Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Optimal control of a functional equation
associated with closed range
selfadjoint operators


Authors: S. C. Gao and N. H. Pavel
Journal: Proc. Amer. Math. Soc. 126 (1998), 2979-2986
MSC (1991): Primary 47N10, 47B25, 49K27.
DOI: https://doi.org/10.1090/S0002-9939-98-04633-4
MathSciNet review: 1473668
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Necessary and sufficient conditions for the optimality of a pair $(y^{*}, u^{*})$ subject to $Ay^{*} = Bu^{*} + f$ are given. Here $A$ is a selfadjoint operator with closed range on a Hilbert space $\mathcal {H}$ and $B \in L(\cal{H})$. The case $B$- unbounded is also discussed, which leads to some open problems. This general functional scheme includes most of the previous results on the optimal control of the $T$-periodic wave equation for all $T$ in a dense subset of $\mathbb{R}$. It also includes optimal control problems for some elliptic equations.


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

  • 1. V. Barbu, Optimal control of the one dimensional periodic wave equation, Appl. Math. Optimiz., 35(1997), 77-90. CMP 97:03
  • 2. V. Barbu and N. H. Pavel, Periodic solutions to nonlinear 1-$D$ wave equation with $x$-dependent coefficients, Trans. Amer. Math. Soc 349 (1997), no. 5, 2035-2048. MR 97h:35129
  • 3. H. Brezis, Periodic solutions of nonlinear vibrating string and duality principles, Bull. AMS, 8(1983), 409-426. MR 84e:35010
  • 4. J. K. Kim and N. H. Pavel, Optimal control of the periodic wave equation, Proceedings of Dynamical Sytems and Applications. Vol.2(1996),09-14, Atlanta, Georgia, May 1995 MR 97j:49042
  • 5. -, $L^{\infty}$-optimal control of the 1-$D$ wave equation with $x$-dependent coefficients, Nonlinear Analysis, TMA (to appear).
  • 6. -, Existence and regularity of weak periodic solutions of the 2-$D$ wave equation, Nonlinear Analysis, TMA (to appear).
  • 7. N. H. Pavel, Periodic solutions to nonlinear 2-$D$ wave equations, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, Vol.178(1996), 243-250 MR 97c:35138
  • 8. -, Nonlinear Evolution Operators and Semigroups. Applications to Partial Differential Equations, Lecture Notes in Pure and Applied Mathematics, Springer-Verlag, Vol 1260(1987). MR 88j:47087

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47N10, 47B25, 49K27.

Retrieve articles in all journals with MSC (1991): 47N10, 47B25, 49K27.


Additional Information

S. C. Gao
Affiliation: Department of Mathematics, Ohio University, Athens, Ohio 45701
Email: shugao@bing.math.ohiou.edu

N. H. Pavel
Affiliation: Department of Mathematics, Ohio University, Athens, Ohio 45701
Email: npavel@bing.math.ohiou.edu

DOI: https://doi.org/10.1090/S0002-9939-98-04633-4
Keywords: Self-adjoint operators with closed range, optimal pairs, maximum principles, periodic waves
Additional Notes: The research of the first author was supported in part by the National Science Foundation of China
The research of the second author was supported in part by the National Research Fund, Korean Research Foundation Project #01-D0406 (jointly with Prof. J. K. Kim)
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1998 American Mathematical Society

American Mathematical Society