COMPUTING EIGENVALUES OF STURM-LIOUVILLE PROBLEMS VIA OPTIMAL-CONTROLTHEORY

In this paper, we shall address three problems arising in the computat ion of eigenvalues of Sturm-Liouville boundary value problems. We firs t consider a well-posed Sturm-Liouville problem with discrete and dist inct spectrum. For this problem, we shall show that the eigenvalues ca n be computed by solving for the zeros of the boundary condition at th e terminal point as a function of the eigenvalue. In the second proble m, we shall consider the case where some coefficients and parameters i n the differential equation are continuously adjustable. For this, the eigenvalues can be optimized with respect to these adjustable coeffic ients and parameters by reformulating the problem as a combined optima l control and optimal parameter selection problem. Subsequently, these optimized eigenvalues can be computed by using an existing optimal co ntrol software, MISER. The last problem extends the first to nonstanda rd boundary conditions such as periodic or interrelated boundary condi tions. To illustrate the efficiency and the versatility of the propose d methods, several non-trivial numerical examples are included.