Quadratic functionals: positivity, oscillation, Rayleigh's principle

Werner Kratz

Archivum Mathematicum (1998)

  • Volume: 034, Issue: 1, page 143-151
  • ISSN: 0044-8753

Abstract

top
In this paper we give a survey on the theory of quadratic functionals. Particularly the relationships between positive definiteness and the asymptotic behaviour of Riccati matrix differential equations, and between the oscillation properties of linear Hamiltonian systems and Rayleigh’s principle are demonstrated. Moreover, the main tools form control theory (as e.g. characterization of strong observability), from the calculus of variations (as e.g. field theory and Picone’s identity), and from matrix analysis (as e.g. l’Hospital’s rule for matrices) are discussed.

How to cite

top

Kratz, Werner. "Quadratic functionals: positivity, oscillation, Rayleigh's principle." Archivum Mathematicum 034.1 (1998): 143-151. <http://eudml.org/doc/248207>.

@article{Kratz1998,
abstract = {In this paper we give a survey on the theory of quadratic functionals. Particularly the relationships between positive definiteness and the asymptotic behaviour of Riccati matrix differential equations, and between the oscillation properties of linear Hamiltonian systems and Rayleigh’s principle are demonstrated. Moreover, the main tools form control theory (as e.g. characterization of strong observability), from the calculus of variations (as e.g. field theory and Picone’s identity), and from matrix analysis (as e.g. l’Hospital’s rule for matrices) are discussed.},
author = {Kratz, Werner},
journal = {Archivum Mathematicum},
keywords = {Quadratic functional; Hamiltonian system; Riccati equation; oscillation; observability; Rayleigh’s principle; eigenvalue problem; linear control system; quadratic functional; Hamiltonian system; Riccati equation; oscillation; Rayleigh's principle; eigenvalue problem; linear control system; observability},
language = {eng},
number = {1},
pages = {143-151},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Quadratic functionals: positivity, oscillation, Rayleigh's principle},
url = {http://eudml.org/doc/248207},
volume = {034},
year = {1998},
}

TY - JOUR
AU - Kratz, Werner
TI - Quadratic functionals: positivity, oscillation, Rayleigh's principle
JO - Archivum Mathematicum
PY - 1998
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 034
IS - 1
SP - 143
EP - 151
AB - In this paper we give a survey on the theory of quadratic functionals. Particularly the relationships between positive definiteness and the asymptotic behaviour of Riccati matrix differential equations, and between the oscillation properties of linear Hamiltonian systems and Rayleigh’s principle are demonstrated. Moreover, the main tools form control theory (as e.g. characterization of strong observability), from the calculus of variations (as e.g. field theory and Picone’s identity), and from matrix analysis (as e.g. l’Hospital’s rule for matrices) are discussed.
LA - eng
KW - Quadratic functional; Hamiltonian system; Riccati equation; oscillation; observability; Rayleigh’s principle; eigenvalue problem; linear control system; quadratic functional; Hamiltonian system; Riccati equation; oscillation; Rayleigh's principle; eigenvalue problem; linear control system; observability
UR - http://eudml.org/doc/248207
ER -

References

top
  1. C. D. Ahlbrandt, Equivalence of discrete Euler equations and discrete Hamiltonian systems, J. Math. Anal. Appl., 180 (1993), 498–517. (1993) Zbl0802.39005MR1251872
  2. C. D. Ahlbrandt S. L. Clark J. W. Hooker W. T. Patula, A discrete interpretation of Reid’s roundabout theorem for generalized differential systems, In: Computers and Mathematics with Applications, Comput. Math. Appl., 28 (1994), 11–21. (1994) MR1284216
  3. H. W. Alt, Lineare Funktionalanalysis, Springer Verlag, Berlin, 1985. (1985) Zbl0577.46001
  4. M. Bohner, Linear Hamiltonian difference systems: Disconjugacy and Jacobi-type conditions, J. Math. Anal. Appl., 199 (1996), 804–826. (199) MR1386607
  5. Z. Došlá O. Došlý, Quadratic functionals with general boundary conditions, Appl. Math. Optim., 36 (1997), 243–262. (1997) MR1457870
  6. J. Klamka, Controllability of dynamical systems, Kluwer, Dordrecht, 1991. (1991) Zbl0732.93008MR1134783
  7. W. Kratz, A substitute of l’Hospital’s rule for matrices, Proc. Amer. Math. Soc., 99 (1987), 395–402. (1987) Zbl0629.15014MR0875370
  8. W. Kratz, A limit theorem for monotone matrix functions, Linear Algebra Appl., 194 (1993), 205–222. (194) MR1243829
  9. W. Kratz, The asymptotic behaviour of Riccati matrix differential equations, Asymptotic Anal., 7 (1993), 67–80. (1993) Zbl0774.34034MR1216453
  10. W. Kratz, On the optimal linear regulator, Intern. J. Control, 60 (1994), 1005–1013. (1994) Zbl0836.49017MR1301896
  11. W. Kratz, An index theorem for monotone matrix-valued functions, SIAM J. Matrix Anal. Appl., 16 (1995), 113–122. (1995) Zbl0820.47012MR1311421
  12. W. Kratz, Characterization of strong observability and construction of an observer, Linear Algebra Appl., 221 (1995), 31–40. (1995) Zbl0821.93017MR1331787
  13. W. Kratz, Quadratic functionals in variational analysis and control theory, Akademie Verlag, Berlin, 1995. (1995) Zbl0842.49001MR1334092
  14. W. Kratz D. Liebscher, A local characterization of observability, Linear Algebra Appl., (1998), to appear. (1998) MR1483524
  15. W. Kratz D. Liebscher R. Schätzle, On the definiteness of quadratic functionals, Ann. Mat. Pura Appl. (4), (1998), to appear. (1998) MR1746539
  16. M. Morse, Variational analysis: Critical extremals and Sturmian theory, Wiley, New York, 1973. (1973) 
  17. M. Picone, Sulle autosoluzioni e sulle formule di maggiorazione per gli integrali delle equazioni differenziali lineari ordinarie autoaggiunte, Math. Z., 28 (1928), 519–555. (1928) MR1544975
  18. W. T. Reid, Ordinary differential equations, Wiley, New York, 1971. (1971) Zbl0212.10901MR0273082
  19. W. T. Reid, Riccati differential equations, Academic Press, London, 1972. (1972) Zbl0254.34003MR0357936
  20. J. Simon, Compact sets in the space L p ( 0 , t ; B ) , Annali di Matematica Pura ed Applicata, 146 (1987), 65–96. (1987) MR0916688
  21. S. A. Swanson, Picone’s identity, Rend. Math., 8 (1975), 373–397. (1975) Zbl0327.34028MR0402188

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.