A new class of inverse problems is considered. In the context of classical theory, inverse problems are concerned with finding a model that has the observed measurements. It is well known that such problems usually are ill-posed. At the same time, it is often the case when there is some a priori information about the system. This naturally leads to the following inverse optimal problem: find data $\hat{F}$ of a model which is the nearest to a priori given data $F_0$ and sufficient to ensure the model has the observed measurements $S$.

In this note, an approach to a complete solution to such a problem is developed. Within the framework of this approach, we consider a model problem of recovering the potential field $\hat{V}$ from the $m$ observed eigenvalues of the Schrödinger operator, provided that such potential field is at the minimum distance from a priori given potential $V_a$. In the main result, we establish a new type of relationship between the linear spectral problems and systems of nonlinear differential equations which enables us to find a solution to the inverse optimal spectral problem and obtain novel results on the existence of solutions to nonlinear problems as well.

考虑了一类新的逆问题。在经典理论的背景下，逆问题与寻找具有观测测量值的模型有关。众所周知，此类问题通常是不适定的。同时，当有一些关于系统的先验信息时，经常会出现这种情况。这自然会导致以下逆最优问题：查找数据$\widehat{F}$ 最接近先验给定数据的模型 $F_0$ 并且足以确保模型具有观察到的测量值 $秒$.

在本说明中，开发了解决此类问题的完整方法。在这种方法的框架内，我们考虑恢复势场的模型问题$\widehat{伏}$ 来自 $米$ 观察到的薛定谔算子的特征值，前提是这种势场与先验给定势的距离最小 ${伏}_{\mathrm{一种}}$. 在主要结果中，我们在线性谱问题和非线性微分方程组之间建立了一种新型关系，这使我们能够找到逆最优谱问题的解，并获得关于非线性问题解存在性的新结果.