Keywords: Optimal Control, Applications in Economics, Management and Environmental Science
Abstract: Infinite-horizon optimal control problems naturally arise in studying different models of optimal dynamic allocation of economic resources, in particular, in growth theory. Typically, the initial state is fixed and the terminal state (at infinity) is free in such problems, while the utility functional to be maximized is given by an improper integral on the time interval [0,∞). Although the state at infinity is not constrained the maximum principle for such problems may not hold in the normal form, and the standard transversality conditions at infinity may fail. Additional difficulties arise when the model involves a natural resource (renewable or not renewable) as an essential factor of production. In this case, typically, admissible controls are only bounded in an integral sense, which precludes the direct application of the standard existence results. The talk is devoted to some recent results in this field of optimal control and their applications in growth theory.

Keywords: Real-Time Control Problems
Abstract: We consider dynamic reconstruction (DR) problems for controlled dynamical systems linear in controls and nonlinear in state variables as inaccurate current information about real motions is known. A solution of this on-line inverse problem is obtained with the help of auxiliary problems of calculus of variations (CV) for integral discrepancy functionals. Key elements of the constructions are solutions of hamiltonian systems obtained with the help of optimality conditions for the CV problems.

An illustrating example is exposed.

Keywords: Numerical Methods for Optimization, Optimization Methods, Real-Time Control Problems
Abstract: Nonlinear Model Predictive Control (NMPC) is an advanced control technique that often relies on a computationally demanding optimization problem and numerical integration algorithms. This paper proposes and investigates a novel method with less computational effort to improve the efficiency of NMPC using a formulation based on discrete mechanics (DM). In contrast to classical NMPC formulations, the proposed method merges the two stages, for solving both the initial value problem (IVP) for prediction as well as the nonlinear programming problem (NLP) for optimization, into a single stage for solving an optimal boundary value problem (BVP) using NLP techniques. By exploiting the structural features of DM a symbolic solution set of the equations of motion are derived offline on each discretization node along the whole optimization horizon. Within an NLP, the optimal solution is efficiently obtained online under the consideration of the boundary constraints. As a benchmark, the widely used NMPC formulation based on direct multiple shooting (MS) method is served to assess the convergence and the excellent real-time performance of this method. The closed-loop performance is demonstrated by the swing-up of an unstable numerical experiment.

Keywords: Optimal Control, Optimization Methods, Real-Time Control Problems
Abstract: The paper presents an approach to design a control of nonlinear dynamical systems based on the extended linearization technique of a mathematical model of an object such that a given quadratic cost function is minimized. In this case, the coefficients of the nonlinear controller are determined by solving Riccati matrix equation with state dependent parameters. A realizability problem of the controller of this sort is computational complexity of the real-time solution. The proposed approach solves the problem by controller coefficients searching at each control time subinterval of the whole control time interval. The presented methodology is illustrated by designing chemotherapy administration for the cancer treatment. The mathematical model of the cancer growth includes interaction between tumor cells, healthy normal cells and activated immune cells. Numerical results show effectiveness of the solutions obtained.

Keywords: Real-Time Control Problems, Robust Control and Stabilization, Numerical Methods for Optimization
Abstract: In this work-in-progress paper we propose a method of stabilizing controller synthesis for unstable linear systems with constrained control using polyhedral Lyapunov functions. Our method is based on the Frank-Wolfe algorithm for quadratic optimization, which is suitable for implementation in embedded microprocessors. The method implicitly constructs an inner approximation of polyhedral controllable set for unstable subsystem of lower dimension and then converts its boundary into polyhedral Lyapunov function without explicit construction of the associated polytope.

Keywords: Differential Games, Optimal Control, Robust Control and Stabilization
Abstract: Differential game of two players with dynamics of the motion of the first player described by the second order equation and the second player controls the movement of the target point is considered. The coordinates of the target point become known at the current time. Conditions are proposed for the parameters of the game under which there is a control first player guaranteeing the end of the game in a finite time. The results of numerical calculations of controls and trajectories for the model parameters of the problem are presented.

Keywords: Robust Control and Stabilization, Stochastic Optimization, Optimization under Uncertainties Including the Theory of Noise Measurements
Abstract: In this paper, the design procedure of anisotropic suboptimal dynamic output regulator for linear discrete time--invariant (LDTI) system is considered. The mean anisotropy level of exogenous disturbance is bounded by the certain positive value. Additional constraints of exogenous disturbance are connected with the first and the second moments. The anisotropic suboptimal regulator design problem is formulated in terms of solvability of linear matrix inequalities (LMI) under convex constraints.

Keywords: Optimal Control, Applications in Economics, Management and Environmental Science, Robust Control and Stabilization
Abstract: A mathematical model of motion of a nonlinear controlled system describing the dynamics of a four-screw helicopter with a rotary device for engines is considered. The equations of motion of the model differ from similar models by the form of the equations describing the dynamics of the model. The problem of terminal control in presence of phase constraints is considered. To solve it, the method of dynamic linearization of the location of the control function is applied. The values ​​of the parameters under which the solution of the problem does not contain singularities are found and satisfies the imposed condition of the problem with constraints on the phase variables. For parameters satisfying these conditions numerically the terminal controls and corresponding trajectories are constructed.

Keywords: Robust Control and Stabilization
Abstract: Alternative approaches to define the anisotropic norm of signals are proposed. The paper is motivated by a definition available in the literature, which is based on Kullback-Leibler divergence and, thus, being non-symmetric, what, at least from a theoretical point of view, stimulates to consider some other possibilities in this field.

Keywords: Optimization Methods, Numerical Methods for Optimization, Robust Control and Stabilization
Abstract: The approach to synthesis and implementation of the controller of ITER plasma stabilization system is considered. The advantage of the presented approach is that the controllers of various dimensions are optimized. Software computes the controllers whose dimension and integral quality criterion are Pareto-optimal. It is described the algorithm to choose the proper controller to final implementation when equations of the plasma stabilization system have a high dimension. The presented integral quality criterion evaluates the upper bound of the ensemble of transient processes for some arbitrary plasma disturbances.

Keywords: Optimal Control, Optimization Methods, Numerical Methods for Optimization
Abstract: Optimization approach to effective radioactive waste management in advanced fuel cycle is proposed in the paper. This approach is based on the optimal control theory. The considered controlled system contains a system of ordinary differential equations, describing radioactive isotopes concentration change in time and a number of switching points in which the system state can be changed. The values of variables in the switching points are the optimizing parameters, subjected to determination. Moreover, because the recycling of nuclear waste is supposed to fulfill in subcritical reactor, the constraints on effective multiplication factor are also taken into account during numerical calculations.

Keywords: Optimal Control, Numerical Methods for Optimization, Optimization Methods
Abstract: We consider two-level systems of heterogeneous structure, in particular, discrete-continuous systems (DCS) for the case in which all homogeneous lower-level subsystems are not only connected by a common functional, but also have their own goal objectives. Based on modification of the previously obtained Krotov’s sufficient optimality conditions, we develop an improvement method that contains various proximity regulators of neighboring approximations at different levels. An illustrative example is provided.

Keywords: Optimal Control, Optimization Methods, Numerical Methods for Optimization
Abstract: In this paper we consider a discrete-time dynamical system consisting from two controllable objects. The dynamics of each object is described by the corresponding vector linear discrete-time recurrent relation. In this dynamical system there are two levels of control. The quality of process implementation at each level of the control system is estimated by the corresponding terminal linear functional. For the dynamical system under consideration, a mathematical formalization of a two-level hierarchical minimax adaptive control problem in the presence of perturbations, and an algorithm for its solving are proposed. The construction of this algorithm can be implemented as a finite sequence of solutions of a linear mathematical programming problems, and a finite discrete optimization problems.

Keywords: Stochastic Optimization, Robust Control and Stabilization, Optimal Control
Abstract: In this work, minimax methods of estimation for discrete time stochastic systems are considered. In the beginning we deal with one-stage systems in order to describe in details two methods of point estimation. After that these methods are transferred on multi-stage systems. Some special cases and examples are also examined.

Keywords: Optimal Control
Abstract: Three-dimensional attainability set ``at~instant'' for a non-linear controlled system is studied, which is often called the ``Dubins car''. A controlled object moves in the plane. It has linear velocity of a constant magnitude and bounded turn radius. The case is explored when the object can turn to one side only. Moreover, a rectilinear motion is prohibited by the constraints onto the control. We prove that the sections of the attainability set by planes orthogonal to the angle coordinate are convex. The geometric structure of these sections is analyzed.

Keywords: Optimal Control
Abstract: We consider a nonlinear affine-control system with constraints given by a level set of an integral functional of the control and the state variables. The convexity of reachable sets for the case when this functional is the sum of a convex quadratic functional of the control and a small functional of the state is studied. We also consider an autonomous control system on a small time interval and prove the convexity of reachable sets assuming appropriate asymptotics for controllability Gramian. A procedure for calculating the reachable sets is described and results of numerical simulations are presented.

Keywords: Optimal Control, Control Design for Hybrid Systems
Abstract: The paper is concerned with a nonlinear impulsive control system with trajectories of bounded variation. Necessary conditions of optimality in a form of the Maximum Principle are derived for a class of infinite horizon impulsive optimal control problems. For the overtaking optimality criterion under the assumption that all gradients of the payoff function are bounded, we construct a transversality condition for the adjoint variable in terms of limit points of the gradient of the payoff function. In the case when this limit point is unique, this condition supplements the system of the Maximum Principle and determines a unique solution of the adjoint system. This solution can be written explicitly with the use of the (Cauchy type) formula proposed earlier by S. M. Aseev and A. V. Kryazhimskii. The key idea of the proof is the application of the convergence of subdifferentials within Halkin's scheme.

Keywords: Differential Games, Optimal Control, Optimization Methods
Abstract: A non-antagonistic positional differential two-person game is considered in which each of the two players, in addition to the usual normal (nor) type of behavior oriented toward maximizing own functional, can use other types of behavior. In particular, it is altruistic (alt), aggressive (agg) and paradoxical (par) types. It is assumed that in the course of the game players can switch their behavior from one type to another. In this game, each player along with the choice of positional strategy also chooses the indicator function dened over the whole time interval of the game and taking values in the set {nor; alt; agg; par}. Player's indicator function shows the dynamics for changing the type of behavior that this player adheres to. The concept of BT-solution for such game is introduced. Using players types of behavior other than normal, can lead to outcomes more preferable to them than in a game with only a normal type of behavior. An example of a game with the dynamics of simple motion in the plane and phase constraints illustrates the procedure for constructing BT-solutions.

Keywords: Differential Games
Abstract: In this paper we present application of i-smooth analysis to approach-evasion linear differential game with delay. The main goal is to show that according to the methodology of i-smooth analysis one can realize extremal shift procedure by the finite dimensional component of the system state.

Keywords: Optimal Control, Differential Games, Numerical Methods for Optimization
Abstract: The paper deals with a control problem for a dynamical system under disturbances. In addition to geometric constraints on the disturbance, it is supposed that all disturbance realizations belong to some unknown L1-compact set. The control is aimed at minimization of a given quality index. Within the game-theoretical approach, the problem of optimizing the guaranteed result is studied. For solving this problem, we use a control procedure with a guide. The paper is focused on the questions of stability of this control procedure with respect to informational and computational errors. The results are illustrated by numerical simulation.

Keywords: Optimization Methods, Optimization under Uncertainties Including the Theory of Noise Measurements, Differential Games
Abstract: A control system with an unknown constant parameter is considered on a finite time interval. The actual value of the parameter in this control system is unknown to the person controlling the system at the moment when the systems starts moving. Finding an unknown parameter is made by applying a trial control to the control system for a short period of time along with monitoring the corresponding change in the movement of the system. After finding the approximate determination of the unknown parameter we can construct resolving control in the usual way, but we must take into account the additional error associated with the process of approximate determination of the parameter. In this paper, we investigate the influence of the error of measuring phase variable on the accuracy of unknown parameter recovery.

Keywords: Optimal Control, Singularities in Optimization
Abstract: The given investigation is oriented to study of generalized elements (GE) for solving problems of attainability under constraints of asymptotic character. But, the development of this direction required a special study of the issues connected with structure of the GE themselves. In considered problems, GE are used for extension of the space of usual solutions or usual controls. This extension has an analogy with extension of topological spaces (TS). So, we use compactification procedures. It is important to know the possibilities for realization of the corresponding compactification of the initial solution space. This investigation is directed at this work. We consider topological constructions realized by ultrafilters of widely understood measurable spaces.

Keywords: Optimal Control, Optimization under Uncertainties Including the Theory of Noise Measurements, Numerical Methods for Optimization
Abstract: We consider bilinear discrete-time systems for the case when interval bounds on the coefficients of the system are imposed, additive input terms are restricted by integral constraints, and initial states are restricted by given sets. Several ways for constructing external parallelepiped-valued estimates of reachable sets of such systems are proposed. One of the above techniques is based on obtaining recurrence relations for reachable sets in the "extended" phase space and constructing corresponding estimates in the form of polytopes of some special shape.

Keywords: Optimal Control
Abstract: The paper is devoted to constructing admissible controls in a problem of optimal control by a nonlinear dynamic system under constraints on the current phase state. The dynamic system under consideration describes the controlled motion of a carrier rocket from the launching point to the time when the carrier rocket enters a given elliptic earth orbit. A problem consists in designing a program control for the carrier rocket that provides the maximal value of the payload mass led to the given orbit and the fulfillment of a number of additional restrictions on the current phase state of the dynamic system at the atmospheric part of the trajectory. The restrictions considered are due to the need to take into account the values of the dynamic velocity pressure, the attack angle and slip angle when the carrier moves in dense layers of the atmosphere. Such a problem is equivalent to a nonlinear time-optimal problem with phase constraints for carrier rockets of some classes. The algorithm for constructing admissible controls ensuring the fulfillment of additional phase constraints is suggested. The methodological basis of this algorithm is the application of some predictive control. This control is constructed in the problem without taking into account the constraints above. For a deterministic model of the atmosphere, such a predictive control is used to predict the values of a part of the phase state of the dynamic system at the next time. The prediction results are applied in the procedure of desired control construction. This procedure essentially takes into account specific features of the additional constraints. The results of numerical modeling are presented.

Keywords: Differential Games, Optimal Control
Abstract: We consider a dynamical system described by linear neutral-type functional-differential equations which is controlled under conditions of unknown disturbances. This system is approximated by a system of ordinary differential equations. An aiming procedure between the initial and approximating systems is elaborated. Using such procedure, results of the control theory for ordinary differential systems can be applied to control of the initial system.

Keywords: Differential Games, Numerical Methods for Optimization, Optimization under Uncertainties Including the Theory of Noise Measurements
Abstract: A dynamical object controlled under conditions of unknown disturbances or counter-actions is considered. The goal is to bring the object to given target points at given times regardless of the disturbance actions. To solve this problem, the quality index is introduced that evaluates the distance between the object and target points at the indicated times, and a zero-sum differential game is considered in which the control actions minimize this quality index while the disturbance actions maximize it. The initial problem is reduced to calculating the game value and constructing a saddle point of the game. The corresponding resolving procedure is proposed that is based on the upper convex hulls method. An example is considered. Results of numerical simulations are presented.

Keywords: Optimal Control, Differential Games
Abstract: The paper is devoted to construction of solutions in dynamic bimatrix games. In the model, the payoffs are presented by discounted integrals on the infinite time horizon. The dynamics of the game is subject to the system of the A.N. Kolmogorov type differential equations. The problem of construction of equilibrium trajectories is analyzed in the framework of the minimax approach proposed by N.N. Krasovskii and A.I. Subbotin in the differential games theory. The concept of dynamic Nash equilibrium developed by A.F. Kleimenov is applied to design the structure of the game solution. For obtaining constructive control strategies of players, the maximum principle of L.S. Pontryagin is used in conjunction with the generalized method of characteristics for Hamilton-Jacobi equations. The impact of the discount index is indicated for equilibrium strategies of the game and demand functions in the dynamic bimatrix game are constructed.

Keywords: Differential Games
Abstract: In this paper, we show an example of two-person pursuit-evasion game with infinite horizon where two kinds of capture turn out to differ. One of them, the infimum capture, is a standard kind of capture, while the limit capture, introduced in this paper, is its stronger variant. Both variants of capture have applications, but it is not known in advance whether they always exist simultaneously. The considered example is based on Lion and Man game, i.e. on the game with equal players' capabilities. The considered space is a compact geodesic space.

Keywords: Differential Games, Optimal Control, Optimization under Uncertainties Including the Theory of Noise Measurements
Abstract: The hereditary selections of multi-functions play an important role in the theory of differential games in connection with the construction of resolving quasi-strategies. The existence of a non-anticipating selection of a non-anticipating multi-function is considered. In most cases important for applications, it is known that any non-anticipating multi-function with non-empty compact values has a non-anticipating selection. Namely, the result is valid when the non-anticipation property is defined by a totally ordered family in the domain of "time" variable. In this note, we show that the condition is essential: when the family is not totally ordered, there exists a hereditary multi-function with non-empty compact values that has no non-anticipating selections.