We consider dynamical models with aftereffect described by functional-differential equations with fractional derivatives. These models encompass processes, in which the system state may change abruptly at certain points in time, which is interpreted as the result of impulse effects (shocks). The trajectories of such systems may have discontinuities at certain points in time, and between these points the behavior of system is described by differentiable functions, which satisfy the equation in the usual sense. We pose a general control problem for a given system. We formulate solvability conditions for this problem in the class of impulse controls, L-2-controls, and their hybrids. The proposed approach to studying systems with fractional derivatives is based on the systematic use of abstract functional-differential equation theory and offers certain advantages for studying systems and processes with aftereffects.

For a wide class of linear systems with aftereffect, the problem of attaining a given system of target values is considered under polyhedral constraints on the control. The aim of the control is set by a finite system of linear functionals ℓ_i , i=1,,N ; this is why the more precise term “ ℓ -attainability” is used in the paper. The general form of the functionals makes it possible to consider terminal, multipoint, and integral target conditions and their linear combinations as special cases. For the class of systems under consideration, the problem of ℓ -attainability is reduced to a variant of the moment problem. One of the features of this problem is the account of random perturbations in elements of the moment matrix. These perturbations result in the distortion of the lower and upper (with respect to inclusion) approximations of the ℓ -attainable set. To obtain a guaranteed result, special procedures are proposed, which allow one to build open-loop controls with the following properties. First, the implementation of such controls produces trajectories on which the objective functionals take reachable values. Second, the calculation of reachable values is accompanied by guaranteed estimates of the errors associated with perturbations of elements of the moment matrix. In this case, each coordinate of the vector of target values corresponds not only to an interval of feasible values but also to the corresponding probability density of their distribution. The latter property allows one to give probabilistic characteristics to the errors.

A linear control system with continuous and discrete times and discrete memory is considered. The model includes an uncertainty in the description of operators implementing control actions. This uncertainty is a consequence of random disturbances under the assumption of their uniform distribution over known intervals. With each implementation a corresponding trajectory arises from random perturbations, and in the aggregate - an ensemble of trajectories, for which a component-by-component probabilistic description is given in the form of a set of probability density functions parametrized by the current time. To construct these functions, the previously obtained representation of the Cauchy operator of the system under consideration is used. The proposed probabilistic description of perturbations for trajectory variables allows one to find their standard characteristics, including expectation and variance, as well as the entire possible range of values. The results are constructive in nature and allow for effective computer implementation. An illustrative example is given.

A class of linear functional differential systems with continuous and discrete times and discrete memory is considered. The paper gives an explicit description of a family of uniquely solvable linear boundary value problems as a neighborhood of a fixed uniquely solvable boundary value problem. The description is based on an explicit representation of the principal components to the general solution representation such as the fundamental matrix and the Cauchy operator. In the study of the problems outside the class under consideration, the systems with discrete memory can be employed as a model or approximating ones. This can be useful as applied to systems with aftereffect under studying rough properties that hold under small disturbances of the parameters.

УДК 517.929 MSC: 34K05, 34K30, 34K35, 93B03, 93C23 DOI: 10.21538/0134-4889-2021-27-3-141-151 Работа выполнена при финансовой поддержке РФФИ (проект 18-01-00332). Рассматривается линейная непрерывно-дискретная функционально-дифференциальная система управления с дискретной памятью. Цель управления задается с помощью конечного набора линейных функционалов, для которых

We consider dynamic models with an aftereffect in the form of functional differential equations with continuous and discrete time. We formulate a general control problem with respect to a given system of target functionals and a brief summary of known results on solvability of this problem under polyhedral point control constraints. In concluding section we present results on estimating the set of attainability under integral restrictions for the control. The proposed version of the synthesis of continuous and discrete systems is based on the systematic use of the theory abstract functional differential equation and has certain advantages in the study of systems and processes with aftereffect. Continuous-discrete functional-differential models allow us to take into consideration the aftereffects when modeling, including cases of complete memory, and effects arising when impulse perturbations (shocks) are taken into consideration and they are leading to jump changes in the phase state by components with continuous time.

Main constructions and relationships for program control actions are proposed as applied to the problem on attainable prescribed values of on-target functionals for continuous-discrete dynamic economic mathematical model with discrete memory under given polyhedral constrains with respect to control. The form of on-target functionals covers widely used kinds of functionals such as multipoint, integral ones and linear combinations of those. The feature of the control system under consideration is the presence of two kinds of the state variables, namely, a part of them depends on continues time, whereas others depend on discrete time. Aftereffect of the system is defined by its discrete memory located at a given collection of instants. The results are obtained on the basis of the principal statements from the general theory of continuous-discrete systems. In the constructive part of the research, the basic idea is the reduction of the original problem to a variant of the general moment problem with taking into account pointwise polyhedral constraints on controls. This allows us to construct estimates of the attainability set and to build program controls on the base of solutions to a series of linear programming problems. Every such a problem provides us with values of the program control on a partial segment. All these values are used while constructing the program control as a whole. The mentioned procedures use in essence the Cauchy operator to the hybrid system under consideration. The property of this operator are studied in the cited previous papers. The obtained results constitute an instrumental basis for efficient studying and constructing solutions to urgent applied problems with constrained resources of control.

For a wide class of linear functional differential systems with Volterra operators, a constructive technique is proposed to obtain estimates of linear functionals values over solutions in conditions of uncertainty of external perturbations. It can be applied to solutions of boundary value problems with arbitrary number of boundary conditions as well as to description of attainability sets in control problems with respect to given on-target functionals. External perturbations are constrained by a given linear inequalities system on the main time segment. The technique is based on the results of general theory of functional differential equations about the solvability of boundary value problems with general linear boundary conditions and the representation of solutions. The problem under consideration is reduced to the generalized moment problem. Therewith the results on the properties of the Cauchy matrix to systems with aftereffect are of essential importance. The general form of functionals allows one to cover many cases being topical in applications such as multipoint, integral ones, as well as hybrids of those.

In this paper, we consider a class of economic dynamics models in the form of linear functional differential systems with continuous and discrete times (hybrid models) that covers many kinds of dynamic models with aftereffect. The focus of attention is periodic boundary value problems with deviating argument, control problems with respect to general on-target vector-functional and questions of stability to solutions. For boundary value problems, some sharp sufficient conditions of the unique solvability are obtained. The attainability of on-target values is under study as applied to control problems with polyhedral constraints with respect to control, some estimates of the attainability set as well as estimates to a number of switch-points of programming control are presented. For a class of hybrid systems, a description of asymptotic properties of solutions is given.

A class of linear functional differential systems with continuous and discrete times and discrete memory is considered. An explicit representation of the principal components to the general solution representation such as the fundamental matrix and the Cauchy operator is derived. The obtained representation is given in terms of the system parameters and opens a way towards efficient studying general linear boundary value problems and control problems with respect to a fixed collection of linear on-target functionals. In the study of the problems mentioned above outside the class under consideration, the systems with discrete memory can be employed as model or approximating ones. This can be useful as applied to systems with aftereffect under studying rough properties that hold under small perturbations of the parameters.

A linear functional differential control system of general form with aftereffect is considered. An optimal control problem with linear constraints on the state and control variables is studied. The control is realized by a linear operator of general form. The cases of distributed and lumped delay in the control loop, as well as the case of impulsive control, are covered. The Cauchy matrix is used to reduce the problem under consideration to a problem formulated only in terms of control variables with the use of some auxiliary variables linked with the defining relations for the Cauchy matrix of the system. In the case where the control is chosen from a finite-dimensional subspace of the control space, a problem effectively solvable by standard software tools is written explicitly. An example of an applied optimal control problem that arises in economic dynamics is presented. A class of hybrid systems (systems with continuous and discrete times) reducible to the system under consideration is described.

The problem of description of attainability sets is considered as applied to a control problem for an economic mathematical model with respect to a family of on-target functionals under some constraints according to control actions. The functionals are given in a general form covering a great many widely used cases. Dynamics of the system under control is governed by equations connecting state variables of continuous and discrete times with taking into account aftereffects. Some constructions and algorithms are proposed which allow to obtain external polyhedral estimates of the attainability sets.

An optimal control problem is considered for the linear system with time delay of the general form. A sufficient and necessary condition of optimality is derived using the Cauchy matrix. The representation of an analog to the Hamilton-Pontryagin function is given as applied to the case of nonlocal control input operator.

A linear functional differential system with aftereffect of general form is considered. Basic relations that define the Cauchy matrix - the kernel of integral representation to solutions of the Cauchy problem - are presented. The role of the Cauchy matrix in the study of a wide range of problems in the theory of functional differential systems, including control problems with respect to a given system of objective functionals and boundary value problems with general boundary conditions, is indicated. The efficiency of solving these problems depends essentially on the possibility of constructing a sufficiently exact approximation to the Cauchy matrix of the system. We propose an approach to the approximate construction of the Cauchy matrix that combines iterative procedures and algorithms for the construction of a rather accurate initial approximation based on a special approximation of parameters of the system. Error estimates are established for the resulting approximations.

In this paper, a class of linear functional differential systems with aftereffect, continuous and discrete times, and impulses (impulse hybrid systems) is considered.The focus of attention is on the structure of the Cauchy operator to the hybrid system under consideration and the representation of their components.Those allow one to give the representation of all trajectories of the hybrid system and to formulate conditions of the solvability for control problems in various classes of controls, to obtain estimates of the attainability sets under constrained control, and to study general linear boundary value problems for the solvability.A detailed description of all components to the Cauchy operator is given and their properties are studied.For the components with continuous time, some conditions of the continuity with respect to the second argument are obtained which is related to deciding on a class of controls.The main results are based on constructions of the Cauchy matrices to systems with continuous time and difference systems.