Non-localization of eigenfunctions for Sturm-Liouville operators and applications. (English) Zbl 1382.49017
The paper deals with localization properties of eigenfunctions of Sturm-Liouville operators of the form:
\[
A_a = -\partial_{xx} + a(\cdot) Id,
\]
where \(a(\cdot)\) is a nonnegative bounded potential defined on the interval \((0,L)\).
Such problems arise when minimizing the \(L^2\)-norm of a function \(e(\cdot)\) on a measurable subset \(\omega\) of \((0,L)\), where \(e(\cdot)\) runs over the set of all eigenfunctions of \(A_a\) and, at the same time, \(\omega\) runs over all measurable subsets of \((0,L)\) with the prescribed measure.
Problems of this kind play important role in Quantum Physics. The one-dimensional problem, that is studied in this paper, arises in control and stabilization of the linear wave equation. Applications of the main result to control and observation theory, along with some simulation results, are also discussed.
Such problems arise when minimizing the \(L^2\)-norm of a function \(e(\cdot)\) on a measurable subset \(\omega\) of \((0,L)\), where \(e(\cdot)\) runs over the set of all eigenfunctions of \(A_a\) and, at the same time, \(\omega\) runs over all measurable subsets of \((0,L)\) with the prescribed measure.
Problems of this kind play important role in Quantum Physics. The one-dimensional problem, that is studied in this paper, arises in control and stabilization of the linear wave equation. Applications of the main result to control and observation theory, along with some simulation results, are also discussed.
Reviewer: Vladimir Jacimovic (Podgorica)
MSC:
| 49K15 | Optimality conditions for problems involving ordinary differential equations |
| 47A75 | Eigenvalue problems for linear operators |
| 49J20 | Existence theories for optimal control problems involving partial differential equations |
| 93B07 | Observability |
| 93B05 | Controllability |
| 34B24 | Sturm-Liouville theory |
Keywords:
Sturm-Liouville operators; eigenfunctions; extremal problems; calculus of variations; control theory; wave equationReferences:
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