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.

MathSciNet review:
1473668

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Abstract | References | Similar Articles | Additional Information

Abstract: Necessary and sufficient conditions for the optimality of a pair subject to are given. Here is a selfadjoint operator with closed range on a Hilbert space and . The case - 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 -periodic wave equation for all in a dense subset of . It also includes optimal control problems for some elliptic equations.

**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 one-dimensional wave equation with 𝑥-dependent coefficients*, Trans. Amer. Math. Soc.**349**(1997), no. 5, 2035–2048. MR**1373628**, 10.1090/S0002-9947-97-01714-5**3.**Haïm Brézis,*Periodic solutions of nonlinear vibrating strings and duality principles*, Bull. Amer. Math. Soc. (N.S.)**8**(1983), no. 3, 409–426. MR**693957**, 10.1090/S0273-0979-1983-15105-4**4.**Jong-Kyu Kim and N. H. Pavel,*Optimal control of the periodic wave equation*, Proceedings of Dynamic Systems and Applications, Vol. 2 (Atlanta, GA, 1995), Dynamic, Atlanta, GA, 1996, pp. 309–314. MR**1419543****5.**-,*-optimal control of the 1- wave equation with -dependent coefficients*, Nonlinear Analysis, TMA (to appear).**6.**-,*Existence and regularity of weak periodic solutions of the 2- wave equation*, Nonlinear Analysis, TMA (to appear).**7.**N. H. Pavel,*Periodic solutions to nonlinear 2-D wave equations*, Theory and applications of nonlinear operators of accretive and monotone type, Lecture Notes in Pure and Appl. Math., vol. 178, Dekker, New York, 1996, pp. 243–249. MR**1386681****8.**Nicolae H. Pavel,*Nonlinear evolution operators and semigroups*, Lecture Notes in Mathematics, vol. 1260, Springer-Verlag, Berlin, 1987. Applications to partial differential equations. MR**900380**

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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:
http://dx.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