyapunov Function (EBLF) is incorporated to achieve fixed-time convergence with state error constraints through the “adding a power integrator” technique and the full-order sliding mode method. During the sliding motion, the system behaves as a desirable full-order dynamics rather than a reduced-order dynamics. The singularity phenomena and chattering problem are avoided while fixed-time convergence without violation of full-state constraints is guaranteed. Simulation results further demonstrate the effectiveness of the proposed method.

Keywords: Multi-Objective Control and Optimization, Optimization Methods, Stochastic Optimization
Abstract: In control problems, it is often required to approximate sets by collections of congruent elements. One of the ways of this approximation is packing a collection of circles of equal radius into figures on a plane. This paper presents two variants of the problem of constructing optimal packing into ellipses of various shapes: in the first variant the number of elements is fixed and it is required to maximize their radius, in the second one the radius of circles is fixed and it is required to maximize their number. Iterative methods simulating the repulsion of centers of the circles from each other and from the boundary of the set are applied in the first variant. Constructions of the Chebyshev center, orthogonal projections and repulsion of points are used in these methods. Near-optimal packing with a hexagonal lattice are considered in the second variant. The software package for construction of packing for eclipses with various axial ratios has been developed.

Keywords: Stochastic Optimization, Applications in Economics, Management and Environmental Science, Multi-Objective Control and Optimization
Abstract: The assembly line is generally known as the last stage of the production processes. It constitutes the main production paradigm of the manufacturing industry. Thus, the performance of the assembly line problem has an important impact on the global performance of the entire production system. Among others, due to demand rate fluctuation. It’s important to quickly rebalance the assembly line and obtain an effective solution for ALB problem. For these reasons, this article proposes an adaptive generalized simulated annealing using fuzzy inference system to solve simple assembly line balancing problem of type I (SALBP-I). The objective of the problem is to minimize the number of stations for a predefined cycle time of workstations in an existing assembly line. Moreover, the performance of our approach is analyzed using a well-known data set of the SALBP-I.

Keywords: Stochastic Optimization, Singularities in Optimization, Robust Control and Stabilization
Abstract: We consider a problem of the construction of feedback regulator which synthesizes an assigned stochastic sensitivity of the equilibrium in stochastically forced nonlinear dynamic system. In the case of complete information, it is shown that this problem can be reduced to the solution of the matrix algebraic equation. A presence of noise in measurements deforms the stochastic sensitivity. We find conditions when such deformation is extremely large, and the considered problem is ill-posed. For this ill-posed problem, a regularization method is suggested. We propose an analytical approach which allows us to take into account a presence of noise in measurements when we construct an optimal feedback regulator. General theoretical results are illustrated by examples.

Keywords: Robust Control and Stabilization, Stochastic Optimization, Applications in Economics, Management and Environmental Science
Abstract: Discrete nonlinear stochastic systems with general parametric noises are considered. To approximate the dispersion of random states, we propose an asymptotic approach based on the stochastic sensitivity analysis. This approach is used for the solution of the stabilization problem for the discrete controlled systems forced by parametric noise. A theory of the synthesis of the stochastic sensitivity by the feedback regulators is elaborated. Regulators minimizing the stochastic sensitivity are used in the problem of the structural stabilization of equilibrium regimes in population dynamics. The efficiency of this technique is demonstrated on the example of the suppression of undesired noisy large-amplitude regular and chaotic oscillations in the Hassell population model.

Keywords: Control Design for Hybrid Systems, Stochastic Optimization, Applications in Economics, Management and Environmental Science
Abstract: This short discussion note is devoted to establishing some connections between the framework of dynamical systems driven by signals of a low regularity (rough differential equations) and the mathematical theory of impulsive control (measure-driven systems), which, as we believe, would essentially enrich one another. More precisely, we attempt to elaborate an impulsive (discontinuous) extension of the theory of rough differential equation for input-affine models with states of unbounded first variation. In the paper, we are focused on a concept of solution for impulsive rough differential equations with vector-valued controls of bounded p-variation, pin (1,2), and its constructive representation through a discrete-continuous Young's integral equation.

Keywords: Stochastic Optimization
Abstract: A polynomial filtering algorithm is proposed to solve a sea vehicle navigation problem using information on the range and radial velocity with respect to a fixed beacon, formulated in the context of the Bayesian approach. An example of practical implementation of the designed algorithm is considered. It is shown that the accuracy of the polynomial algorithm is close to that calculated by method of statistical tests with the use of a particle filter; in comparison, the Extended Kalman filter (EKF) yields a worse result.

Keywords: Optimal Control, Applications in Economics, Management and Environmental Science, Generalized Solutions of Hamilton-Jacobi Equations
Abstract: Properties of the value function are examined in an infinite horizon optimal control problem with an unlimited integrand index appearing in the quality functional with a discount factor. Optimal control problems of such type describe solutions in models of economic growth. Necessary and sufficient conditions are derived to ensure that the value function satisfies the infinitesimal stability properties. It is proved that value function coincides with the minimax solution of the Hamilton{Jacobi equation. Description of the growth asymptotic behavior for the value function is provided for the logarithmic, power and exponential quality functionals. An example is given to illustrate construction of the value function in economic growth models.

Keywords: Optimal Control
Abstract: The problem of maximization of the horizontal coordinate and minimization of the fuel expenditures of mass-point moving in the vertical plane driven by gravity, linear and quadratic viscous drag, and thrust is considered. The slope angle and the thrust are considered as a control variables. The problem is related to the modified brachistochrone problem. Principle maximum procedure allows to reduce the optimal control problem to the boundary value problem for a set of systems of two nonlinear differential equations. Thrust control depending on the velocity and slope angle is designed. Thus, the structure of optimal synthesis and the qualitative behavior of the optimal trajectories are investigated by means of methods of the theory of dynamical systems. The extremal synthesis of the thrust is designed. It is established that the extreme thrust control program consists either of single arc with intermediate thrust control, or two arcs, starting with maximum thrust and ending with the intermediate thrust or three arcs: ”intermediate-maximum-intermediate”.

Keywords: Optimal Control
Abstract: The algorithm for determining the optimal route for moving and controlling the mechanisms for reloading fuel assemblies of fast neutron reactors is proposed. It allows increasing the efficiency of the Nuclear Power Plant operation by reducing the stopping time for nuclear fuel transhipment.

Keywords: Real-Time Control Problems, Optimal Control, Numerical Methods for Optimization
Abstract: Increasing the density of aircraft traffic and complication of schemes of the air traffic control (ATC) create difficulties for the air traffic control operator to make ``by-hands'' decisions for organization of non-conflict motions and providing their optimality on some criteria. Under this, the operator needs fore-handed analysis of his possible decisions and recommendations (from the automated ATC System) for detecting and solving possible conflict situations (of dangerous closing or approach). The paper is devoted to elaboration of algorithms of using the procedures for straightening the aircraft flight-paths w.r.t. its previous flight plan trajectories. Possible induced conflict situations are detected and necessary recommendations for their exclusion are given.

Keywords: Large Scale Optimization Problems, Optimization under Uncertainties Including the Theory of Noise Measurements, Stochastic Optimization
Abstract: Due to significant advancements in embedded systems, sensor devices and wireless communication technology, sensor networks have been attracting widespread attention in areas such as target tracking, monitoring, and surveillance. Technological advancements made it possible to deploy a large number of inexpensive but technically advanced sensors to cover wide areas. However, when a tracking system has to track a large number of targets, the computation and communication loads arise. In this paper we compare two task assignment methods that might be used in the multiple target tracking problem. The first one is the brute force method and the second one is based on linear matrix inequalities. We provide performance and load testing results for these methods.

Keywords: Differential Games, Optimization under Uncertainties Including the Theory of Noise Measurements, Optimization Methods
Abstract: The paper is concerned with the mean field type differential game that describes the behavior of the large number of similar agents governed by the unique decision maker and influenced by disturbances. It is assumed that the decision maker wishes to bring the distribution of agents onto the target set in the space of probabilities within the state constraints. The solution of this problem is obtained based on the notions u- and v-stability first introduced for the finite dimensional differential games.

Keywords: Numerical Methods for Optimization, Optimization under Uncertainties Including the Theory of Noise Measurements
Abstract: The problem of object localization in multilateration systems is considered in the case when measurements (times of arrival, or TOA) of several consecutive signal transmissions are processed together. The suggested solution of the problem is based on minimization of the sum of residuals between TOA and their model. We proposed an effective numerical method for this optimization task, which accuracy is close to the Cramer-Rao lower bound for the corresponding observation equations. The results of work of the algorithm on simulated data with real locations of the receiving stations are presented.

Keywords: Robust Control and Stabilization, Optimization under Uncertainties Including the Theory of Noise Measurements, Real-Time Control Problems
Abstract: This paper is devoted to frequency estimation of a non-stationary sinusoidal signal. The amplitude is supposed to be a known function within a constant factor, the phase should be known. Example of such problem statement is sensorless angular velocity estimation for permanent magnet synchronous motors. On the first step by reparametrization, a third order linear regression model is obtained. On the next step, an estimation algorithm is constructed based on a standard gradient approach. The frequency estimate can be computed from one of the model parameters using inverse trigonometric functions. To improve estimates quality for noisy measurements we propose a new identification method, which can be tuned to attenuate the noise influence. It is shown that the frequency estimation error converges to zero exponentially fast. The described algorithm does not require measuring or calculating derivatives of the input signal. The efficiency of the proposed approach is demonstrated through the set of numerical simulations.

Keywords: Control of Partial Differential Equations
Abstract: Optimal control problems solvability are researched for infinite dimensional control systems, described by semilinear evolution equations in Banach spaces with degenerate linear operator at the Caputo fractional derivative. The pair of linear operators in the equation is relatively bounded and the nonlinear operator satisfies some smoothness conditions, in particular the condition of uniform Lipschitz continuity, and one of two types additional conditions: independence of degeneracy subspace elements or non-belonging of the operator image to the degeneracy subspace. The control system is endowed by the generalized Showalter - Sidorov initial conditions, which are natural for degenerate evolution equations. Optimal control have to belong to a convex closed set of admissible controls and to minimize a convex, bounded from below, lower semicontinuous and coercive cost functional. Solvability conditions are found for the optimal control problem of this class. If the existence of the initial problem solution with an admissible control is obvious, it is shown that the local Lipschitz continuity in phase variables that uniform with respect to time is sufficient for the optimal control existence. Abstract results are illustrated by optimal control problem for the equations system of the fractional viscoelastic Kelvin - Voigt fluid dynamics.

Keywords: Control of Partial Differential Equations
Abstract: The approximate controllability of a class of infinite dimensional control systems, described by the equation not solved with respect to the fractional Caputo derivative, is studied. Under the supposition of relative p-boundedness of the pair of operators in the equation the control system is reduced to two subsystems on mutually complement subspaces. One of subsystem is solved with respect to the fractional derivative, another subsystem has a nilpotent operator at the derivative. It is proved the equivalence of the approximate controllability of the original system and of every of two subsystems by the same control. This fact applied to the deriving of a criterion of the approximate controllability of the degenerate control system after research of every subsystem approximate controllability conditions. Application of the criterion is demonstrated on an example of a system described by partial differential equations.

Keywords: Optimal Control, Control of Partial Differential Equations, Singularities in Optimization
Abstract: The problem of occurrence of singular sets of solution of the velocity control problem for one class of dynamic problems on a plane with a nonconvex target set is studied. The theoretical apparatus is developed for determining pseudo-vertices of a target set in case when its boundary is a curve with a minimum order of smoothness. Finding pseudo-vertices is a necessary element of the procedure for constructing branches of the singular set of the optimal result function. Necessary conditions for the existence of pseudo-vertices, expressed in terms of one-sided partial limits of differential relations dependent on properties of local diffeomorphisms that determine these singular points, are obtained. Examples of constructing a nonsmooth solution to the velocity control problem are given. The developed procedures for determining the disturbance of smoothness of the solution to the dynamic control problem are also applicable when constructing generalized solutions of Hamilton-Jacobi type equations, as well as when forming a generalized eikonal in geometrical optics.

Keywords: Control of Partial Differential Equations, Optimal Control
Abstract: A boundary control problem for the one-dimensional wave equation is considered. The goal of the closed-loop control is to bring the system from the unknown initial state to the rest state. The main specificity of the problem is the assumption that the coefficients from Robin boundary conditions are unknown and must be indentified using additional boundary information on the controlled side of the space interval. In the paper a method for numerical solution of this problem is proposed. The method is a combination of the method developed previously for the case of known boundary coefficients and a special identification procedure based on smoothness analysis.

Keywords: Differential Games, Optimal Control
Abstract: Minimax filtering algorithm based on approximation of the information sets and reachability ones by parallelepipeds guaranteed to contain exact values of the measured parameters is considered. To calculate the parallelepipeds, approximating the reachability set of the high-speed aircraft, it is proposed to use the non-linear system of differential equations. The results of filtration are given for various unknown program controls of an unmanned aerial vehicle.

Keywords: Applications in Economics, Management and Environmental Science, Multi-Objective Control and Optimization, Optimal Control
Abstract: A hierarchical model of a large-scale system including a control Centre and a transport monopoly is considered. The mechanism of control of the transport monopoly is defined. The decisions of the game of monopoly with the control Centre are considered. The task of the optimal synthesis of the control mechanism is set. Sufficient conditions for the optimality of this mechanism are found when driving vehicles along a ring road with a prohibited overtaking (for example, by rail or a single-lane highway). The cases of the optimal balance of supply and demand, as well as scarce demand are considered. For the case of excessive demand, sufficient conditions for the optimality of the number of vehicles and the carrying capacity have been determined. Some conditions for the optimality of control mechanisms for road and rail transport have been found.

Keywords: Optimal Control, Generalized Solutions of Hamilton-Jacobi Equations
Abstract: For a time-dependent control system we consider a "reversed" minimum time problem, which consists in finding the minimum time needed by the system, whose state is initially located in a given set, to reach a given point. We show that the minimum time function constructed in this way is a unique viscosity solution of a static first order PDE, provided that, at every point of the extended phase space, admissible velocities form a convex set containing zero in the interior. We also describe a version of the Fast Marching Method (FMM) that effectively solves this PDE.

Keywords: Optimal Control, Singularities in Optimization, Optimization Methods
Abstract: On the linear control system, we consider an integral quadratic functional with a degenerate coefficient at the square of control. The problem is to find its minimum under a given initial value of the state variable and a free terminal value, which comes into the terminal part of the functional. Using a change of variable, the so-called Goh transformation, a passage to a new control and thus to an extended space of admissible controls is performed. Rewritten in the new variables, the functional may be now nondegenerate, i.e., may satisfy the strengthened Legendre condition with respect to the new control. Assuming the positive definiteness of the transformed functional for zero initial state value, we prove the existence of its minimum for any initial values of the state variable and then show that this minimum is a quadratic form of the initial value, the matrix of which satisfies the corresponding Riccati equation. It is also proved that the minimum of the transformed functional is equal to the infinum of the initial one, and a minimization sequence for the later one is constructed.

Keywords: Optimal Control, Real-Time Control Problems, Robust Control and Stabilization
Abstract: Background. Despite the wide use of controllers with proportional-integral-differential control law (PID controllers), the issue of expanding the limited functionality of not taking into account the trend of changing technological parameters is open. Methods. The control action is formed taking into account the predicted value of the technological parameter. Verification. The proposed method is verified for the reactor that processes the waste gases in sulfur production. Conclusions. Using a PID controller with a predictive component significantly improves the quality of control, reduces the maximum deviations of process parameters from specified values, contributing to additional energy saving and a significant increase in the efficiency of automated process units in the oil and gas industry and mechanical engineering.

Keywords: Optimization under Uncertainties Including the Theory of Noise Measurements
Abstract: The article deals with linear dynamical system state estimation problem under uncertainty when disturbance and measurement error statistics are unknown but the sets of their possible values are available. The approach to adaptive algorithm development of guaranteed estimation is proposed. The approach is based on the processing of innovation sequence values in the Kalman filter under conditions of a small number of available measurements. The Kalman filter implementation is performed for measurement data preprocessing the result of which is the mathematical model development and refining the estimates of unknown measurement errors.

Keywords: Optimization under Uncertainties Including the Theory of Noise Measurements, Numerical Methods for Optimization
Abstract: In this talk we discuss an approach for analysis of new class of optimal control problems under uncertainty. A particular case of such problem is the control system which dynamics is evolved in accordance with one of the possible scenarios of changing of parameters of the system. A bifurcating nature of these scenarios requires a use of non-anticipating (causal) control strategies or feedback controls.

We describe optimization theory of characterization of such optimal non-anticipating strategies. Assuming that the set of possible evolution scenarios is finite and is known along with probabilities of realization of such scenarios, we derive necessary conditions for optimal non-anticipating strategies in terms of some non-standard maximum principle. The main feature of this maximum principle is a new extended adjoint system which is significantly different from the one in classical Pontryagin maximum principle. We demonstrate that these new optimality conditions can be used for a construction of optimal discontinuous feedback control. Robustness properties of this feedback with respect to small measurement errors of state vector and small perturbations of the dynamics are also studied.

Keywords: Real-Time Control Problems, Optimal Control, Optimization under Uncertainties Including the Theory of Noise Measurements
Abstract: The problem of guaranteed closed-loop guidance at a given time is studied for a nonlinear dynamical control system. The initial state is unknown, but belongs to a given nite set of admissible initial states. The problem of package guidance is formulated for such system and a theorem of its solvability is proved by using the representation in the form of Takagi-Sugeno fuzzy model. An example illustrated the proposed technique by a specic nonlinear mechanical system is given.

Keywords: Optimal Control
Abstract: An optomechanical derotator is an optical system that can be used to extend various measurement methods so they can be applied to rotating measurement objects. The function principle is to manipulate the image of the rotating measurement object by means of rotating reflective components inside the derotator. Thus it appears stationary in the measurement data. For this it is inevitably that the angular position and velocity of the derotator are controlled in such way that they always amount to half that of the measurement object. It is therefore necessary to adjust the derotator with the help of an implemented controller. Since we want to acquire the reference input of the control with a high-speed camera and image processing algorithms the sampling time of the control must be adapted to the rather long computation time of the image processing. As a result, the use of analog control techniques is prohibited so that a digital controller has to be implemented. This paper proposes an approach where an image-based discrete-time linear-quadratic regulator is used to satisfy the requirements concerning the angular position and velocity of the derotator. To improve its performance, it is supplemented by integral action and a model-based feedforward control. Furthermore, the image processing algorithms by which the reference input of the control is determined are described. Finally, it is demonstrated that this controller concept produces the desired result by generating a standing image of a rotating object with a camera and the derotator.

Keywords: Differential Games, Optimal Control, Control of Partial Differential Equations
Abstract: The problem of controlling the process of heating the rod by changing the temperature at the left end of the rod is considered. The temperature at the right end of the rod is formed by a disturbance. The exact value of the heat density function is unknown, and only the boundaries of the range of its possible values are given. The goal of the control process is that at a fixed time the average value of the temperature of the rod belongs to the given interval. Necessary and sufficient conditions are found that the initial temperature of the rod must satisfy, so that the goal can be achieved for any admissible realizations of disturbance and uncertainty. The corresponding heating control of the left end of the rod is constructed.

Keywords: Numerical Methods for Optimization, Control of Partial Differential Equations
Abstract: In this paper, we introduce a numerical scheme for fractional differential equations with feedback control. Due to the possibility of dealing with a feedback control as a functional delay, we construct a numerical method based on Euler method accompanied with piecewise constant interpolation. The method is based on the idea of separating the current state and the prehistory function. The convergence of the method is stated and proved. Numerical experiments are given to clarify the good agreement between numerical and theoretical results.

Keywords: Numerical Methods for Optimization, Control of Partial Differential Equations, Optimization Methods
Abstract: For the Euler--Bernoulli plate equation the Dirichlet control problem is considered. To solve this problem numerically, a combination of classical regularization methods and a scheme of compact embedding of the spaces of optimal regularity into an auxiliary pair of spaces is used. Estimates of the uniform proximity of the exact and approximate operators in the new pair of spaces are obtained.

Keywords: Optimal Control, Control of Partial Differential Equations
Abstract: In this paper we consider the synthesis problem of optimal point control in the optimization of oscillatory processes in the case when the equation of the boundary value problem contains the Fredholm integral operator. The investigation is conducted according to the methodology of Professor A.I. Egorov developed by him on the basis of the Bellman scheme. Using the concepts of a generalized solution of the boundary value problem and the concept of the Fr'echet differential for the Bellman functional, we obtain the integro-differential Bellman---Egorov equation in partial derivatives. The structure of the solution of the Cauchy---Bellman---Egorov problem is found.

Keywords: Control of Partial Differential Equations, Optimal Control, Optimization Methods
Abstract: A survey of the results obtained in the theory of optimization of distributed systems by the method of Volterra functional-operator equations is given. Topics are considered: the conditions for preserving the global solvability of controllable initial-boundary value problems, optimality conditions, singular controlled systems in the sense of J.L. Lions, singular optimal controls, numerical optimization methods substantiation and others.