Advertisement

Differential Equations

, Volume 55, Issue 5, pp 718–728 | Cite as

Solvability of an Operator Riccati Integral Equation in a Reflexive Banach Space

  • N. V. ArtamonovEmail author
Integral Equations
  • 7 Downloads

Abstract

We show that if X is a reflexive Banach space, then a nonautonomous operator Riccati integral equation has a unique strongly continuous self-adjoint nonnegative solution P(t)L(X,X*).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Bensoussan, A., Da Prato, G., Delfour, M.C., and Mitter, S.K., Representation and Control of Infinite Dimensional Systems, Boston: Birkhäuser, 2007.CrossRefzbMATHGoogle Scholar
  2. 2.
    Curtain, R. and Pritchard, A.J., The infinite-dimensional Riccati equation for systems defined by evolution operators, SIAM J. Control Optim., 1976, vol. 14, no. 5, pp. 951–983.MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Lasiecka, I., Optimal control problems and Riccati equations for systems with unbounded controls and partially analytic generators, in Functional Analytic Methods for Evolution Equations, Berlin, 2004, pp. 313–371.CrossRefGoogle Scholar
  4. 4.
    Gibson, J.S., The Riccati integral equations for optimal control problems on Hilbert spaces, SIAM J. Control Optim., 1979, vol. 17, no. 4, pp. 537–565.MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Pritchard, A.J. and Salamon, D., The linear quadratic control problem for infinite dimensional systems with unbounded input and output operators, SIAM J. Control Optim., 1987, vol. 25, no. 1, pp. 121–144.MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Lasiecka, I. and Triggiani, R., Riccati differential equations with unbounded coefficients and non-smooth terminal condition—the case of analytic semigroups, SIAM J. Math. Anal., 1992, vol. 23, no. 2, pp. 449–481.MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Da Prato, G. and Ichikawa, A., Riccati equations with unbounded coefficients, Ann. Mat. Pura Appl., 1985, vol. 40, no. 1, pp. 209–211.MathSciNetCrossRefGoogle Scholar
  8. 8.
    Artamonov, N.V., On the solvability of a system of forward–backward linear equations with unbounded operator coefficients, Math. Notes, 2016, vol. 100, nos. 5–6, pp. 747–750.MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Yong, J., Forward–backward evolution equations and applications, 2015, arXiv: 1508.03550v1.Google Scholar
  10. 10.
    Engel, K. and Nagel, R., One-parameter semigroups for linear evolution equations, in Grad. Texts in Math., Berlin: Springer-Verlag, 2000, Vol. 194.Google Scholar
  11. 11.
    Koshkin, S., Positive semigroup and algebraic Riccati equations in Banach spaces, Positivity, 2016, vol. 20, pp. 541–563.MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2019

Authors and Affiliations

  1. 1.MGIMO UniversityMoscowRussia

Personalised recommendations