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Volume 15
Issue 6
10.3390/axioms15060436
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This is an early access version, the complete PDF, HTML, and XML versions will be available soon.
Article

Critical Problem of Optimal Stabilization Without Control Constraints

by
Volodymyr Kapustyan
1,
Anna Sukretna
2,*,
Zhanna Chernousova
1 and
Yuriy Kharkevych
3
1
Department of Economic Cybernetics, Faculty of Management and Marketing, National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, 03056 Kyiv, Ukraine
2
Department of Integral and Differential Equations, Faculty of Mechanics and Mathematics, Taras Shevchenko National University of Kyiv, 01601 Kyiv, Ukraine
3
Department of Function Theories and Mathematics Teaching Methods, Faculty of Information Technology and Mathematics, Lesya Ukrainka Volyn National University, 43025 Lutsk, Ukraine
*
Author to whom correspondence should be addressed.
Axioms 2026, 15(6), 436; https://doi.org/10.3390/axioms15060436
Submission received: 28 April 2026 / Revised: 2 June 2026 / Accepted: 9 June 2026 / Published: 11 June 2026
(This article belongs to the Special Issue Modern Trends and Application of Decision-Making Theory, Stability and Control, 2nd Edition)
Versions Notes

Abstract

This paper investigates the linear–quadratic optimal stabilization problem in the so-called critical case, that is, the situation in which the spectrum of the system matrix contains purely imaginary eigenvalues or the standard positive-definiteness conditions on the weight matrices of the objective functional are violated. To address these challenges, new regularization methods for critical problems via perturbation of the system matrices and the functional are studied, and novel algorithms for decomposing multidimensional problems into a collection of one-dimensional canonical systems are developed. The main contribution of this work lies in providing a systematic framework for critical cases where standard methods fail. The results are of practical significance for the construction of optimal synthesis in various engineering and applied systems; in particular, they are applicable to the stabilization of unmanned aerial vehicles, robotic complexes, and intelligent power grids.
Keywords: optimal stabilization problem; optimal synthesis; feedback control; linear–quadratic regulator (LQR); critical case of optimal stabilization problem; regularization optimal stabilization problem; optimal synthesis; feedback control; linear–quadratic regulator (LQR); critical case of optimal stabilization problem; regularization

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MDPI and ACS Style

Kapustyan, V.; Sukretna, A.; Chernousova, Z.; Kharkevych, Y. Critical Problem of Optimal Stabilization Without Control Constraints. Axioms 2026, 15, 436. https://doi.org/10.3390/axioms15060436

AMA Style

Kapustyan V, Sukretna A, Chernousova Z, Kharkevych Y. Critical Problem of Optimal Stabilization Without Control Constraints. Axioms. 2026; 15(6):436. https://doi.org/10.3390/axioms15060436

Chicago/Turabian Style

Kapustyan, Volodymyr, Anna Sukretna, Zhanna Chernousova, and Yuriy Kharkevych. 2026. "Critical Problem of Optimal Stabilization Without Control Constraints" Axioms 15, no. 6: 436. https://doi.org/10.3390/axioms15060436

APA Style

Kapustyan, V., Sukretna, A., Chernousova, Z., & Kharkevych, Y. (2026). Critical Problem of Optimal Stabilization Without Control Constraints. Axioms, 15(6), 436. https://doi.org/10.3390/axioms15060436

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