On the compactness of solutions to certain operator inequalities arising from the Likhtarnikov — Yakubovich frequency theorem
DOI:
https://doi.org/10.21638/spbu01.2020.405Abstract
We study the compactness property of operator solutions to certain operator inequalities arising from the frequency theorem of Likhtarnikov — Yakubovich for C0-semigroups. We show that the operator solution can be described through solutions of an adjoint problem as it was previously known under some regularity condition. Thus we connect some regularity properties of the semigroup with the compactness of the operator in the general case. We also prove several results useful for checking the non-compactness of operator solutions to Lyapunov inequalities and equations, into which the operator Riccati equation degenerates in certain cases arising in applications. As an example, we apply these theorems for a scalar delay equation posed in a proper Hilbert space and show that the operator solution cannot be compact. This results are related to the author recent work on a non-local reduction principle of cocycles (non-autonomous dynamical systems) in Hilbert spaces.Keywords:
frequency theorem, Lyapunov inequality, compact operator, delay equations
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References
Литература
1. Krein S. G. Linear differential equations in Banach space. AMS, 1971.
2. Лихтарников А. Л., Якубович В. А. Частотная теорема для сильно непрерывных однопараметрических полугрупп // Известия Российской академии наук. Серия математическая. 1977. Т. 41, №4. P. 895–911.
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5. Anikushin M. M. A non-local reduction principle for cocycles in Hilbert spaces // J. Differ. Equations. 2020. Vol. 269, no. 9. P. 6699–6731.
6. Anikushin M. M. The Poincar´e-Bendixson theory for certain semi-flows in Hilbert spaces. URL: https://arxiv.org/abs/2001.08627 (дата обращения: 25.10.2020)
7. Smith R. A. Orbital stability and inertial manifolds for certain reaction diffusion systems // Proceedings of the London Mathematical Society. 1994. Vol. 3, no. 1. P. 91–120.
8. Datko R. Extending a theorem of A. M. Liapunov to Hilbert space // J. Math. Anal. Appl. 1970. Vol. 32, no. 3. P. 610–616.
9. Triggiani R. An optimal control problem with unbounded control operator and unbounded observation operator where the algebraic Riccati equation is satisfied as a Lyapunov equation // Applied Mathematics Letter. 1997. Vol. 10, no. 2. P. 95–102.
10. Barbu V. Nonlinear semigroups and differential equations in Banach spaces. Springer, 1976.
11. Likhtarnikov A. L., Yakubovich V. A. The frequency theorem for equations of evolutionary type // Sib. Math. J. 1976. Vol. 17, no. 5. P. 790–803.
12. Lions J. L. Optimal control of systems governed by partial differential equations. Springer-Verlag, 1971.
13. Gusev S. V. Extended Kalman — Yakubovich — Popov Lemma in a Hilbert Space and Fenchel Duality // Proceedings of the 44th IEEE Conference on Decision and Control. 2005.
14. Anikushin M. M. Frequency theorem for the regulator problem with unbounded cost functional and its applications to nonlinear delay equations. URL: https://arxiv.org/abs/2003.12499v2 (дата обращения: 25.10.2020).
15. Proskurnikov A. V. A new extension of the infinite-dimensional KYP lemma in the coercive case // IFAC-PapersOnLine. 2015. Vol. 48, no. 11. P. 246–251.
16. Kuznetsov N. V., Reitmann V. Attractor Dimension Estimates for Dynamical Systems: Theory and Computation. Switzerland: Springer International Publishing AG, 2020.
17. Anikushin M. M. On the Smith reduction theorem for almost periodic ODEs satisfying the squeezing property // Rus. J. Nonlin. Dyn. 2019. Vol. 15, no. 1. P. 97–108.
18. Аникушин М. М., Райтманн Ф., Романов А. О. Аналитические и численные оценки фрактальнойразмерности вынужденных квазипериодических колебанийв системах управления // Дифференциальные уравнения и процессы управления. 2019. Т. 87, №2.
19. Anikushin M. M. On the Liouville Phenomenon in Estimates of Fractal Dimensions of Forced Quasi-Periodic Oscillations // Vestnik St. Petersburg University, Mathematics. 2019. Vol. 52, no. 3. P. 234–243.
20. Helemskii A. Ya. Lectures and exercises on functional analysis. Providence. RI: American Mathematical Society, 2006.
21. Wexler D. On frequency domain stability for evolution equations in Hilbert spaces via the algebraic Riccati equation // SIAM Journal on Mathematical Analysis. 1980. Vol. 11, no. 6. P. 969–983.
References
1. Krein S. G., Linear Differential Equations in Banach Space (AMS, 1971).
2. Likhtarnikov A. L., Yakubovich V. A., “The frequency theorem for continuous one-parameter semigroups”, Math. USSR-Izv. 41(4), 895–911 (1977). (In Russian)
3. Louis J.-Cl., Wexler D., “The Hilbert space regulator problem and operator Riccati equation under stabilizability”, Annales de la Soci´et´e Scientifique de Bruxelles 105(4), 137–165 (1991).
4. Arov D. Z., Yakubovich V. A., “Semiboundedness conditions for quadratic functionals on Hardy spaces”, Vestn. Leningr. Univ., Ser. Mat. Mekh. Astron. iss. 1, 7–13 (1982).
5. Anikushin M. M., “A non-local reduction principle for cocycles in Hilbert spaces”, J. Differ. Equations 269(9), 6699–6731 (2020).
6. Anikushin M. M., “The Poincar´e — Bendixson theory for certain semi-flows in Hilbert spaces”. Available at: https://arxiv.org/abs/2001.08627 (accessed: October 25, 2020).
7. Smith R. A., “Orbital stability and inertial manifolds for certain reaction diffusion systems”, Proceedings of the London Mathematical Society 3(1), 91–120 (1994).
8. Datko R., “Extending a theorem of A. M. Liapunov to Hilbert space”, J. Math. Anal. Appl. 32(3), 610–616 (1970).
9. Triggiani R., “An optimal control problem with unbounded control operator and unbounded observation operator where the algebraic Riccati equation is satisfied as a Lyapunov equation”, Applied Mathematics Letters 10(2), 95–102 (1997).
10. Barbu V., Nonlinear Semigroups and Differential Equations in Banach Spaces (Springer, 1976).
11. Likhtarnikov A. L., Yakubovich V. A., “The frequency theorem for equations of evolutionary type”, Sib. Math. J. 17(5), 790–803 (1976).
12. Lions J. L., Optimal Control of Systems Governed by Partial Differential Equations Springer-Verlag, 1971).
13. Gusev S. V., “Extended Kalman — Yakubovich — Popov Lemma in a Hilbert Space and Fenchel Duality”, Proceedings of the 44th IEEE Conference on Decision and Control (2005).
14. Anikushin M. M., “Frequency theorem for the regulator problem with unbounded cost functional and its applications to nonlinear delay equations”. Available at: https://arxiv.org/abs/2003.12499v2 (accessed: October 25, 2020).
15. Proskurnikov A. V., “A new extension of the infinite-dimensional KYP lemma in the coercive case”, IFAC-PapersOnLine 48(1), 246–251 (2015).
16. Kuznetsov N. V., Reitmann V., Attractor Dimension Estimates for Dynamical Systems: Theory and Computation (Springer International Publishing AG, Switzerland, 2020).
17. Anikushin M. M., “On the Smith reduction theorem for almost periodic ODEs satisfying the squeezing property”, Rus. J. Nonlin. Dyn. 15(1), 97–108 (2019).
18. Anikushin M. M., Reitmann V., Romanov A. O., “Analytical and numerical estimates of the fractal dimension of forced quasiperiodic oscillations in control systems”, Differential Equations and Control Processes 87(2) (2019). (In Russian)
19. Anikushin M. M., “On the Liouville Phenomenon in Estimates of Fractal Dimensions of Forced Quasi-Periodic Oscillations”, Vestnik St. Petersburg University, Mathematics 52(3), 234–243 (2019).
20. Helemskii A. Ya., Lectures and Exercises on Functional Analysis (American Mathematical Society, Providence, RI, 2006).
21. Wexler D., “On frequency domain stability for evolution equations in Hilbert spaces via the algebraic Riccati equation”, SIAM Journal on Mathematical Analysis 11(6), 969–983 (1980).
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Published
2020-12-27
How to Cite
Anikushin, M. M. (2020). On the compactness of solutions to certain operator inequalities arising from the Likhtarnikov — Yakubovich frequency theorem. Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy, 7(4), 622–635. https://doi.org/10.21638/spbu01.2020.405
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Mathematics
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Articles of "Vestnik of Saint Petersburg University. Mathematics. Mechanics. Astronomy" are open access distributed under the terms of the License Agreement with Saint Petersburg State University, which permits to the authors unrestricted distribution and self-archiving free of charge.
