The solution of the optimal control synthesis problem based on the Bellman equation is considered. The Bellman optimal control problem allows us to obtain the control function as a function of the state space coordinates. The essential difficulty of this problem is that the function is unknown. To solve the problem, one of the symbolic regression methods, the network operator method, is used to search for the Bellman function. This approach enables an automatic search for the structure and parameters of the mathematical expression using a special genetic algorithm. The result of the solution is a mathematical expression for the Bellman function. The optimal control found includes the gradient of the Bellman function. This function is described by code in the form of a network operator matrix. Therefore, the gradient of Bellman function is calculated numerically. An example of solving the control synthesis problem via Bellman equation for a mobile robot subjected to disturbance of initial conditions is presented.

It is known that solving the classical problem of optimal control leads to obtaining a control function, as a function of time, which cannot be implemented directly in the control system of a real object, since the resulting control system is open-loop one. It is proposed to use the method of the extended model of the control object. Initially, a universal system for stabilizing the movement of an object along any trajectory in the space of states from a certain class is synthesized for the object model. This stabilization system is built into the control object. The original reference model of the control object is then added to the object with a free control vector on the right side. Thus, the extended object model includes an object model with a motion stabilization system and a reference model. The optimal control problem is solved for the extended model. In the synthesis of a universal stabilization system, machine learning by symbolic regression is used. An example of solving the problem of optimal control of a wheel robot with a differential drive is given.

The area monitoring problem is considered as optimization task. The problem belongs to the class of global optimization. First, the problem of finding optimal trajectories is solved so that the control objects, when moving along these trajectories, observe the entire area, do not collide with each other and do not violate phase constraints. One of the main quality criteria when searching for trajectories is their minimum total length, and the closeness of the trajectory lengths of each control object. Subsequently, the optimal control problem is solved in order to ensure the movement of the object along the obtained optimal trajectory. To solve the problem in class of implemented in real object control functions the stabilisation system of control object motion along a given trajectory is synthesized. When solving the optimal control problem, the same criteria are considered as when finding optimal trajectories, with the addition of the accuracy of passing points on trajectories. An illustrative example of solving the optimal area monitoring problem by two quadcopters with one circular phase constraint is provided. Symbolic regression is employed to synthesise a system for stabilising the movement of an object along the optimal trajectory.

The paper deals with the problem of traffic flow control in the urban road networks. The control is performed by traffic lights at intersections. The statement of the optimal control problem is given. A model-based approach is applied. The mathematical model of traffic flow based on the theory of controlled networks is used. A quality criterion is set. It is pointed out that a feature of the control of this object is its cyclical nature. Control is repeated at a certain time interval, which is called a control cycle. The duration of traffic light phases is considered as control. The phases switch sequentially in a certain order. Additional constraints are given that must be taken into account when choosing phase duration. As a result of solution of the optimal control problem the coordination plan for traffic lights is obtained. Coordination plan is cyclically repeated. The control synthesis problem is formulated that accounts for change in initial state. It is proposed to solve the control synthesis problem as a problem of selecting an optimal coordination plan from a set of plans calculated in advance for various initial flow states and model parameters.

Traffic lights at nearby intersections have a notable impact on the traffic flow. To ensure effective performance of the entire traffic system signal timing should be coordinated and synchronized. In this paper, the signal timing problem is considered as an optimal control problem. A universal recurrent traffic flow model is applied. The assumption is that information on the topology of road network, manoeuvre parameters, initial conditions, input flows, capacity constraints on road sections, and traffic light phase duration constraints is available. The mentioned parameters may be obtained from detectors of road infrastructure such as video cameras, loop and radio detectors and documentation. The objective is to develop an optimal traffic coordination plan for all monitored intersections within a specific time frame and minimizing a specified quality criterion. Coordination plan consists of ordered set of phases and their duration. The search space is big enough, so the problem is solved by evolutionary approach. The proposed optimization method is a variational genetic algorithm. The algorithm employs a principle of small variations of a basic solution to generate the population of possible solutions. Using this principle, we select a basic solution that is a current coordination plan. The set of codes that represent small variations of the basic solution are then used to determine all other feasible solutions. The proposed method is implemented to resolve the optimal control problem in an X-type intersection which experiences heavy traffic.

The problem of numerical solution of the optimal control problem for a group of robots is considered. The main complexity of the problem is the presence of dynamic phase constraints that describe the condition to prevent collisions between robots. Since there are several possibilities to avoid collision, and each of these possibilities corresponds to a local minimum, therefore the objective function has many local minima, thus the problem belongs to the class of global optimization. Precise methods for solving global optimization problems in this case cannot be applied due to the large dimension of the search space. It is proposed to use evolutionary algorithms to solve the optimal control problem of a group of robots. In computational experiment a complex optimal control problem for four quadcopters moving in space with four obstacles was solved by genetic and hybrid algorithms.

A multi-objective optimization problem for traffic flows in the urban road network is considered. A mathematical model of control object is given as a system of recurrent finite-difference equations. The control is a vector that determines the moments of switching the working phases of traffic lights at intersections. To solve the problem, it is proposed to use variational nondominated sorting genetic algorithm. Particular attention is paid to the presentation of possible solutions. The current signal timing for intersection is considered to be a basic solution. The search is performed on the space of variations of basic solution. An example of solving the multi-objective optimal control problem for sample X-type intersection is given.

The optimal control problem of object motion along the program trajectory is considered. The problem is stated as the optimal control problem with phase constraints in the form of equalities. Initial and terminal states are constrained. The numerical solution to the problem is proposed. Some points are defined on the program trajectory. The quality criterion includes time and hitting accuracy to the terminal state, integral error of motion along the program trajectory and sum of minimal distances to defined points on the program trajectory. To solve the problem direct approach on the base of control approximation by piecewise linear function and synthesized control are used. For synthesized control a method of machine learning control by symbolic regression is applied. In computational experiment the optimal control problem is solved for mobile robot moving along a program trajectory. Solutions obtained by both methods are compared in the presence of perturbations of the mathematical model of control object.

This paper concerns the quasi time-optimal motion in a simplified model of a mobile robot with constraints imposed on the state variables. As it is known, in this kind of problems involving the so-called unicycle type of dynamics, the classical assumptions of regularity with respect to the state constraints are not fulfilled. This fact greatly complicates the analysis of such control problems through the use of Pontryagin’s maximum principle. Moreover, the difficulties are also due to the presence of a singular control mode which is related to the angular velocity. The paper proposes a certain regularization approach in order to overcome the obstacles above and to develop a tool for the subsequent numerical implementation.

This paper is addressed to application of SOMA to the optimal control problem of a group of objects with static and dynamic phase constraints. The optimization problem is not convex and unimodal. In the practical part a synthesized optimal control problem is considered. Firstly, it is necessary to solve a control synthesis problem to stabilize each object relatively a point on the state space. To solve the synthesis problem a network operator method was used. For each control object the coordinates of some stabilization points were found. While moving from point to point in given time interval all objects reach their terminal states without violation of constraints and with optimal quality criterion. A modified SOMA is proposed for the search of stabilization points.

A study is focused on formulation of traffic flow control problems in the urban roads network. The problems of parametric identification, optimal control, and control synthesis are considered. Initially, the problem of obtaining a mathematical model of traffic flows as a control object is described. To obtain a model, it is proposed to use an approach based on the controlled networks theory. Evolutionary algorithms are used to solve the considered control problems.

The use of various optimization methods and the efficiency of their work strongly depends on the type of functional under investigation. It turned out that in the case of solving optimization problems with a nonlinear compound functional, it is not at all easy to estimate its convexity. And the absence of the unimodality property of the objective function means the instability and low efficiency of the application of gradient extremum search methods. This paper is devoted to the study of the properties of unimodality and convexity of functionals. The nontriviality of the problem of estimating the unimodality of a functional is shown, the concept of a fundamental sequence of functions that are the arguments of the objective functional is introduced, theorems on sufficient conditions for the absence of unimodality of the objective function are formulated and proved.

An experimental comparison of evolutionary algorithms and gradient-based methods for the optimal control problem is carried out. The problem is solved separately by Particle swarm optimization, Grey wolf optimizer, Fast gradient descent method, Marquardt method and Adam method. The simulation is performed on a jet aircraft model. The results of each algorithm performance are compared according to the best found value of the fitness function, the mean value and the standard deviation.

A reliability index of various electronics is determined by the experimental data of tests for different values of parameters of the equipment. The received data are collected in bulky tables and references. This paper presents modern numerical approach, allowing to compile the experimental data on changes of reliability index not in the form of tables but as a function of the operating parameters of the devices. The methodology is based on the method of network operator for the design of the optimal structure of function and selection of its parameters. The network operator method belongs to a class of methods of symbolic regression and provides an evolutionary search for the best compositions of mathematical expressions on the space of elementary structures. The method allows you to automatically receive the required description of the functional dependencies. The effectiveness of the method is demonstrated by the example of searching the law, which describes the change in the failure rate depending on three parameters that characterize its constructive and technological performance and operating conditions.

A problem of optimal urban traffic flows control is considered. A mathematical model of control by the traffic lights at intersections using the controlled networks theory is given. It is a system of nonlinear finite-differential equations. To present a large scale road networks the model contains the connection matrices that describe interactions between input and output roads in subnetworks. The traffic flow control is performed by the coordination of active phases of traffic lights. A control goal is to minimize the difference between the total input flow and total output flow for all subnetworks. In this paper, a neural network approach for traffic road network parameters adjustment is presented. A simulation is conducted under a microscopic traffic simulation software CTraf. Results demonstrate that neural network reinforcement training obtained good parameters of the network model.

The paper proposes a new method for synthesis of optimal control systems. The network operator method is used. The intellectual evolution algorithm is applied for searching. This algorithm combines advantages of several evolutionary algorithms. A numerical example of control system synthesis for a wheeled mobile robot is presented.