In this talk we consider systems described by the heat equation on the interval $[0,\pi]$ with $L^2$ boundary controls and study the reachable space at some instant $\tau>0$. The main results assert that this space is generally sandwiched between two Hilbert spaces of holomorphic functions defined on a square in the complex plane and which has $[0,\pi]$ as one of the diagonals. More precisely, in the case of Dirichlet boundary controls acting at both ends we prove that the reachable space contains the Hardy-Smirnov space and it is contained in the Bergman space associated to the above mentioned square.
For this we discuss the (explicit) operator associating with both boundary control functions (the
so-called input)
the corresponding
state of the system. This operator is commonly called the input-to-state map. A key step, based
on the Poisson summation formula, is the
representation of this operator as a convolution with a sum of Gaussians. As it turns out, only two terms
of this sum corresponding to both boundary controls need to be considered.
These leading terms give rise to
integral transforms associated with the heat kernel (or weighted
Laplace transforms on sectors) which have
been investigated by Aikawa, Hayashi and Saitoh. Another key ingredient is given by
Riesz bases for Hardy-Smirnov spaces in polygons which have been discussed
by Levin and Lyubarskii.