Quantitative Stability for Eigenvalues of Schrödinger Operator, Quantitative Bathtub Principle, and Application to the Turnpike Property for a Bilinear Optimal Control Problem
Idriss Mazari, Domènec Ruiz-Balet · SIAM Journal on Mathematical Analysis · 2022
This work is concerned with two optimization problems that we tackle from a qualitative perspective. The first one deals with quantitative inequalities for spectral optimization problems for Schrödinger operators in general domains, and 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 optimization 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 crucially relies on a quantitative bathtub principle. We believe our approach can be generalized to other steady elliptic optimization 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 optimization 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 optimization 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 $T$.