Remote Access Mathematics of Computation
Green Open Access

Mathematics of Computation

ISSN 1088-6842(online) ISSN 0025-5718(print)

 
 

 

Numerical approximations of algebraic Riccati equations for abstract systems modelled by analytic semigroups, and applications


Authors: I. Lasiecka and R. Triggiani
Journal: Math. Comp. 57 (1991), 639-662, S13
MSC: Primary 47N70; Secondary 47D06, 65J10, 65L99, 93C25
DOI: https://doi.org/10.1090/S0025-5718-1991-1094953-1
MathSciNet review: 1094953
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: This paper provides a numerical approximation theory of algebraic Riccati operator equations with unbounded coefficient operators A and B, such as arise in the study of optimal quadratic cost problems over the time interval $ [0,\infty ]$ for the abstract dynamics $ \dot y = Ay + Bu$. Here, A is the generator of a strongly continuous analytic semigroup, and B is an unbounded operator with any degree of unboundedness less than that of A. Convergence results are provided for the Riccati operators, as well as for all the other relevant quantities which enter into the dynamic optimization problem. The present numerical theory is the counterpart of a known continuous theory. Several examples of partial differential equations with boundary/point control, where all the required assumptions are verified, illustrate the theory. They include parabolic equations with $ {L_2}$-Dirichlet control, as well as plate equations with a strong degree of damping and point control.


References [Enhancements On Off] (What's this?)

  • [1] H. T. Banks and K. Kunisch, The linear regulator problem for parabolic systems, SIAM J. Control Optim. 22 (1984), 684-699. MR 755137 (86h:49036)
  • [2] I. Babuška and A. Aziz, The mathematical foundations of the finite element method with applications to partial differential equations, Academic Press, New York, 1972. MR 0347104 (49:11824)
  • [3] J. H. Bramble, A. H. Schatz, V. Thomée, and L. B. Wahlbin, Some convergence estimates for semidiscrete Galerkin type approximations for parabolic equations, SIAM J. Numer. Anal. 14 (1977), 218-241. MR 0448926 (56:7231)
  • [4] S. Chen and R. Triggiani, Proof of two conjectures of G. Chen and D. L. Russell on structural damping for elastic systems: The case $ \alpha = 1/2$, Proc. Seminar in Approximation and Optimization (University of Havana, Cuba, January 12-14, 1987), Lecture Notes in Math., vol. 1354, Springer-Verlag, 1988, pp. 234-256. MR 996678 (90j:34088)
  • [5] -, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math. 136 (1989), 15-55. MR 971932 (90g:47071)
  • [6] -, Gevrey class semigroups arising from elastic systems with gentle dissipation: The case $ 0 < \alpha < 1/2$, Proc. Amer. Math. Soc. 110 (1990), 401-415. MR 1021208 (90m:47054)
  • [7] -, Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications, J. Differential Equations 88 (1990), 279-293. MR 1081250 (91m:47054)
  • [8] G. DaPrato and A. Ichikawa, Riccati equations with unbounded coefficients, Ann. Mat. Pura Appl. 140 (1985), 209-221. MR 807638 (87b:49052)
  • [9] F. Flandoli, Algebraic Riccati equations arising in boundary control problems, SIAM J. Control Optim. 25 (1987), 612-636. MR 885189 (88i:49006)
  • [10] -, Riccati equation arising in a boundary control problem with distributed parameters, SIAM J. Control Optim. 22 (1984), 76-86. MR 728673 (85b:49006)
  • [11] J. S. Gibson, The Riccati integral equations for optimal control problems on Hilbert spaces, SIAM J. Control Optim. 17 (1979), 537-565. MR 534423 (81b:93029)
  • [12] J. Gardiner and A. Laub, Matrix sign function implementations on a hypercube multiprocessor, Proc. CDC Conf., December 1988, pp. 1466-1471.
  • [13] T. Kato, Perturbation theory of linear operators, Springer-Verlag, New York, 1966. MR 0203473 (34:3324)
  • [14] I. Lasiecka, Convergence estimates for semidiscrete approximations of nonselfadjoint parabolic equations, SIAM J. Numer. Anal. 21 (1984). MR 760624 (85m:65100)
  • [15] -, Galerkin approximations of abstract parabolic boundary value problems with rough boundary data; $ {L_p}$ theory, Math. Comp. 47 (1986), 55-75. MR 842123 (87i:65187)
  • [16] J. Lagnese, Boundary stabilization of thin plates, SIAM Stud. Appl. Math., vol. 10, SIAM, Philadelphia, PA, 1989. MR 1061153 (91k:73001)
  • [17] I. Lasiecka and R. Triggiani, The regulator problem for parabolic equations with Dirichlet boundary control. I, Appl. Math. Optim. 16 (1987), 147-168. MR 894809 (88g:93063a)
  • [18] -, The regulator problem for parabolic equations with Dirichlet boundary control. II, Appl. Math. Optim. 16 (1987), 187-216. MR 901813 (88g:93063b)
  • [19] -, Algebraic Riccati equations with applications to boundary/point control problems: Continuous theory and approximation theory, Perspectives in Control Theory (B. Jakubczyk, K. Malanowski, and W. Respondek, eds.), Birkhäuser, 1990, pp. 175-235; Expanded version to appear as a volume in Springer Verlag Lecture Notes in Control and Information Sciences.
  • [20] -, Dirichlet boundary control problem for parabolic equations with quadratic cost: Analyticity and Riccati's feedback synthesis, SIAM J. Control Optim. 21 (1983), 41-67. MR 688439 (84h:93039)
  • [21] A. Manitius and R. Triggiani, Function space controllability of linear retarded systems: A derivation from abstract operator conditions, SIAM J. Control Optim. 16 (1978), 599-645. MR 0482505 (58:2571)
  • [22] A. Pazy, Semigroups of operators and applications to partial differential equations, Springer-Verlag, 1983. MR 710486 (85g:47061)
  • [23] C. Sadosky, Interpolation of operators and singular integrals, Marcel Dekker, New York, 1979. MR 551747 (81d:42001)
  • [24] V. Thomée, Galerkin finite element methods for parabolic problems, Lecture Notes in Math., vol. 1054, Springer, Berlin, 1984.
  • [25] R. Triggiani, On the stabilizability problem in Banach space, J. Math. Anal. Appl. 52 (1975), 383-403. MR 0445388 (56:3730)
  • [26] -, Regularity of structurally damped systems with point/boundary control, preprint 1989; J. Math. Anal. Appl. (to appear).
  • [27] -, Boundary feedback stabilization of parabolic equations, Appl. Math. Optim. 6 (1980), 201-220. MR 576260 (81m:35020)
  • [28] H. Triebel, Interpolation theory, function spaces, differential operators, VEB, North-Holland, 1978. MR 503903 (80i:46032b)

Similar Articles

Retrieve articles in Mathematics of Computation with MSC: 47N70, 47D06, 65J10, 65L99, 93C25

Retrieve articles in all journals with MSC: 47N70, 47D06, 65J10, 65L99, 93C25


Additional Information

DOI: https://doi.org/10.1090/S0025-5718-1991-1094953-1
Article copyright: © Copyright 1991 American Mathematical Society

American Mathematical Society