Bimodal solutions in eigenvalue optimization problems

The problem of maximizing the minimum eigenvalue of a selfadjoint operator is examined. An isoperimetric condition is imposed on the control variable. This problem has interesting applications in the optimal design of structures. In papers on the optimization of the critical stability parameters and the frequencies of the natural oscillations of elastic systems /1–12/ it was shown that in a number of cases the optimal solutions are characterized by two or more forms of loss of stability or natural oscillations. In the case of conservative systems described by selfadjoint equations this signifies multiplicities of eigenvalues, i.e., of critical loads, under which loss of stability or of natural oscillation frequencies occurs. The necessary conditions for an extremum are obtained in the case when the optimal solution is characterized by a double eigenvalue. These conditions have a constructive character and can be used for the numerical and analytical solution of optimization problems. Both discrete and continuous cases of the specification of the original system are analyzed. Examples are given.