Literature Cited
V. I. Plotnikov, “Necessary and sufficient optimality conditions and uniqueness conditions for optimizing functions for controllable systems of a general form,” Izv. Akad. Nauk SSSR, Ser. Mat.,36, No. 3, 652–679 (1972).
A. P. Rudin, Nuclear Reactors and the Pontryagin Maximum Principle [in Russian], Atomizdat, Moscow (1971).
C. P. Tzanos, E. P. Gyftopoulos, and M. J. Driscoll, “Optimization of material distribution in fast reactor cores,” Nucl. Sci. Eng.,52, No. 1, 84–94 (1973).
A. I. Egorov, Optimal Control of Thermal and Diffusion Processes [in Russian], Nauka, Moscow (1978).
V. Prager, Introduction to Structural Optimization, Springer-Verlag (1975).
J.-L. P. Armand, Applications of the Optimal Control Theory for Systems with Distributed Parameters to Construction Optimization Problems [Russian translation], Mir, Moscow (1977).
V. A. Troitskii, Optimal Oscillation Processes of Mechanical Systems [in Russian], Mashinostroenie, Leningrad (1976).
N. Olhoff, “Optimization of vibrating beams with respect to higher order natural frequencies,” J. Struct. Mech.,4, No. 1, 87–122 (1976).
P. B. Hagedorn, “Minimum mass structures with given natural frequencies,” in: Proceedings of III Congress Brasilian Mech. Engineers, Vol. D, Rio de Janeiro, 1129–1134 (1975).
Yu. A. Kuznetsov, “On necessary optimality conditions in control problems for systems described by elliptic equations of type,” Sib. Mat. Zh.,20, No. 3, 586–596 (1979).
O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and Quasilinear Elliptic Equations, Academic Press (1968).
T. Kato, Perturbation Theory for Linear Operators, Springer-Verlag (1966).
G. M. Fikhtengol'ts, A Course on Differential and Integral Calculus [in Russian], Vol. I, Fizmatgiz, Moscow (1962).
N. Dunford and J. T. Schwartz, Linear Operators, Wiley-Interscience (1971).
M. M. Day, Normed Linear Spaces, Springer-Verlag (1973).
A. E. Taylor, “Theorems of ascent, descent, nullity, and defect of linear operators,” Math. Ann.,163, 18–49 (1966).
E. Bishop, “Spectral theory for operators on a Banach space,” Trans. Am. Math. Soc.,86, No. 2, 414–445 (1957).
Additional information
Translated from Sibirskii Matematicheskii Zhurnal, Vol. 23, No. 3, pp. 118–135, May–June, 1982.
Rights and permissions
About this article
Cite this article
Kuznetsov, Y.A. Necessary optimality conditions in control problems for systems described by eigenfunctions of an elliptic operator. Sib Math J 23, 386–399 (1982). https://doi.org/10.1007/BF00973496
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00973496