Abstract

We consider operator-valued differential Lyapunov and Riccati equations, where the operators $B$ and $C$ may be relatively unbounded with respect to $A$ (in the standard notation). In this setting, we prove that the singular values of the solutions decay fast under certain conditions. In fact, the decay is exponential in the negative square root if $A$ generates an analytic semigroup and the range of $C$ has finite dimension. This extends previous similar results for algebraic equations to the differential case. When the initial condition is zero, we also show that the singular values converge to zero as time goes to zero, with a certain rate that depends on the degree of unboundedness of $C$. A fast decay of the singular values corresponds to a low numerical rank, which is a critical feature in large-scale applications. The results reported here provide a theoretical foundation for the observation that, in practice, a low-rank factorization usually exists.

Keywords

  1. differential Riccati equations
  2. differential Lyapunov equations
  3. operator-valued
  4. infinite dimensional
  5. singular value decay
  6. low rank

MSC codes

  1. 47A62
  2. 47A11
  3. 49N10

Get full access to this article

View all available purchase options and get full access to this article.

References

1.
H. Abou-Kandil, G. Freiling, V. Ionescu, and G. Jank, Matrix Riccati Equations in Control and Systems Theory, Birkhäuser, Basel, Switzerland, 2003.
2.
A. C. Antoulas, D. C. Sorensen, and Y. Zhou, On the decay rate of Hankel singular values and related issues, Systems Control Lett., 46 (2002), pp. 323--342, https://doi.org/10.1016/S0167-6911(02)00147-0.
3.
T. Başar and P. Bernhard, $H^{\infty}$-Optimal Control and Related Minimax Design Problems: A Dynamic Game Approach, Systems Control Found. Appl., 2nd ed., Birkhäuser, Boston, MA, 1995, https://doi.org/10.1007/978-0-8176-4757-5.
4.
J. Baker, M. Embree, and J. Sabino, Fast singular value decay for Lyapunov solutions with nonnormal coefficients, SIAM J. Matrix Anal. Appl., 36 (2015), pp. 656--668, https://doi.org/10.1137/140993867.
5.
U. Baur, P. Benner, and L. Feng, Model order reduction for linear and nonlinear systems: A system-theoretic perspective, Arch. Comput. Methods Eng., 21 (2014), pp. 331--358, https://doi.org/10.1007/s11831-014-9111-2.
6.
P. Benner and T. Breiten, Low rank methods for a class of generalized Lyapunov equations and related issues, Numer. Math., 124 (2013), pp. 441--470, https://doi.org/10.1007/s00211-013-0521-0.
7.
P. Benner and Z. Bujanović, On the solution of large-scale algebraic Riccati equations by using low-dimensional invariant subspaces, Linear Algebra Appl., 488 (2016), pp. 430--459, https://doi.org/10.1016/j.laa.2015.09.027.
8.
P. Benner, P. Kürschner, and J. Saak, Frequency-limited balanced truncation with low-rank approximations, SIAM J. Sci. Comput., 38 (2016), pp. A471--A499, https://doi.org/10.1137/15M1030911.
9.
P. Benner, J.-R. Li, and T. Penzl, Numerical solution of large-scale Lyapunov equations, Riccati equations, and linear-quadratic optimal control problems, Numer. Linear Algebra Appl., 15 (2008), pp. 755--777, https://doi.org/10.1002/nla.622.
10.
A. Bensoussan, G. Da Prato, M. C. Delfour, and S. K. Mitter, Representation and Control of Infinite Dimensional Systems, Systems Control Found. Appl., 2nd ed., Birkhäuser, Boston, MA, 2007.
11.
S. C. Brenner and L. R. Scott, The Mathematical Theory of Finite Element Methods, Texts Appl. Math. 15, 3rd ed., Springer, New York, 2008, https://doi.org/10.1007/978-0-387-75934-0.
12.
R. Courant and D. Hilbert, Methods of Mathematical Physics. Vol. I, Interscience, New York, 1953.
13.
R. F. Curtain and A. J. Sasane, Compactness and nuclearity of the Hankel operator and internal stability of infinite-dimensional state linear systems, Internat. J. Control, 74 (2001), pp. 1260--1270, https://doi.org/10.1080/00207170110061059.
14.
K. Fan, Maximum properties and inequalities for the eigenvalues of completely continuous operators, Proc. Natl. Acad. Sci. USA, 37 (1951), pp. 760--766.
15.
W. Gawronski and J.-N. Juang, Model reduction in limited time and frequency intervals, Internat. J. Systems Sci., 21 (1990), pp. 349--376, https://doi.org/10.1080/00207729008910366.
16.
L. Grubišić and D. Kressner, On the eigenvalue decay of solutions to operator Lyapunov equations, Systems Control Lett., 73 (2014), pp. 42--47, https://doi.org/10.1016/j.sysconle.2014.09.006.
17.
F. Hecht, New development in freefem++, J. Numer. Math., 20 (2012), pp. 251--265.
18.
A. Ichikawa and H. Katayama, Remarks on the time-varying $H\sb \infty$ Riccati equations, Systems Control Lett., 37 (1999), pp. 335--345.
19.
P. Kürschner, Balanced truncation model order reduction in limited time intervals for large systems, Adv. Comput. Math., to appear.
20.
N. Lang, H. Mena, and J. Saak, On the benefits of the $LDL^T$ factorization for large-scale differential matrix equation solvers, Linear Algebra Appl., 480 (2015), pp. 44--71, https://doi.org/10.1016/j.laa.2015.04.006.
21.
I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories I. Abstract Parabolic Systems, Cambridge University Press, Cambridge, 2000.
22.
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, in Encyclopedia Math. Appl. 75, Cambridge University Press, Cambridge, 2000, pp. 645--1067.
23.
J.-R. Li and J. White, Low rank solution of Lyapunov equations, SIAM J. Matrix Anal. Appl., 24 (2002), pp. 260--280, https://doi.org/10.1137/S0895479801384937.
24.
J.-L. Lions and E. Magenes, Non-homogeneous Boundary Value Problems and Applications. Vol. I, Grundlehren Math. Wiss. 181, Springer, New York, 1972.
25.
J. Lund and K. L. Bowers, Sinc Methods for Quadrature and Differential Equations, Other Titles Appl. Math., SIAM, Philadelphia, 1992, https://doi.org/10.1137/1.9781611971637.
26.
A. M\aalqvist, A. Persson, and T. Stillfjord, Multiscale differential Riccati equations for linear quadratic regulator problems, SIAM J. Sci. Comput., 40 (2018), pp. A2406--A2426.
27.
K. M. Mikkola, Infinite-Dimensional Linear Systems, Optimal Control and Algebraic Riccati Equations, Dissertation, Helsinki University of Technology, Helsinki, Finland, 2002, http://lib.tkk.fi/Diss/2002/isbn9512260794/.
28.
L. S. D. Morley, The triangular equilibrium element in the solution of plate bending problems, Aeronaut. Quart., 19 (1968), pp. 149--169.
29.
M. Opmeer, Decay of singular values of the Gramians of infinite-dimensional systems, in Proceedings 2015 European Control Conference (ECC), Linz, Austria, IEEE, Piscataway, NJ, 2015, pp. 1183--1188, https://doi.org/10.1109/ECC.2015.7330700.
30.
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Appl. Math. Sci. 44, Springer, New York, 1983.
31.
T. Penzl, Eigenvalue decay bounds for solutions of Lyapunov equations: The symmetric case, Systems Contol Lett., 40 (2000), pp. 139--144, https://doi.org/10.1016/S0167-6911(00)00010-4.
32.
I. R. Petersen, V. A. Ugrinovskii, and A. V. Savkin, Robust Control Design Using $H^{\infty}$ Methods, Springer, London, 2000.
33.
D. Salamon, Infinite-dimensional linear systems with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), pp. 383--431, https://doi.org/10.2307/2000351.
34.
D. C. Sorensen and Y. Zhou, Bounds on Eigenvalue Decay Rates and Sensitivity of Solutions to Lyapunov Equations, Technical report TR02-07, Rice University, Houston, TX, 2002, https://scholarship.rice.edu/handle/1911/101987.
35.
O. Staffans, Well-Posed Linear Systems, Encyclopedia Math. Appl. 103, Cambridge University Press, Cambridge, 2005, https://doi.org/10.1017/CBO9780511543197.
36.
F. Stenger, Integration Formulae Based on the Trapezoidal Formula, J. Inst. Math. Appl., 12 (1973), pp. 103--114.
37.
F. Stenger, Numerical Methods Based on Sinc and Analytic Functions, Springer Ser. Comput. Math. 20, Springer, New York, 1993, https://doi.org/10.1007/978-1-4612-2706-9.
38.
T. Stillfjord, Low-rank second-order splitting of large-scale differential Riccati equations, IEEE Trans. Automat. Control, 60 (2015), pp. 2791--2796, https://doi.org/10.1109/TAC.2015.2398889.
39.
T. Stillfjord, Adaptive high-order splitting schemes for large-scale differential Riccati equations, Numer. Algorithms, 78 (2017), pp. 1129--1151, https://doi.org/10.1007/s11075-017-0416-8.
40.
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Adv. Texts Basler Lehrbücher, Birkhäuser, Basel, 2009, https://doi.org/10.1007/978-3-7643-8994-9.

Information & Authors

Information

Published In

cover image SIAM Journal on Control and Optimization
SIAM Journal on Control and Optimization
Pages: 3598 - 3618
ISSN (online): 1095-7138

History

Submitted: 3 April 2018
Accepted: 31 July 2018
Published online: 9 October 2018

Keywords

  1. differential Riccati equations
  2. differential Lyapunov equations
  3. operator-valued
  4. infinite dimensional
  5. singular value decay
  6. low rank

MSC codes

  1. 47A62
  2. 47A11
  3. 49N10

Authors

Affiliations

Metrics & Citations

Metrics

Citations

If you have the appropriate software installed, you can download article citation data to the citation manager of your choice. Simply select your manager software from the list below and click Download.

Cited By

View Options

View options

PDF

View PDF

Media

Figures

Other

Tables

Share

Share

Copy the content Link

Share with email

Email a colleague

Share on social media