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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 F ˆ 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 V ˆ 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. • A new class of inverse optimal spectral problem. • The relationship between the spectral problems and the nonlinear problem. • Existence result for the system of NLS equations. [ABSTRACT FROM AUTHOR]