-
Previous Article
Stability of the anisotropic Maxwell equations with a conductivity term
- EECT Home
- This Issue
-
Next Article
Generation of semigroups for the thermoelastic plate equation with free boundary conditions
Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data
Technische Universität Berlin, Institut für Mathematik, Straẞe des 17. Juni 136, 10623 Berlin, Germany |
For initial value problems associated with operator-valued Riccati differential equations posed in the space of Hilbert–Schmidt operators existence of solutions is studied. An existence result known for algebraic Riccati equations is generalized and used to obtain the existence of a solution to the approximation of the problem via a backward Euler scheme. Weak and strong convergence of the sequence of approximate solutions is established permitting a large class of right-hand sides and initial data.
References:
[1] |
H. W. Alt, Linear Functional Analysis, Springer, London, 2016.
doi: 10.1007/978-1-4471-7280-2. |
[2] |
U. M. Ascher, R. Mattheij and R. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, SIAM Publications, Philadelphia, PA, 2nd edition, 1995.
doi: 10.1137/1.9781611971231. |
[3] |
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leyden, 1976. |
[4] |
V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010.
doi: 10.1007/978-1-4419-5542-5. |
[5] |
P. Benner and H. Mena, BDF methods for large-scale differential Riccati equations, Proc. of Mathematical Theory of Network and Systems, MTNS, 2004. Google Scholar |
[6] |
P. Benner and H. Mena,
Numerical solution of the infinite-dimensional LQR-problem and the associated differential Riccati equations, Journal of Numerical Mathematics, 26 (2018), 1-20.
doi: 10.1515/jnma-2016-1039. |
[7] |
J. Bergh and J. Löfström, Interpolation Spaces. An introduction, Springer, Berlin, 1976. |
[8] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. |
[9] |
R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Springer, Berlin, 1978. |
[10] |
G. Da Prato, Quelques résultats d'existence, unicité et régularité pour un problème de la théorie du contrôle, Journal de Mathématiques Pures et Appliquées, 52 (1973), 353–375. |
[11] |
G. Da Prato and A. Ichikawa,
Riccati equation with unbounded coefficients, Annali di Matematica Pura ed Applicata, 140 (1985), 209-221.
doi: 10.1007/BF01776850. |
[12] |
G. Da Prato, I. Lasiecka and R. Triggiani,
A direct study of the Riccati equation arising in hyperbolic boundary control problems, Journal of Differential Equations, 64 (1986), 26-47.
doi: 10.1016/0022-0396(86)90069-0. |
[13] |
E. DiBenedetto, Real Analysis, Birkhäuser, Bosten, 2002.
doi: 10.1007/978-1-4612-0117-5. |
[14] |
N. Dunford and J. Schwartz, Linear Operators Part Ⅰ: General Theory, Interscience Publishers, New York, 1957. Google Scholar |
[15] |
N. Dunford and J. Schwartz, Linear Operators Part Ⅱ: Spectral Theory, Interscience Publishers, New York, 2nd edition, 1963. |
[16] |
F. Flandoli,
Riccati equation arising in a boundary control problem with distributed parameters, SIAM Journal on Control and Optimization, 22 (1984), 76-86.
doi: 10.1137/0322006. |
[17] |
F. Flandoli,
On the direct solution of Riccati equations arising in boundary control theory, Annali di Matematica Pura ed Applicata, 163 (1993), 93-131.
doi: 10.1007/BF01759017. |
[18] |
H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974. |
[19] |
E. Hansen and T. Stillfjord,
Convergence analysis for splitting of the abstract Riccati equation, SIAM Journal on Numerical Analysis, 52 (2014), 3128-3139.
doi: 10.1137/130935501. |
[20] |
T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1995. |
[21] |
I. Lasiecka and R. Triggiani,
Dirichlet boundary control problem for parabolic equations with quadratic cost: Analyticity and Riccati's feedback synthesis, SIAM Journal on Control and Optimization, 21 (1983), 41-67.
doi: 10.1137/0321003. |
[22] |
I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Ⅰ: Abstract Parabolic Systems, Cambridge University Press, Cambridge, 2000.
![]() |
[23] |
I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Ⅱ: Abstract Hyperbolic-like Systems over a Finite Time Horizon., Cambridge University Press, Cambridge, 2000.
doi: 10.1017/CBO9780511574801.002.![]() ![]() |
[24] |
J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971. |
[25] |
W. Reid, Riccati Differential Equation, Academic Press, New York, 1972.
![]() |
[26] |
T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2nd edition, 2013.
doi: 10.1007/978-3-0348-0513-1. |
[27] |
I. G. Rosen,
Convergence of Galerkin approximations for operator Riccati equations – A nonlinear evolution equation approach, Journal of Mathematical Analysis and Applications, 155 (1991), 226-248.
doi: 10.1016/0022-247X(91)90035-X. |
[28] |
B. Simon, Operator Theory. A Comprehensive Course in Analysis, Part 4, American Mathematical Society, Providence, RI, 2015.
doi: 10.1090/simon/004. |
[29] |
L. Tartar,
Sur l'étude directe d'équations non linéaires intervenant en théorie du contrôle optimal, Journal of Functional Analysis, 17 (1974), 1-47.
doi: 10.1016/0022-1236(74)90002-0. |
[30] |
L. Tartar, An Introduction to Navier-Stokes Equation and Oceanography, Springer, Berlin, 2006.
doi: 10.1007/3-540-36545-1. |
[31] |
R. Temam, Étude directe d'une équation d'évolution du type de Riccati, associée à des opérateurs non bornés, Comptes Rendus de l'Académie des Sciences, 268 (1969), 1335–1338. |
[32] |
R. Temam,
Sur l'équation de Riccati associeé à des opérateurs non bornés, en dimension infinie, Journal of Functional Analysis, 7 (1971), 85-115.
doi: 10.1016/0022-1236(71)90046-2. |
[33] |
K. Yosida, Functional Analysis, Springer, Berlin, 6th edition, 1980. |
show all references
References:
[1] |
H. W. Alt, Linear Functional Analysis, Springer, London, 2016.
doi: 10.1007/978-1-4471-7280-2. |
[2] |
U. M. Ascher, R. Mattheij and R. Russell, Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, SIAM Publications, Philadelphia, PA, 2nd edition, 1995.
doi: 10.1137/1.9781611971231. |
[3] |
V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Noordhoff International Publishing, Leyden, 1976. |
[4] |
V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer, New York, 2010.
doi: 10.1007/978-1-4419-5542-5. |
[5] |
P. Benner and H. Mena, BDF methods for large-scale differential Riccati equations, Proc. of Mathematical Theory of Network and Systems, MTNS, 2004. Google Scholar |
[6] |
P. Benner and H. Mena,
Numerical solution of the infinite-dimensional LQR-problem and the associated differential Riccati equations, Journal of Numerical Mathematics, 26 (2018), 1-20.
doi: 10.1515/jnma-2016-1039. |
[7] |
J. Bergh and J. Löfström, Interpolation Spaces. An introduction, Springer, Berlin, 1976. |
[8] |
H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. |
[9] |
R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Springer, Berlin, 1978. |
[10] |
G. Da Prato, Quelques résultats d'existence, unicité et régularité pour un problème de la théorie du contrôle, Journal de Mathématiques Pures et Appliquées, 52 (1973), 353–375. |
[11] |
G. Da Prato and A. Ichikawa,
Riccati equation with unbounded coefficients, Annali di Matematica Pura ed Applicata, 140 (1985), 209-221.
doi: 10.1007/BF01776850. |
[12] |
G. Da Prato, I. Lasiecka and R. Triggiani,
A direct study of the Riccati equation arising in hyperbolic boundary control problems, Journal of Differential Equations, 64 (1986), 26-47.
doi: 10.1016/0022-0396(86)90069-0. |
[13] |
E. DiBenedetto, Real Analysis, Birkhäuser, Bosten, 2002.
doi: 10.1007/978-1-4612-0117-5. |
[14] |
N. Dunford and J. Schwartz, Linear Operators Part Ⅰ: General Theory, Interscience Publishers, New York, 1957. Google Scholar |
[15] |
N. Dunford and J. Schwartz, Linear Operators Part Ⅱ: Spectral Theory, Interscience Publishers, New York, 2nd edition, 1963. |
[16] |
F. Flandoli,
Riccati equation arising in a boundary control problem with distributed parameters, SIAM Journal on Control and Optimization, 22 (1984), 76-86.
doi: 10.1137/0322006. |
[17] |
F. Flandoli,
On the direct solution of Riccati equations arising in boundary control theory, Annali di Matematica Pura ed Applicata, 163 (1993), 93-131.
doi: 10.1007/BF01759017. |
[18] |
H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974. |
[19] |
E. Hansen and T. Stillfjord,
Convergence analysis for splitting of the abstract Riccati equation, SIAM Journal on Numerical Analysis, 52 (2014), 3128-3139.
doi: 10.1137/130935501. |
[20] |
T. Kato, Perturbation Theory for Linear Operators, Springer, Berlin, 1995. |
[21] |
I. Lasiecka and R. Triggiani,
Dirichlet boundary control problem for parabolic equations with quadratic cost: Analyticity and Riccati's feedback synthesis, SIAM Journal on Control and Optimization, 21 (1983), 41-67.
doi: 10.1137/0321003. |
[22] |
I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Ⅰ: Abstract Parabolic Systems, Cambridge University Press, Cambridge, 2000.
![]() |
[23] |
I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Ⅱ: Abstract Hyperbolic-like Systems over a Finite Time Horizon., Cambridge University Press, Cambridge, 2000.
doi: 10.1017/CBO9780511574801.002.![]() ![]() |
[24] |
J. L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer, Berlin, 1971. |
[25] |
W. Reid, Riccati Differential Equation, Academic Press, New York, 1972.
![]() |
[26] |
T. Roubíček, Nonlinear Partial Differential Equations with Applications, Birkhäuser, Basel, 2nd edition, 2013.
doi: 10.1007/978-3-0348-0513-1. |
[27] |
I. G. Rosen,
Convergence of Galerkin approximations for operator Riccati equations – A nonlinear evolution equation approach, Journal of Mathematical Analysis and Applications, 155 (1991), 226-248.
doi: 10.1016/0022-247X(91)90035-X. |
[28] |
B. Simon, Operator Theory. A Comprehensive Course in Analysis, Part 4, American Mathematical Society, Providence, RI, 2015.
doi: 10.1090/simon/004. |
[29] |
L. Tartar,
Sur l'étude directe d'équations non linéaires intervenant en théorie du contrôle optimal, Journal of Functional Analysis, 17 (1974), 1-47.
doi: 10.1016/0022-1236(74)90002-0. |
[30] |
L. Tartar, An Introduction to Navier-Stokes Equation and Oceanography, Springer, Berlin, 2006.
doi: 10.1007/3-540-36545-1. |
[31] |
R. Temam, Étude directe d'une équation d'évolution du type de Riccati, associée à des opérateurs non bornés, Comptes Rendus de l'Académie des Sciences, 268 (1969), 1335–1338. |
[32] |
R. Temam,
Sur l'équation de Riccati associeé à des opérateurs non bornés, en dimension infinie, Journal of Functional Analysis, 7 (1971), 85-115.
doi: 10.1016/0022-1236(71)90046-2. |
[33] |
K. Yosida, Functional Analysis, Springer, Berlin, 6th edition, 1980. |
[1] |
Yingdan Ji, Wen Tan. Global well-posedness of a 3D Stokes-Magneto equations with fractional magnetic diffusion. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3271-3278. doi: 10.3934/dcdsb.2020227 |
[2] |
Masahiro Ikeda, Ziheng Tu, Kyouhei Wakasa. Small data blow-up of semi-linear wave equation with scattering dissipation and time-dependent mass. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021011 |
[3] |
Changpin Li, Zhiqiang Li. Asymptotic behaviors of solution to partial differential equation with Caputo–Hadamard derivative and fractional Laplacian: Hyperbolic case. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021023 |
[4] |
Haili Qiao, Aijie Cheng. A fast high order method for time fractional diffusion equation with non-smooth data. Discrete & Continuous Dynamical Systems - B, 2021 doi: 10.3934/dcdsb.2021073 |
[5] |
Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825 |
[6] |
Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409 |
[7] |
Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018 |
[8] |
Jiangxing Wang. Convergence analysis of an accurate and efficient method for nonlinear Maxwell's equations. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2429-2440. doi: 10.3934/dcdsb.2020185 |
[9] |
Marion Darbas, Jérémy Heleine, Stephanie Lohrengel. Numerical resolution by the quasi-reversibility method of a data completion problem for Maxwell's equations. Inverse Problems & Imaging, 2020, 14 (6) : 1107-1133. doi: 10.3934/ipi.2020056 |
[10] |
Lunji Song, Wenya Qi, Kaifang Liu, Qingxian Gu. A new over-penalized weak galerkin finite element method. Part Ⅱ: Elliptic interface problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2581-2598. doi: 10.3934/dcdsb.2020196 |
[11] |
Kaifang Liu, Lunji Song, Shan Zhao. A new over-penalized weak galerkin method. Part Ⅰ: Second-order elliptic problems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2411-2428. doi: 10.3934/dcdsb.2020184 |
[12] |
Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213 |
[13] |
Prasanta Kumar Barik, Ankik Kumar Giri, Rajesh Kumar. Mass-conserving weak solutions to the coagulation and collisional breakage equation with singular rates. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2021009 |
[14] |
Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175 |
[15] |
Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control & Optimization, 2021 doi: 10.3934/naco.2021008 |
[16] |
Yuan Gao, Jian-Guo Liu, Tao Luo, Yang Xiang. Revisit of the Peierls-Nabarro model for edge dislocations in Hilbert space. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3177-3207. doi: 10.3934/dcdsb.2020224 |
[17] |
J. Frédéric Bonnans, Justina Gianatti, Francisco J. Silva. On the convergence of the Sakawa-Shindo algorithm in stochastic control. Mathematical Control & Related Fields, 2016, 6 (3) : 391-406. doi: 10.3934/mcrf.2016008 |
[18] |
Fernando P. da Costa, João T. Pinto, Rafael Sasportes. On the convergence to critical scaling profiles in submonolayer deposition models. Kinetic & Related Models, 2018, 11 (6) : 1359-1376. doi: 10.3934/krm.2018053 |
[19] |
Alberto Bressan, Carlotta Donadello. On the convergence of viscous approximations after shock interactions. Discrete & Continuous Dynamical Systems, 2009, 23 (1&2) : 29-48. doi: 10.3934/dcds.2009.23.29 |
[20] |
Caifang Wang, Tie Zhou. The order of convergence for Landweber Scheme with $\alpha,\beta$-rule. Inverse Problems & Imaging, 2012, 6 (1) : 133-146. doi: 10.3934/ipi.2012.6.133 |
2019 Impact Factor: 0.953
Tools
Metrics
Other articles
by authors
[Back to Top]