Quantitative stability for eigenvalues of Schr\"{o}dinger operator, Quantitative bathtub principle \& Application to the turnpike property for a bilinear optimal control problem

TL;DR

This paper establishes quantitative inequalities for the first eigenvalue of Schrödinger operators and applies these results to demonstrate a turnpike property in bilinear optimal control problems, linking spectral optimization to control theory.

Contribution

It introduces a new method based on a quantitative bathtub principle for spectral optimization and applies it to analyze the turnpike property in bilinear control systems.

Findings

Quantitative inequalities for eigenvalues under $L^ Infty$ and $L^1$ constraints.

A new approach using the quantitative bathtub principle.

Uniform bounds on the distance of optimal controls to the eigenvalue optimizer set.

Abstract

This work is concerned with two optimisation problems that we tackle from a qualitative perspective. The first one deals with quantitative inequalities for spectral optimisation problems for Schr\"{o}dinger operators in general domains, the second one deals with the turnpike property for optimal bilinear control problems. In the first part of this article, we prove, under mild technical assumptions, quantitative inequalities for the optimisation of the first eigenvalue of $-\Delta-V$ with Dirichlet boundary conditions with respect to the potential $V$, under $L^\infty$ and $L^1$ constraints. This is done using a new method of proof which relies on in a crucial way on a quantitative bathtub principle. We believe our approach susceptible of being generalised to other steady elliptic optimisation problems. In the second part of this paper, we use this inequality to tackle a turnpike problem. Namely, considering a bilinear control system of the form $u_t-\Delta u=\mathcal V u$, $\mathcal V=\mathcal V(t,x)$ being the control, can we give qualitative information, under $L^\infty$ and $L^1$ constraints on $\mathcal V$, on the solutions of the optimisation problem $\sup \int_\Omega u(T,x)dx$? We prove that the quantitative inequality for eigenvalues implies an integral turnpike property: defining $\mathcal I^*$ as the set of optimal potentials for the eigenvalue optimisation problem and $\mathcal V_T^*$ as a solution of the bilinear optimal control problem, the quantity $\int_0^T \operatorname{dist}_{L^1}(\mathcal V_T^*(t,\cdot)\,, \mathcal I^*)^2$ is bounded uniformly in $T$.