A unified optimality condition for eigenvalue problems. (English) Zbl 0790.49026

Summary: An optimality condition is derived for the problem of maximizing the first eigenvalue of an abstract elliptic differential operator in a Hilbert space. This condition unifies some known criteria in optimal design and gives some new ones as well. The result is obtained by using some techniques of nonsmooth optimization.


49K27 Optimality conditions for problems in abstract spaces
49Q10 Optimization of shapes other than minimal surfaces
49J52 Nonsmooth analysis
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[1] A. S. Bratus, S. Seiranian: Bimodel solutions in eigenvalue optimization problems. Appl. Math. Mech. 43 (1983), 451-457.
[2] A. S. Bratus: Condition of extremum for eigenvalues of some elliptic boundary value problems. J. Optim. Theory Appl. 68 (1991), 423-437. · Zbl 0725.35066
[3] F. Clarke: Optimization and Nonsmooth Analysis. J. Wiley, New York 1983. · Zbl 0582.49001
[4] K. Choi, E. J. Haug: Optimization of structures with repeated eigenvalues. Optimal Design of Distributed Structures (E. J. Haug and J. Cea, Sijthoff-Noordhoff, Leyden 1981, pp. 219-277. · Zbl 0543.73117
[5] S. J. Cox, M. L. Overton: On the optimal design of columns against buckling. Preprint of Rice university 1990, TR90-13.
[6] M. Delfour, J. P. Zolesio: Shape sensitivity analysis via min max differentiability. SIAM J. Control Optim. 26 (1988), 835-863. · Zbl 0654.49010
[7] Z. Z. Guang, G. C. Zhang: Linear Functional Analysis. Science Press of Shangai, Shanghai 1979
[8] E. J. Haug, B. Rousselet: Design sensitivity of eigenvalue variations. Optimal Design of Distributed Structures (E. J. Haug and J. Cea, Sijthoff-Noordhoff, Leyden 1981, pp. 1370-1396.
[9] A. D. loffe: Necessary and sufficient conditions for a local minimum I. SIAM J. Control Optim. 17 (1979), 245-250. · Zbl 0417.49027
[10] F. Lempio, H. Maurer: Differential stability in infinite-dimensional nonlinear programming. Appl. Math. Optim. 6 (1980), 139-152. · Zbl 0426.90072
[11] E. Masur: Optimal structural design under multiple eigenvalue constraints. Internat. J. Solids and Structures 20 (1984), 211-231. · Zbl 0544.73117
[12] N. Olhoff, S. Rasmussen: On single and bimodal optimum buckling loads of clamped columns. Internat. J. Solids and Structures 75 (1977), 605-614. · Zbl 0357.73041
[13] I. Tadjbakhsh, J. Keller: Strongest columns and isoperimetric inequalities for eigenvalues. J. Appl. Mech. 29 (1962), 159-164. · Zbl 0106.38301
[14] J. P. Zalesio: Semi-derivatives of repeated eigenvalues. Optimal Design of Distributed Structures (E.J. Haug and J. Cea, Sijthoff-Noordhoff, Leyden 1981, pp. 1457-1473.
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