Reachable space of the Hermite heat equation with boundary control

Andreas Hartmann (1) , Marcu-Antone Orsoni (1)

1 IMB - Institut de Mathématiques de Bordeaux

We discuss reachable states for the Hermite heat equation on a segment with boundary $L^2$-controls. The Hermite heat equation corresponds to the heat equation to which a quadratic potential is added. We will discuss two situations: when one endpoint of the segment is the origin and when the segment is symmetric with respect to the origin. One of the main results is that reachable states extend to functions in a Bergman space on a square one diagonal of which is the segment under consideration, and that functions holomorphic in a neighborhood of this square are reachable.

Mots clés: Boundary control; reachable space; Hermite-Heat equation; Mehler kernel; method of images; holomorphic functions; Bergman space

Domaines: Variables complexes [math.CV]; Equations aux dérivées partielles [math.AP]

Andreas Hartmann, Marcu-Antone Orsoni. Reachable space of the Hermite heat equation with boundary control. SIAM Journal on Control and Optimization, 2022, 60 (6). ⟨hal-03330217⟩