Small-time estimates for a real moment problem with two-term Weyl spectral law

In this work, we first study the solvability of moment problems involving real exponentials and provide explicit estimates of the associated control cost. The result holds when the increasing sequence of distinct real numbers satisfies a suitable two-term Weyl asymptotic law, without imposing any uniform spacing condition on blocks of its elements. We then deduce a corresponding controllability result for a linear control problem. Next, we present an exponential family fitting our hypotheses that cannot be treated by existing results of this type. Finally, we show how to deduce new exact controllability results for suitable fractional bilinear heat equations in higher-dimensional domains.