We consider a class of problems in which it is required to determine unknown constant coefficients appearing in the free terms of non-autonomous systems of ordinary differential equations. The basic and additional (overdetermining) conditions are, in the general case, of a nonlocal nature: they involve cumulative characteristics of the unknown state functions – both their values at selected points and their integral values over specified sub-intervals. Depending on the relation between the number of unknown coefficients and the number of additional conditions, several problem settings and corresponding solution methods are discussed. Results of computer experiments are presented, together with an analysis of how errors in the prescribed conditions affect the accuracy of the solutions for test problems.

The problem of optimal synthesizing the power control of point heat sources moving along given trajectories for heating a two-dimensional plate is investigated. The motion of sources is described by ordinary differential equations. The current values of the power and movement velocity of the sources are determined based on the plate temperature values measured by the sensing devices. The source power, their movement velocity, and the locations of the measurement points are optimized. Necessary optimality conditions are derived for the feedback parameters that define the current values of the source power and movement velocity based on the measured temperature values on the plate and the positions of the measuring sensors. These conditions include formulas for the components of the gradient of the objective functional with respect to the optimized parameters. Results from computer experiments obtained using first-order numerical optimization methods are presented.

We consider the problem of identifying the constant parameters involved in the right-hand sides of a linear nonautonomous system of differential equations with first-order ordinary derivatives. The specificity of the problem lies in the fact that additional conditions for identifying the unknown parameters, firstly, are nonlocal, and, secondly, include derivatives of an unknown function. The work examines the conditions for existence and uniqueness of a solution to the problem and proposes two different approaches to the numerical solution of the problem. The results of computer experiments are presented.

The authors study the problem of identifying constant parameters involved in the right-hand sides of a linear, nonautonomous system of differential equations with first-order ordinary derivatives. The specificity of the problem lies in the fact that additional conditions for identifying parameters, firstly, are nonlocal, and secondly, they include derivatives of an unknown function. The authors examine the conditions for the existence and uniqueness of a solution to the problem and propose two different approaches to the numerical solution of the problem. They also present the results of computer experiments.

The numerical solution of optimal control problems through second-order methods is examined in this paper. Controlled processes are described by a system of nonlinear ordinary differential equations. There are two specific characteristics of the class of control actions used. The first one is that controls are searched for in a given class of functions, which depend on unknown parameters to be found by minimizing an objective functional. The parameter values, in general, may be different at different time intervals. The second feature of the considered problem is that the boundaries of time intervals are also optimized with fixed values of the parameters of the control actions in each of the intervals. The special cases of the problem under study are relay control problems with optimized switching moments. In this work, formulas for the gradient and the Hessian matrix of the objective functional with respect to the optimized parameters are obtained. For this, the technique of fast differentiation is used. A comparison of numerical experiment results obtained with the use of first- and second-order optimization methods is presented.

In the paper, we investigate the optimal control problem for a linear system of ordinary differential equations with linear boundary conditions. The boundary conditions involve the values of the phase variable both at separate intermediate points and their integral values over individual intervals of the independent variable. In the problem, we determine the controls involved in the differential equations and the values of the right-hand sides of multipoint and integral boundary conditions. The necessary conditions for the existence and uniqueness of the solution to the boundary value problem, the convexity of the objective functional and the necessary optimality conditions for the optimizable parameters are obtained. The conditions contain constructive formulas of the gradient components of the functional. The numerical solution of an illustrative problem is considered.

Numerical methods are employed to address an inverse problem associated with a water distribution network characterized by a complex loopback structure. The problem is to ascertain the locations and magnitudes of leaks based on measurements of unsteady flow characteristics at certain points in the pipeline. Key aspects of the problem include the involvement of impulse functions within a system of hyperbolic differential equations, the lack of traditional initial conditions, and the specification of nonseparated boundary conditions between states at the endpoints of adjacent pipeline segments. The problem is transformed into a parametric optimal control problem, devoid of initial conditions but featuring nonseparated boundary conditions. The latter problem is tackled using first-order optimization methods. The paper presents the outcomes of numerical experiments. Notably, this research distinguishes itself from others by addressing the inverse problem of determining leak locations and magnitudes within an unsteady flow scenario in a water distribution network with a complex (loopback) structure, as opposed to studies focusing on steady flow or transient flow in simpler pipeline configurations.

We investigate a system of linear ordinary differential equations containing point and integral loadings with nonlocal boundary conditions. Boundary conditions include integral and point values of the unknown function. An essential feature of the problem is that the kernels of the integral terms in the differential equations depend only on the integration variable. It is shown that similar problems arise during feedback control of objects with both lumped and distributed parameters during point and integral measurements of the current state for the controllable object. The problem statement considered in the paper generalizes a lot of previously studied problems regarding loaded differential equations with nonlocal boundary conditions. By introducing auxiliary parameters, we obtain necessary conditions for the existence and uniqueness of a solution to the problem under consideration. To solve the problem numerically, we propose to use a representation of the solution to the original problem, which includes four matrix functions that are solutions to four auxiliary Cauchy problems. Using solutions to the auxiliary problems in boundary conditions, we obtain the values of the unknown function at the loading points. This is enough to get the desired solution. The paper describes the application of the method using the example of solving a test model problem.

We consider the problem of determining the parameters of a system consisting of a large number of dynamic objects, connected in an arbitrary order only by boundary conditions. The state of each object is described by a system of ordinary linear differential equations. The identification problem of unknown parameters involved in the differential equations and non-local boundary conditions is formulated as an optimization problem, and first-order numerical optimization methods are proposed for its solution. To achieve this, the convexity and differentiability of the objective function of the problem are investigated, and formulas for its gradient components are obtained. The results of numerical experiments are presented on a test problem.

In the article, the conception of the student’s free self-study, self-test and assessment system is elaborated. The main principles of the functions of the system built on the basis of this concept are shown.

In this paper, we study the problem of synthesizing the optimal control of the power of point-wise heat sources for heating a two-dimensional plate. The current power values are determined depending on the plate temperature values measured by the measuring devices. At the same time, these devices move along predefined individual trajectories. The movement velocities of the devices depend on the results of the current measurements. Necessary optimality conditions are obtained for the feedback parameters determined by the current values of the source powers and the velocities of the measuring devices. The conditions contain formulas for the gradient components of the objective functional with respect to the optimized feedback parameters. The results of computer experiments obtained with the use of first-order numerical optimization methods are presented.

The problem of synthesis the control of the motion of lumped sources is studied on the example of the problem of control with feedback by moving heat sources when the rod is heated. The speeds of point sources are assigned depending on the state of the process at the measurement points. Formulas for the components of the functional gradient are obtained, which make it possible to use first-order optimization methods for the numerical solution of the problem.

The synthesis problem of control of objects with distributed parameters with feedback is investigated on the example of the process of heating a rod in a furnace. To generate the values of control actions, we propose to use their linear de- pendence on the values of the state at the measurement points, both at the current and previous moments of time. The unknown coefficients involved in this dependence of control on the measured state values are feedback parameters. They are deter- mined by minimizing the objective functional using numerical methods of first-order optimization. Formulas for the gradient of the objective functional with respect to the feedback parameters are obtained. The results of numerical experiments are presented.

The problem of synthesis of power control of point-wise heat sources of heating of a rod moving along the rod by given trajectories and optimization of the placements of temperature measurement points is considered. To form the current power values of each of the heat sources, it is proposed to use the formula of their linear dependence on the temperature of the rod at the measured points. In general, the original optimal control problem is reduced to finding a finite-dimensional vector of feedback parameters and coordinates of measurement points that optimize the given objective functional. Regarding the feedback parameters and the coordinates of the measurement points, the necessary conditions for the optimality of the functional of the problem are formulated, containing formulas for the components of the gradient of the objective functional. The obtained formulas make it possible to use effective numerical first-order optimization methods for solving the problem. The results of numerical experiments obtained on initial test data are presented, and the analysis of the results is carried out. In particular, the influence of the temperature measurement errors at the measurement points on the quality of process control, namely on the value of the objective functional is analyzed.

A numerical approach to the investigation of optimal control problems of oscillatory processes with boundary and intermediate concentrated (lumped) control actions has been proposed in this paper. The corresponding analytical formulas for the components of the target functional gradient with respect to control actions considered on the class of piecewise continuous functions are obtained. The results of numerical experiments on the examples of solving model problems of optimal control of oscillatory processes with boundary and intermediate concentrated controls are provided. These results illustrate the dependence of the minimum settling time of oscillatory processes on the number and locations of concentrated control actions, on the process parameters, on the resistance coefficient of the medium (dissipation factor), and other factors.

The problem of control synthesis for the heating process of a rod by lumped sources moving along the rod is studied. The problem of feedback control of moving heat sources during rod heating is considered. The speed of point-wise sources are assigned depending on the state of the processes at the measurement points. The formulas for the gradient components of the objective functional allowing for the numerical solution of the problem using of the first-order optimization methods are obtained.

In the work proposes an approach to solving the problem of synthesis of motion and power control of lumped sources with optimization of the locations of the points of the measurements. For specificity, the problem of feedback control of moving heat sources during rod heating is considered. The power and speed of point-wise sources are assigned depending on the state of the processes at the measurement points. The formulas for the gradient components of the objective functional, allowing for the numerical solution of the problem using of the first-order optimization methods are obtained.

The paper studies inverse source problems for a parabolic equation with nonlocal initial and boundary conditions. The specificity of these problems is that the identifiable coefficients depend on only space or time variable. A numerical method for solution to problems is proposed based on reducing the initial problems to the parametric inverse problems with respect to ordinary differential equations using the method of lines. Then, we propose a non-iterative method based on using a special representation of the solutions to the obtained problems. We prove the existence and uniqueness of their solutions, and substantiate the existence of special representation of the solutions to these problems. Computation schemes, formulae, and the results of numerical experiments on test problems are provided.

The problem of calculating flow regimes of transient processes in complex hydraulic networks with loops is considered in the chapter. Fluid flow in each linear segment of pipeline network is described by a system of two linear partial differential equations of the first order. Non-separated boundary conditions are satisfied at the nodes of the network. These conditions are determined by the first Kirchhoff’s law and by the continuity of flow. The scheme of numerical solution to the problem based on the application of grid method is suggested. The formulas analogous to the formulas of sweep method are derived. The obtained formulas are independent of the number of nodes, segments, and structure of the pipeline network. Numerical experiments are carried out with the use of the suggested approach, and the obtained results are analyzed.

This paper studies the problem of optimizing the right-hand sides of nonlocal conditions with respect to a system of linear differential equations. Nonseparated nonlocal conditions linearly depend on the point and integral values of the unknown function. In the problem, it is required to determine the optimal values of the right-hand sides of nonlocal conditions. The issues of existence and uniqueness of a solution to a system of differential equations with nonlocal conditions under consideration are investigated, the convexity of the objective functional is proved. The obtained necessary optimality conditions for the values of the right-hand side terms of the nonlocal conditions allow using first-order optimization methods for the numerical solution of the problem. Computational experiments on solving a test problem are given and the results obtained are analyzed.