# BOUNDARY VALUE PROBLEMS FOR THE LYAPUNOV EQUATION

### Abstract

The boundary value problems for the Lyapunov equation in the resonant (irregular) case in Banach and Hilbert spaces, when the solution of the equation does not exist for all right-hand sides and its uniqueness may be violated, have been investigated. The conditions for bifurcation and branching of solutions in linear and nonlinear cases, including with a moving right end of the segment on which the corresponding boundary value problem is considered, are found.

### References

Daletsky Yu. L., Krein M. G. Stability of solutions of differential equations in a Banach space. M.: Nauka. 1970. 534 p.

Bamieh B., Dahleh M. Energy amplification in channel flows with stochastic excitation. Physics of Fluids. 2001. 13. P. 3258–3269.

Bhatia R. A note on the Lyapunov equation. Linear algebra and its applications. 1997. 259. p. 71–76.

Krein M. G. Stability of solutions of differential equations in a Banach space. M.: Nauka, 1970. 534 p.

Bryson A., Ho Yu. Applied Optimal Control Theory. M.: Mir, 1972. 544 p.

Boichuk A. A., Krivosheya S. A. A critical periodic boundary-value problem for a matrix Riccati equation. Differential Equations.2001. Vol.37, No 4. P. 464–471.

Boichuk A. A., Zhuravlev V. F., Samoilenko A. M. Generalized inverse operators and Noetherian boundary value problems. K.: Inst. math. NASU, 1995. 320 p.

Krein S. G. Linear equations in a Banach space. M.: Nauka, 1971.104 p.

Krein S. G. Linear differential equations in a Banach space. M.: Nauka, 1967. 464 p.

Panasenko E. V., Pokutnyi O. O. Boundary value problems for differential equations in Banach space with an unbounded operator in the linear part. Non-linear oscillations. 2013. Vol. 16, No. 4. P. 518–526.

Pokutnyi O. O. Generalized inverse operator in Frechet, Banach, and Hilbert spaces. Bulletin of Taras Shevchenko Kyiv National University, Series: physical and mathematical sciences. 2013, No. 4. P. 158–161.

Trenogin V. A. Functional analysis. M.: Nauka, 1980. 496 p.

Arnold V. I. Catastrophe theory. M.: Nauka, 1990. 129 p.

Vainberg M. M., Trenogin V. A. Branching theory of solutions of nonlinear equations. M.: Nauka, 1969. 527 p.

Naife A. Perturbation methods. M.: Mir, 1976. 454 p.

Chuiko S. M. The generalized Green's operator of a Noetherian boundary value problem for a matrix differential equation. News of higher educational institutions. Mathematics. 2016. No 8. P. 74–83.

Boichuk A. A., Samoilenko A. M. Generalized Inverse Operators and Fredholm Boundary-Value Problems. II edition, De Gruyter. 2016. 296 p.

Boichuk O. A., Krivosheya S. A. Criterion for the solvability of matrix equations of the Lyapunov type. Ukr. Math. J.1998. 50, No. 8. P. 1162–1169.

Bondarev A. N., Laptinskii V. N. Multipoint boundary value problem for the Lyapunov equation in the case of strong degeneration of the boundary conditions. Differential Equations. 2011. Vol. 47, No 6. P. 778–786.

Chuiko S. M. On the solution of matrix Lyapunov equations. Visn. Kharkiv. Univ., Ser. Mat. Prikl. Mat. Mekh. 2014. No. 1120. P. 85–94.

Datko R. Extending a theorem of a A. M. Lyapunov to Hilbert space. Journal of mathematical analysis and applications.1970. 32. P. 610–616.

Druskin V., Knizhnerman L., Simoncini V. Analysis of the rational Krylov subspace and ADI methods for solving the Lyapunov equation. SIAM J. Numer. Anal. 2011. Vol. 49, No. 5. P. 1875–1898.

Duncana T. E., Maslowski B., Pasik-Duncana B. Stochastic equations in Hilbert space with a multiplicative fractional Gaussian noise. Stochastic Processes and their Applications. 2005. 115. P. 1357–1383.

Kielhofer H. On the Lyapunov-Stability of Stationary Solutions of Semilinear Parabolic Differential Equations. Journal of diff. equations. 22. 1976. P. 193–208.

Man’ko V. I., Vilela Mendes R. Lyapunov exponent in quantum mechanics. Aphase-space approach. Physica D.2000. 145. P. 330–348.

Panasenko E. V., Pokutnyi O. O. Boundary-Value Problems for the Lyapunov Equation in Banach Spaces. J. Math. Sci.2017. v. 223. P. 1–7.

Panasenko E. V., Pokutnyi O. O. Boundary-value problems for differential equations in a Banach space with unbounded operator in the linear part. J. Math. Sci. 2014. 203, No. 3. P. 366–374.

Pazy A. On the applicability of Lyapunov’s theorem in Hilbert space. SIAM J. Math. Anal.1972. Vol. 3, No. 2. P. 291–294.

Pokutnyi O. O. Generalized inverse operator in Frechet, Banach, and Hilbert spaces. Visn. Kyiv. Nats. Univ. Ser. Fiz. Mat. Nauk. 2013. No. 4. p. 158–161.

Maci Przyluski K. The Lyapunov Equation and the Problem of Stability for Li-near Bounded Discrete-Time Systems in Hilbert Space. Applied mathematics and optimization. 1980. 6. P. 97–112.

Rosen I. G. and Wang C. A multilevel technique for the approximate solution of operator Lyapunov and algebraic Riccati equations. SIAM J. Numer. Anal. 1995. Vol. 32, No. 2. P. 514–541.

Sather D. Branching of Solutions of an Equation in Hilbert Space. Arch. Rational M. Anal.1970. Vol. 36. P. 47–64.

Vu N. P., Tran T. K. On the Lyapunov equation in Banach spaces and applications to control problems. IJMMS. 2002. 29:3. P. 155–166.

Wen John Ting-Yung, Balas Mark J. Robust Adaptive Control in Hilbert Space. Journal of mathematical analysis and applications. 1989. 143. P. 1–26.

Dragicevic D., Preda C. Lyapunov theorems for exponential dichotomies in Hilbert spaces. International journal of mathematics. 2016. Vol. 27, No. 4. P. 1–12.

Preda C., Preda P. Lyapunov operator inequalities for exponential stability of Banach space semigroups of operators. Appl. math. letters. 2012. 25. P. 401–403.

Gil’ M. Solution estimates for the discrete Lyapunov equation in a Hilbert space and applications to difference equations. Axioms. 2019. 8, 20. 22 pages.

Lucas J. An algorithm for solving generalized algebraic Lyapunov equations in Hilbert space, applications to boundary value problems. Proceedings of the Edinburgh mathematical society. 1988. 31. P. 99–105.

Latushkin Y., Montgomery-Smith S. Lyapunov theorems for Banach spaces. Bulletin of the American mathematical society. 1994. Vol.31, No. 1. P. 44–49.

Gahinet P., Sorine M., Laub A.J., Kenney C. 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.

Maciej Przyluski K. The Lyapunov equation and the problem of stability for linear bounded discrete-time systems in Hilbert space. Appl. Math. Optim. 1980. 6. P. 97–112.

Ivanov R. P., Raykov I. L. Parametric Lyapunov function method for solving nonlinear systems in Hilbert spaces. Numer. Funct. Anal. and Optimiz. 1996. 17,(9, 10). P. 893–901.

Polyakov A. On homogeneous Lyapunov function theorem for evolution equations. IFAC 2020 International federation of automatic control, 21st world congress. Jul 2020, Berlin / Virtual, Germany.

Gil’ M. Stability of linear equations with differentiable operators in a Hilbert space. IMA journal of mathematical control and information. 2018. P. 1–8.

Boichuk A. A., Pokutnyi A. A. Perturbation theory of operator equations in Fréchet and Hilbert spaces. Ukr. mat. journal 2015. No 9. P. 1181–1188.

Panasenko E. V., Pokutnyi O.O. The bifurcation condition of solutions of the Lyapunov equation in Hilbert space. Non-linear oscillations. 2017. 20, No. 3. P. 373–390.

Panasenko E. V., Pokutnyi O. O. Nonlinear boundary value problems for the Lyapunov equation in the space L_p. Non-linear oscillations. 2018. 21, No. 4. P. 523–536.

Deutch E. Semi-inverses, reflexive semi-inverses, and pseudoinverses of an arbitrary linear transformation. Linear algebra and its applications. 1971. 4. P. 313–322.

Tikhonov A. N., Arsenin V. Ya. Methods for solving ill-posed problems. M.: Nauka. 1979. 285 p.