Boundary-value problems for the Lyapunov equation. І
Abstract
UDC 517.9
We study boundary-value problems for the Lyapunov operator-differential equation. By using the theory of Moore–Penrose pseudoinverse operators and its development, we establish conditions for the existence of generalized solutions and propose algorithms for their construction.
References
A. E. Bryson (Jr.), Yu-Chi Ho, Applied optimal control: optimization, estimation and control, 1st ed., Taylor Francis Group, New York, London (1975).
A. A. Boichuk, S. A. Krivosheya, A critical periodic boundary-value problem for a matrix Riccati equation, Different. Equat., 37, № 4, 464–471 (2001). DOI: https://doi.org/10.1023/A:1019267220924
А. А. Бойчук, В. Ф. Журавлев, А. М. Самойленко, Обобщенно-обратные операторы и нетеровы краевые задачи, Институт математики НАН Украины, Киев (1995).
I. A. Bondar, Linear boundary-value problems for systems of integrodifferential equations with degenerate kernel. Resonance case for a weakly perturbed boundary-value problem, J. Math. Sci., 274, № 6, 822–832 (2023); DOI: https://doi.org/10.1007/s10958-023-06645-1. DOI: https://doi.org/10.1007/s10958-023-06645-1
Ju. L. Daleckii, M. G. Krein, Stability of solutions of differential equations in Banach space, Amer. Math. Soc. (2002).
S. G. Krein, Linear equations in Banach spaces, Birkhäuser (1982); DOI: https://doi.org/10.1007/978-1-4684-8068-9. DOI: https://doi.org/10.1007/978-1-4684-8068-9
S. G. Krein, Linear differential equations in Banach space, Amer. Math. Soc. (1972).
Є. В. Панасенко, О. О. Покутний, Крайові задачі для диференціальних рівнянь у банаховому просторі з необмеженим оператором у лінійній частині, Нелінійні коливання, 16, № 4, 518–526 (2013).
О. О. Покутний, Узагальнено-обернений оператор у просторах Фреше, Банаха та Гільберта, Вісн. Київ. нац. ун-ту ім. Тараса Шевченка, Сер. фіз.-мат. науки, № 4, 158–161 (2013).
V. I. Arnold, Catastrophe theory, Springer (1992);
DOI: https://doi.org/10.1007/978-3-642-58124-3. DOI: https://doi.org/10.1007/978-3-642-58124-3
M. M. Vainberg, Theory of branching of solutions of nonlinear equations, Monogr. and Textbooks Pure and Appl. Math., Noordhoff International Publ. (1974).
A. H. Nayfeh, Perturbation methods, Wiley, New York (1973).
B. Bamieh, M. Dahleh, Energy amplification in channel flows with stochastic excitation, Phys. Fluids, 13, 3258–3269 (2001). DOI: https://doi.org/10.1063/1.1398044
Bhatia Rajendra, A note on the Lyapunov equation, Linear Algebra and Appl., 259, 71–76 (1997). DOI: https://doi.org/10.1016/S0024-3795(96)00242-X
A. A. Boichuk, A. M. Samoilenko, Generalized inverse operators and Fredholm boundary-value problems, 2nd ed., De Gruyter (2016). DOI: https://doi.org/10.1515/9783110378443
O. A. Boichuk, S. A. Krivosheya, Criterion for the solvability of matrix equations of the Lyapunov type, Ukr. Math. J., 50, № 8, 1162–1169 (1998). DOI: https://doi.org/10.1007/BF02513089
S. M. Chuiko, On the solution of matrix Lyapunov equations, Visn. Kharkiv. Univ., Ser. Mat., Prikl. Mat., Mekh., № 1120, 85–94 (2014).
R. Datko, Extending a theorem of a A. M. Lyapunov to Hilbert space, J. Math. Anal. and Appl., 32, 610–616 (1970). DOI: https://doi.org/10.1016/0022-247X(70)90283-0
V. Druskin, L. Knizhnerman, V. Simoncini, Analysis of the rational Krylov subspace and ADI methods for solving the Lyapunov equation, SIAM J. Numer. Anal., 49, № 5, 1875–1898 (2011). DOI: https://doi.org/10.1137/100813257
T. E. Duncana, B. Maslowski, B. Pasik-Duncana, Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise, Stochast. Process. and Appl., 115, 1357–1383 (2005). DOI: https://doi.org/10.1016/j.spa.2005.03.011
H. Kielhöfer, On the Lyapunov-stability of stationary solutions of semilinear parabolic differential equations, J. Different. Equat., 22, 193–208 (1976). DOI: https://doi.org/10.1016/0022-0396(76)90011-5
V. I. Man’ko, R. Vilela, Mendes Lyapunov exponent in quantum mechanics. A phase-space approach, Physica D, 145, 330–348 (2000). DOI: https://doi.org/10.1016/S0167-2789(00)00117-2
E. V. Panasenko, O. O. Pokutnyi, Boundary-value problems for the Lyapunov equation in Banach spaces, J. Math. Sci., 223, 1–7 (2017). DOI: https://doi.org/10.1007/s10958-017-3356-x
Є. В. Панасенко, О. О. Покутний, Крайові задачі для диференціальних рівнянь у банаховому просторі з необмеженим оператором у лінійній частині, Нелінійні коливання, 16, № 4, 518–526 (2013); English translation: J. Math. Sci., 203, № 3, 366–374 (2014).
A. Pazy, On the applicability of Lyapunov's theorem in Hilbert space, SIAM J. Math. Anal., 3, № 2, 291–294 (1972). DOI: https://doi.org/10.1137/0503028
K. Maciej Przyluski, The Lyapunov equation and the problem of stability for linear bounded discrete-time systems in Hilbert space, Appl. Math. and Optim., 6, 97–112 (1980). DOI: https://doi.org/10.1007/BF01442886
I. G. Rosen, C. Wang, A multilevel technique for the approximate solution of operator Lyapunov and algebraic Riccati equations, SIAM J. Numer. Anal., 32, № 2, 514–541 (1995). DOI: https://doi.org/10.1137/0732022
D. Sather, Branching of solutions of an equation in Hilbert space, Arch. Ration. Mekh. and Anal., 36, 47–64 (1970). DOI: https://doi.org/10.1007/BF00255746
Vu Ngoc Phat, Tran Tin Kiet, On the Lyapunov equation in Banach spaces and applications to control problems, Int. J. Math. and Math. Sci., 29, № 3, 155–166 (2002). DOI: https://doi.org/10.1155/S0161171202010840
Wen John Ting-Yung, J. M. Balas, Robust adaptive control in Hilbert space, J. Math. Anal. and Appl., 143, 1–26 (1989). DOI: https://doi.org/10.1016/0022-247X(89)90025-5
Davor Dragicevic, Ciprian Preda, Lyapunov theorems for exponential dichotomies in Hilbert spaces, Int. J. Math., 27, № 4 (2016). DOI: https://doi.org/10.1142/S0129167X16500336
Ciprian Preda, Petre Preda, Lyapunov operator inequalities for exponential stability of Banach space semigroups of operators, Appl. Math. Lett., 25, 401–403 (2012). DOI: https://doi.org/10.1016/j.aml.2011.09.022
M. Gil', Solution estimates for the discrete Lyapunov equation in a Hilbert space and applications to difference equations, Axioms, 8, № 1 (2019). DOI: https://doi.org/10.3390/axioms8010020
Lucas Jodar, An algorithm for solving generalized algebraic Lyapunov equations in Hilbert space, applications to boundary value problems, Proc. Edinburgh Math. Soc., 31, 99–105 (1988). DOI: https://doi.org/10.1017/S0013091500006611
Y. Latushkin, S. Montgomery-Smith, Lyapunov theorems for Banach spaces, Bull. Amer. Math. Soc., 31, № 1, 44–49 (1994). DOI: https://doi.org/10.1090/S0273-0979-1994-00495-1
P. Gahinet, M. Sorine, A. J. Laub, C. Kenney, Stability margins and Lyapunov equations for linear operators in Hilbert space, Proceedings of the 29th Conference on Decision and Control Honolulu (1990), p. 2638–2639. DOI: https://doi.org/10.1109/CDC.1990.203444
R. P. Ivanov, I. L. Raykov, Parametric Lyapunov function method for solving nonlinear systems in Hilbert spaces, Numer. Funct. Anal. and Optim., 17, 893–901 (1996). DOI: https://doi.org/10.1080/01630569608816732
A. Polyakov, On homogeneous Lyapunov function theorem for evolution equations, IFAC 2020 – International Federation of Automatic Control, 21st World Congress, Jul 2020, Berlin/Virtual, Germany (2020).
M. Gil', Stability of linear equations with differentiable operators in a Hilbert space, IMA J. Math. Control and Inform., 1–8 (2018). DOI: https://doi.org/10.1093/imamci/dny035
А. А. Бойчук, А. А. Покутний, Теория возмущений операторных уравнений в пространствах Фреше и Гильберта, Укр. мат. журн., 67, № 9, 1181–1188 (2015).
Є. В. Панасенко, О. О. Покутний, Умова біфуркації розв’язків рівняння Ляпунова у просторі Гільберта, Нелінійні коливання, 20, № 3, 373–390 (2017); English translation: J. Math. Sci., 236, № 3, 313–332 (2019).
Є. В. Панасенко, О. О. Покутний, Нелінійні крайові задачі для рівняння Ляпунова у просторі $L_{p}$, Нелінійні коливання, 21, № 4, 523–536 (2018); English translation: J. Math. Sci., 246, № 4, 394–409 (2020).
E. Deutch, Semi-inverses, reflexive semi-inverses, and pseudoinverses of an arbitrary linear transformation, Linear Algebra and Appl., 4, 313–322 (1971). DOI: https://doi.org/10.1016/0024-3795(71)90002-4
A. N. Tikhonov, V. Y. Arsenin, Solutions of ill-posed problems, John Wiley, New York (1977).
H. Harbrecht, M. Schmidlin, C. Schwab, The Gevrey class implicit mapping theorem with application to $UQ$ of semilinear elliptic PDEs}; arXiv:2310.01256 (2023). DOI: https://doi.org/10.1142/S0218202524500179
H. Harbrecht, I. Kalmykov, Sparse grid approximation of the Riccati operator for closed loop parabolic control problems with Dirichlet boundary control, SIAM J. Control and Optim., 59, № 6, 4538–4562 (2021). DOI: https://doi.org/10.1137/20M1370604
V. Kaloshin, Ke. Zhang, Arnold diffusion for smooth systems of two and a half degrees of freedom, Princeton and Oxford (2020). DOI: https://doi.org/10.23943/princeton/9780691202525.001.0001
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