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DOI: https://doi.org/10.20537/vm160409

Abstract: We consider the nonlinear controlled system in a finite-dimensional Euclidean space in a finite time interval. We study the problem of a system approaching a given compact set in finite time. Approximate solution of the approaching problem is discussed. The approach used to construct an approximate solution is based on constructions based on the notion of a set of solvability of the approaching problem. The concept of correcting control with and without additional operating influences is introduced. We propose a scheme of approximate backward construction of the solvability set, as well as the scheme of control software, which allows finding approximately a solution to the approaching problem. In it, the operating influence breaks down into “main” and “correcting”. An estimate of the deviation of the operated system from the target set at the final moment is constructed and it is shown that the use of additional correcting control in the control process can essentially improve the result of control.

Keywords: control problem, approaching problem, correcting control, controlled system, integral funnel, set of solvability.

Funding agency	Grant number
Russian Science Foundation	15-11-10018

Received: 17.10.2016

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Citation: V. N. Ushakov, A. A. Ershov, “On the solution of control problems with fixed terminal time”, Vestn. Udmurtsk. Univ. Mat. Mekh. Komp. Nauki, 26:4 (2016), 543–564

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This publication is cited in the following articles:

1. K. A. Schelchkov, “K nelineinoi zadache presledovaniya s diskretnym upravleniem”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 27:3 (2017), 389–395      
2. K. A. Schelchkov, “Ob odnoi nelineinoi zadache presledovaniya s diskretnym upravleniem i nepolnoi informatsiei”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 28:1 (2018), 111–118      
3. A. G. Chentsov, D. M. Khachai, “Relaxation of the Pursuit–Evasion Differential Game and Iterative Methods”, Proc. Steklov Inst. Math. (Suppl.), 308, suppl. 1 (2020), S35–S57          
4. V. N. Ushakov, A. A. Ershov, G. V. Parshikov, “O privedenii dvizheniya upravlyaemoi sistemy na mnozhestvo Lebega lipshitsevoi funktsii”, Vestn. Udmurtsk. un-ta. Matem. Mekh. Kompyut. nauki, 28:4 (2018), 489–512      
5. A. A. Ershov, A. V. Ushakov, V. N. Ushakov, “An approach problem for a control system and a compact set in the phase space in the presence of phase constraints”, Sb. Math., 210:8 (2019), 1092–1128              
6. Alexander Chentsov, Daniel Khachay, Studies in Systems, Decision and Control, 203, Advanced Control Techniques in Complex Engineering Systems: Theory and Applications, 2019, 129  
7. V. N. Ushakov, A. V. Ushakov, “O navedenii integralnoi voronki upravlyaemoi sistemy na tselevoe mnozhestvo v fazovom prostranstve”, Izv. IMI UdGU, 56 (2020), 79–101    
8. Vladimir Ushakov, Aleksandr Ershov, Lecture Notes in Control and Information Sciences - Proceedings, Stability, Control and Differential Games, 2020, 225  
9. Shchelchkov K., “Epsilon-Capture in Nonlinear Differential Games Described By System of Order Two”, Dyn. Games Appl., 12:2 (2022), 662–676        
10. A. A. Ershov, “Linear Parameter Interpolation of a Program Control in the Approach Problem”, J Math Sci, 260:6 (2022), 725  
11. K. A. Shchelchkov, “Estimate of the Capture Time and Construction of the Pursuer's Strategy in a Nonlinear Two-Person Differential Game”, Diff Equat, 58:2 (2022), 264  
12. K. A. Shchelchkov, “Relative optimality in nonlinear differential games with discrete control”, Sb. Math., 214:9 (2023), 1337–1350            
13. A. A. Ershov, “Bilinear interpolation of program control in approach problem”, Ufa Math. J., 15:3 (2023), 41–53    
14. K. A. Shchelchkov, “One-Sided Capture in Nonlinear Differential Games”, Int. Game Theory Rev., 25:02 (2023)  
15. Vladimir Nikolaevich Ushakov, Aleksandr Anatol'evich Ershov, Anna Aleksandrovna Ershova, Aleksandr Vladimirovich Alekseev, Communications in Computer and Information Science, 1881, Mathematical Optimization Theory and Operations Research: Recent Trends, 2023, 324  
16. K. A. Shchelchkov, “On the Problem of Controlling a Nonlinear System by a Discrete Control under Disturbance”, Diff Equat, 60:1 (2024), 127  
17. A. V. Alekseev, A. A. Ershov, “Target-Point Interpolation of a Program Control in the Approach Problem”, Comput. Math. and Math. Phys., 64:3 (2024), 585  

Citing articles in Google Scholar: Russian citations, English citations
Related articles in Google Scholar: Russian articles, English articles