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June  2019, 8(2): 315-342. doi: 10.3934/eect.2019017

Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data

Monika Eisenmann , Etienne Emmrich , and  Volker Mehrmann

Technische Universität Berlin, Institut für Mathematik, Straẞe des 17. Juni 136, 10623 Berlin, Germany

* Corresponding author

Received  May 2018 Revised  August 2018 Published  June 2019 Early access  March 2019

Fund Project: The authors gratefully acknowledge financial support from the Deutsche Forschungsgemeinschaft through the Collaborative Research Center 901 "Control of self-organizing nonlinear systems: Theoretical methods and concepts of application" (projects A2, A8).

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.

Keywords: Riccati differential equation, Hilbert-Schmidt operator, semi-definite data, weak solution, well-posedness, backward Euler method, convergence.
Mathematics Subject Classification: Primary: 47J35, 35K58, 65J15, 65M12; Secondary: 34H05, 49J99.
Citation: Monika Eisenmann, Etienne Emmrich, Volker Mehrmann. Convergence of the backward Euler scheme for the operator-valued Riccati differential equation with semi-definite data. Evolution Equations and Control Theory, 2019, 8 (2) : 315-342. doi: 10.3934/eect.2019017
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.

[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.

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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.

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N. Dunford and J. Schwartz, Linear Operators Part Ⅰ: General Theory, Interscience Publishers, New York, 1957.

[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.

[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.

[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.

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