CONFLICT-CONTROLLED PROCESSES WITH TIME DELAY

Russia's full-scale invasion caused extraordinary damage to Ukraine, but at the same time became an extraordinary impetus for the rapid development of scientific and technological innovations in Ukraine. 

In recent years, the drone market has expanded significantly, both in the consumer sector and in the defense sector, due to its relatively low cost. Additionally, the ability to carry a payload enables unmanned aerial vehicles (UAVs) to be utilized in various fields, ranging from recreational to professional applications. From a research perspective, the use of UAVs facilitates the development of technologies that have a positive impact on the community, such as search and rescue operations, intelligent logistics, environmental monitoring, and precision agriculture.

The improvement of UAV control systems, especially in swarm applications, also plays a major role in security policy. Their use can cause harm to society through failures, improper use, or criminal activity. It is also worth noting that the number of accidents involving UAVs is on the rise. For example, improper use near an airport can pose a serious threat to public safety, as evidenced by hundreds of canceled flights.

Conflict-controlled processes are a branch of applied mathematics that studies the theory of control under conditions of conflict and uncertainty. A wide range of fundamental mathematical methods has been developed in theory to study various confrontation scenarios. In the theory of dynamic pursuit-evasion games, alongside methods focused on making an optimal decision, several methods provide a guaranteed result. Among the former are, first, the Isaacs method [1] associated with the basic equation of the theory of differential games, Pontryagin’s method of alternating integral, and the method of Pshenichnyi -operators. The guaranteed result is provided by Pontryagin’s First Direct Method, the Krasovskii Extreme Aiming Rule, and Chikrii’s Method of Resolving Functions. 

The second group of methods, without focusing on the problem of optimality, justifies the well-known practice methods of Euler's running curve and parallel pursuit [2]. Academician Arkadii Chikrii heads the Ukrainian branch of the International Society of Dynamic Games and the Ukrainian Association of Scientists and Specialists in Automatic Control, a branch of the International Federation of Automatic Control. Based on his optimal control method [3], scientists have obtained significant results in the field of automated control systems, particularly in the manipulation of moving objects operating in uncertain and conflicting conditions, which are extremely relevant.

Today, unmanned aerial vehicles acquire increasingly wider functional capabilities, and their variety of technical characteristics complicates their simultaneous control in a group. There is an urgent need to develop control systems for unmanned aerial vehicles in swarm applications. The approach problems of controlled objects and strategies for intercepting targets in the game, as well as problems of dynamics with many participants, remain complex and insoluble. Such control methods are proposed to be developed in the project for dynamic systems with time delays, which are relevant for long-distance flights, modeling dynamic processes for underwater controlled objects, and other applications. UAVs have recently acquired wide functional capabilities, which has caused the complexity of interacting groups of moving objects with different inertia in conflict conditions. Today, in most cases, it is stated that pursuers and evaders have different inertias (for example, a missile and an airplane, a torpedo/underwater drone and a surface vessel, a fighter and an aircraft carrier - the so-called "soft landing problem," etc.). Additionally, in practice, combined swarms are created, comprising underwater, ground, and/or airborne unmanned vehicles. Field tests are currently being conducted with existing UAV prototypes featuring different inertias, as well as experiments utilizing computer modeling. Thus, there is a need for theoretical substantiation of the results of these works, namely, the creation of a method for forming a control strategy, which makes it possible to develop an effective mathematical apparatus for its application in real conditions to solve the approach problem of conflict-controlled objects with different inertia and the strategy of intercepting targets in the game problems of dynamics with many participants [4, 5].

The key condition in classical pursuit methods is the Pontryagin condition, which is often not satisfied in many practical situations, particularly for objects with different inertias (e.g., an FPV drone, a reconnaissance aircraft, a missile, and an air target). In such cases, modifications of the Chikrii’s Method of Resolving Functions are used using: 1) Upper and lower resolution functions - a new approach that allows you to bypass the failure of classical conditions and can be effectively implemented in real-time systems; 2) Terminal set modifications - introducing a solid-state component of the cylindrical terminal set to compensate for the limitations; 3) Special matrix functions - to equalize player control resources.

All mathematical constructions must be developed with the possibility of direct implementation in computer vision system software in mind. Control is developed in two classes of strategies, both of which are oriented towards fully autonomous operation and integration with artificial intelligence systems:

• Quasistrategies consider the entire history of the fugitive's control, which allows for more accurate predictions and optimal trajectories in automatic mode. This is naturally consistent with the architectures of recurrent neural networks and attention mechanisms, which also work with data sequences. Such synergy allows for the effective integration of mathematical strategies with computer vision systems for processing temporal information.
• Stroboscopic strategies - do not require information about the full history of the fugitive; the switching moment does not depend on its previous actions. This approach is the most practical for implementation in real autonomous control systems with limited computing resources, as it minimizes memory requirements and provides a fast response to changing situations, which is critically important for UAV onboard systems.

A critical requirement for the developed control strategies is their ability to operate in a fully autonomous mode without the need for human operator intervention. This is due to the specific requirements of combat UAV use, where the speed of decision-making is critical, as well as the need to ensure resistance to electronic warfare means that can disrupt communication with the operator.

Therefore, the orientation of the developed strategies and algorithms for direct application in computer vision systems with elements of artificial intelligence is relevant today. The mathematical apparatus must be developed to ensure seamless integration of theoretical control methods with practical systems for video stream processing, object detection, and decision-making based on neural networks.

A systematic review of previous research on real-time object detection using UAVs, considering application scenarios, equipment selection, real-time detection paradigms, detection algorithms, and their optimization technologies, is presented in the papers [6, 7].

Obtaining the most accurate and complete information about the location of moving objects of certain types over a large area requires many aerial surveying facilities. If the coverage by such facilities is sufficiently dense, the problem of data redundancy arises. To improve the efficiency of moving object identification and classification by analyzing multiple video streams, a method for moving object identification based on the analysis of merged video streams has recently been developed [8], enabling the creation of an automated fugitive identification system.

Fundamental control, autopilot, and navigation methods form the methodological basis for developing adaptive unmanned aerial vehicle (UAV) control algorithms that ensure robust performance across heterogeneous operating environments, while maintaining the safety and reliability of autonomous flight operations [9].

Thus, the development of mathematical tools to enhance control systems for swarming UAVs has a significant impact on science, economics, and society.

The present theoretical framework naturally leads to the consideration of a discrete-time control model and the development of algorithms for fundamental pursuit methods, which constitutes a promising direction for future research.

I am very grateful to the International Research Fellowship Programme in Regensburg, a component of the project “Denkraum Ukraine / Think Space Ukraine” (DU), which is funded by the German Academic Exchange Service (DAAD) and the German Federal Foreign Office, for supporting my research work.

1. Isaacs R. (1965). Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization. John Wiley and Sons, New York.
2. Chikrii A., Vyshenskyy V. (2025). Pascal’s snail, Apollonius circle, and Cartesian oval in classical game problems of moving controlled objects interception. International Scientific Technical Journal «Problems of Control and Informatics», 70(5), 19–32. https://doi.org/10.34229/1028-0979-2025-5-2
3. Chikrii A. (2013). Conflict-Controlled Processes. Springer Science & Business Media, 556 p. https://link.springer.com/book/10.1007/978-94-017-1135-7 
4. Baranovska L. V. (2022). Method of upper and lower resolving functions for pursuit differential-difference games with pure delay. In: Y. Kondratenko, V. Kuntsevich, A. Chikrii, V. Gubarev (eds). Recent Developments in Automatic Control Systems. River Publishers Series in Automation, Control and Robotics, 131–144. https://doi.org/doi:10.1201/9781003339229-7  
5. Baranovska L. V. (2021). Differential-Difference Games of Approach with Multiple Delays. Cybern Syst Anal, 57, 787–795. https://doi.org/10.1007/s10559-021-00403-4
6. Cao, Z., Kooistra, L., Wang, W., Guo, L., & Valente, J. (2023). Real-Time Object Detection Based on UAV Remote Sensing: A Systematic Literature Review. Drones, 7(10), 620. https://doi.org/10.3390/drones7100620
7. Arafat, M. Y., Alam, M. M., & Moh, S. (2023). Vision-Based Navigation Techniques for Unmanned Aerial Vehicles: Review and Challenges. Drones, 7(2), 89. https://doi.org/10.3390/drones7020089
8. Chykrii A., Chikrii O., Baranovska L. (2025). Method for identification of moving objects based on analysis of combined video streams. International Scientific Technical Journal "Problems of Control and Informatics", 70(3), 20–32. https://doi.org/10.34229/1028-0979-2025-3-2
9. Li H. (2025). Adaptive algorithms for drone flight control under communication constraints and information incompleteness. The Aeronautical Journal, 129(1332), 282–295. https://doi.org/10.1017/aer.2024.112