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

Lasiecka, I.; Triggiani, R.

doi:10.1090/S0025-5718-1991-1094953-1

Numerical approximations of algebraic Riccati equations for abstract systems modelled by analytic semigroups, and applications
HTML articles powered by AMS MathViewer

by I. Lasiecka and R. Triggiani PDF

Math. Comp. 57 (1991), 639-662 Request permission

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

H. T. Banks and K. Kunisch, The linear regulator problem for parabolic systems, SIAM J. Control Optim. 22 (1984), no. 5, 684–698. MR 755137, DOI 10.1137/0322043
A. K. Aziz (ed.), The mathematical foundations of the finite element method with applications to partial differential equations, Academic Press, New York-London, 1972. MR 0347104
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), no. 2, 218–241. MR 448926, DOI 10.1137/0714015
Shu Ping Chen and Roberto Triggiani, Proof of two conjectures by G. Chen and D. L. Russell on structural damping for elastic systems, Approximation and optimization (Havana, 1987) Lecture Notes in Math., vol. 1354, Springer, Berlin, 1988, pp. 234–256. MR 996678, DOI 10.1007/BFb0089601
Shu Ping Chen and Roberto Triggiani, Proof of extensions of two conjectures on structural damping for elastic systems, Pacific J. Math. 136 (1989), no. 1, 15–55. MR 971932
Shu Ping Chen and Roberto Triggiani, Gevrey class semigroups arising from elastic systems with gentle dissipation: the case $0<\alpha <\frac 12$, Proc. Amer. Math. Soc. 110 (1990), no. 2, 401–415. MR 1021208, DOI 10.1090/S0002-9939-1990-1021208-4
Shu Ping Chen and Roberto Triggiani, Characterization of domains of fractional powers of certain operators arising in elastic systems, and applications, J. Differential Equations 88 (1990), no. 2, 279–293. MR 1081250, DOI 10.1016/0022-0396(90)90100-4
G. Da Prato and A. Ichikawa, Riccati equations with unbounded coefficients, Ann. Mat. Pura Appl. (4) 140 (1985), 209–221. MR 807638, DOI 10.1007/BF01776850
Franco Flandoli, Algebraic Riccati equation arising in boundary control problems, SIAM J. Control Optim. 25 (1987), no. 3, 612–636. MR 885189, DOI 10.1137/0325035
Franco Flandoli, Riccati equation arising in a boundary control problem with distributed parameters, SIAM J. Control Optim. 22 (1984), no. 1, 76–86. MR 728673, DOI 10.1137/0322006
J. S. Gibson, The Riccati integral equations for optimal control problems on Hilbert spaces, SIAM J. Control Optim. 17 (1979), no. 4, 537–565. MR 534423, DOI 10.1137/0317039

Matrix sign function implementations on a hypercube multiprocessor

Tosio Kato, Perturbation theory for linear operators, Die Grundlehren der mathematischen Wissenschaften, Band 132, Springer-Verlag New York, Inc., New York, 1966. MR 0203473
Irena Lasiecka, Convergence estimates for semidiscrete approximations of nonselfadjoint parabolic equations, SIAM J. Numer. Anal. 21 (1984), no. 5, 894–909. MR 760624, DOI 10.1137/0721058
I. Lasiecka, Galerkin approximations of abstract parabolic boundary value problems with rough boundary data—$L_p$ theory, Math. Comp. 47 (1986), no. 175, 55–75. MR 842123, DOI 10.1090/S0025-5718-1986-0842123-X
John E. Lagnese, Boundary stabilization of thin plates, SIAM Studies in Applied Mathematics, vol. 10, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 1989. MR 1061153, DOI 10.1137/1.9781611970821
I. Lasiecka and R. Triggiani, The regulator problem for parabolic equations with Dirichlet boundary control. I. Riccati’s feedback synthesis and regularity of optimal solution, Appl. Math. Optim. 16 (1987), no. 2, 147–168. MR 894809, DOI 10.1007/BF01442189
I. Lasiecka and R. Triggiani, The regulator problem for parabolic equations with Dirichlet boundary control. I. Riccati’s feedback synthesis and regularity of optimal solution, Appl. Math. Optim. 16 (1987), no. 2, 147–168. MR 894809, DOI 10.1007/BF01442189

Algebraic Riccati equations with applications to boundary/point control problems: Continuous theory and approximation theory

I. Lasiecka and R. Triggiani, Dirichlet boundary control problem for parabolic equations with quadratic cost: analyticity and Riccati’s feedback synthesis, SIAM J. Control Optim. 21 (1983), no. 1, 41–67. MR 688439, DOI 10.1137/0321003
A. Manitius and R. Triggiani, Function space controllability of linear retarded systems: a derivation from abstract operator conditions, SIAM J. Control Optim. 16 (1978), no. 4, 599–645. MR 482505, DOI 10.1137/0316041
A. Pazy, Semigroups of linear operators and applications to partial differential equations, Applied Mathematical Sciences, vol. 44, Springer-Verlag, New York, 1983. MR 710486, DOI 10.1007/978-1-4612-5561-1
Cora Sadosky, Interpolation of operators and singular integrals, Monographs and Textbooks in Pure and Applied Mathematics, vol. 53, Marcel Dekker, Inc., New York, 1979. An introduction to harmonic analysis. MR 551747

Galerkin finite element methods for parabolic problems

Roberto Triggiani, On the stabilizability problem in Banach space, J. Math. Anal. Appl. 52 (1975), no. 3, 383–403. MR 445388, DOI 10.1016/0022-247X(75)90067-0

Regularity of structurally damped systems with point/boundary control

Roberto Triggiani, Boundary feedback stabilizability of parabolic equations, Appl. Math. Optim. 6 (1980), no. 3, 201–220. MR 576260, DOI 10.1007/BF01442895
H. Triebel, Interpolation theory, function spaces, differential operators, VEB Deutscher Verlag der Wissenschaften, Berlin, 1978. MR 500580