This study focuses on solving the optimal control problem for a wheeled car-like robot by leveraging its neural network-based dynamic model. The neural network model captures nonlinear dynamic effects—such as variations in wheel-surface adhesion and battery charge depletion—enhancing the model's accuracy and realism. However, employing such a model restricts the applicability of classical optimal control methods. To overcome this limitation, the paper introduces a direct numerical approach for solving the optimal control problem under phase constraints for a neural network-represented system. The proposed method approximates control inputs as piecewise polynomial functions, with parameter optimization performed using state-of-the-art evolutionary algorithms. An experimental analysis of the most widely used algorithms demonstrates that hybridizing them improves the efficiency of the solution.

This paper presents a new motion planning algorithm for mobile robots that uses synthesized optimal control and time-adaptive particle swarm optimization (PSO). The proposed approach aims to address the challenge of creating collision-free paths in complex environments with static obstacles, while optimizing both spatial and temporal constraints. The algorithm dynamically adjusts the intermediate waypoints and time budget to minimize a combined cost function that includes factors such as positional error, travel time, and obstacle avoidance costs. The effectiveness of the proposed algorithm is demonstrated through simulations in Gazebo using the ROSBot 2.0 model. The algorithm successfully navigates the robot through the optimized waypoints in scenarios with obstacles, avoiding collisions and minimizing the travel time. In environments without obstacles, the trajectory is collapsed into a direct path, significantly reducing the total travel time. The results highlight the adaptability and efficiency of the algorithm, making it a promising solution for real-world applications. Key contributions of this work include the integration of optimal control with time-adaptive particle swarm optimization (PSO), the dynamic adjustment of waypoints and time constraints, and the demonstration of algorithm performance in both complex and simplified environments.

The work presents original approach for solving the most popular computational traveling salesman problem. This is a modified genetic algorithm built on the basis of the principle of small variations of the basic solution. Its advantage is that the time of its operation does not depend on the complexity of the problem in this case on the number of cities, but only depends on the parameters of the algorithm. In this algorithm, a possible solution is not an ordered set of visited cities but a set of small variations of some one possible solution. This solution is called basic. It is determined by the researcher as the closest to the optimal solution. In this case, the basic solution is made by a greedy algorithm. During the search process, the basic solution changes to the best current solution found. Together with the genetic algorithm, an accurate overlap algorithm is used, which has polynomial complexity and improves solutions by eliminating the self-intersections of the path.

The paper argues that modern machine learning methods make it possible to find mathematical expressions for various multidimensional functions. The article formulates a list of problems in which it is necessary to find a mathematical expression of a multidimensional function as a solution. Searching for control functions also belongs to the class of such problems. The presented list of problems is a proposal to researchers in the field of computational mathematics to develop new effective general methods of machine learning in order to obtain mathematical expressions of the desired functions. To solve the stated problems, the paper presents a structural-parametric approach based on evolutionary machine learning methods using symbolic regression. The paper presents solutions for three example problems of the function search.

The problem of control synthesis for a mobile robot with phase constraints in the form of an arbitrarily located obstacle is formulated. To solve the problem, a numerical method of machine learning based on symbolic regression is used. According to the approach, in addition to its state, the object receives information about the distance to the obstacle and the direction of its location. As a result of solving the synthesis problem, we obtain a nonlinear feedback control function that provides a better estimate of the obstacle bypass according to a given criterion. An example for a mobile robot with a differential drive is considered.

The problem of optimal control of Mecanum-wheeled robot is considered. Mecanum wheels allow to move along the plane in any direction without turning the body of the robot. A feature of this robot model as a control object is that the control vector dimension is greater than the state vector. As a result, the same optimal robot movement can be realized by different control means. To solve the problem of optimal control, two numerical methods are used: direct control and synthesized. To check the efficiency of the obtained control functions, the mathematical model of the control object with both control functions was simulated under disturbances. The evaluation criterion was the change in the value of the quality criterion of the optimal solution depending on the level of perturbation. Experiments have shown that the synthesized optimal control for the Mecanum-wheeled robot is less sensitive to disturbances than the direct optimal control.

A numerical solution of the problem of the general synthesis of a stabilization system by a symbolic regression method is considered. The goal is to automatically find a feedback control function using a computer so that the control object can reach a given terminal position from anywhere in a given region of the initial conditions with an optimal value of the quality criterion. Usually, the control synthesis problem is solved analytically or technically taking into account the specific properties of the mathematical model. We suppose that modern numerical approaches of symbolic regression can be applied to find a solution without reference to specific model equations. It is proposed to use the numerical method of Cartesian genetic programming (CGP). It was developed for automatic writing of programs but has never been used to solve the synthesis problem. In the present work, the method was modified with the principle of small variations in order to reduce the search area and increase the rate of convergence. To apply the general principle of small variations to CGP, we developed special types of variations and coding. The modified CGP searches for the mathematical expression of the feedback control function in the form of a code and, at the same time, the optimal value of the parametric vector which is also a new feature-simultaneous tuning of the parameters inside the search process. This approach enables working with objects and functions of any type, which is not always possible with analytical methods. The need to use the received solution on the onboard processor of the control object imposes certain restrictions on the used basic set of elementary functions. This article proposes the theoretical foundations of the study of these functions, and the concept of the space of machine-made functions is introduced. The capabilities of the approach are demonstrated on the numerical solution of the control system synthesis problems for a mobile robot and a Duffing model.

This paper presents a new two-step numerical approach to a solution of the optimal control problem with phase constraints named a synthesized optimal control. The stated problem combines two well-known tasks: the optimal control and the control system synthesis. Initially, the synthesis problem is considered and a feedback control is received that provides a steady state for the control object relative to some point in a state space. Then a sequence of points is searched in the state space, so that each of the points is stable in the state space of the object, so the object moves from the initial condition to terminal one by switching from one stabilization point to another in some time interval. It is shown in the paper that such approach allows to receive a solution that does not differ much from those of the optimal control problem considering the value of the quality criterion, but it works more stable in the presence of disturbances. At the same time such approach is much more applicable in real engineering tasks since some distinctions between the mathematical model of the control object and the real control object are smoothed over due to the first stabilization procedure of the approach. The paper includes a mathematical problem statement, the description of modern computational methods for its solution and computational example.

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.

This paper considers the optimal control problem for a group of interacting robots. To solve the problem, a hybrid evolutionary algorithm is applied that consists of two popular evolutionary algorithms: particle swarm optimization and gray wolf optimizer. When solving the problem, we use the approach of synthesized optimal control. Initially, we make controlled objects stable in the state space with respect to a certain point. Then we look for coordinates of the stabilization points using the hybrid algorithm. Points should be located so that when switching stabilization points after each fixed time interval, the objects reach the control goal without violating the phase constraints with the optimal value of the quality criterion.

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.

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 synthesis of optimal control for a group of robots is considered. Suppose every robot of group has full information about other robots. A two-stage numerical approach for optimal control synthesis is applied. Firstly, the task of stability in a state space for one robot is decided. For this purpose multilayer network operator is used. Secondly, a problem of designing optimal trajectories is considered. Trajectories in a form of set of points in a state space are set. It is necessary to find these points the way that robots could not encounter in the process of movement. Computational experiment is presented for group of three mobile robots are parking at spatial restricts.

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.