On a constrained approximate controllability problem for the heat equation

• Published in: Journal of optimization theory and applications Vol. 108; no. 1; pp. 29 - 64
• Main Authors: ORTEGA, J. H, ZUAZUA, E
• Format: Journal Article
• Language: English
• Published: New York, NY Springer 2001; Springer Nature B.V
• Subjects: Cauchy problems; Exact sciences and technology; Mathematical analysis; Mathematics; Ordinary differential equations; Sciences and techniques of general use; Boundary control; Controllability; Heat equation; Spectral theory
• ISSN: 0022-3239; 1573-2878
• DOI: 10.1023/A:1026409821088

In this work, we study an approximate control problem for the heatequation, with a nonstandard but rather natural restriction on thesolution. It is well known that approximate controllability holds. On theother hand, the total mass of the solutions of the uncontrolled system isconstant in time. Therefore, it is natural to analyze whether approximatecontrollability holds supposing the total mass of the solution to be a givenconstant along the trajectory. Under this additional restriction,approximate controllability is not always true. For instance, this propertyfails when Ω is a ball. We prove that the system is genericallycontrollable; that is, given an open regular bounded domain Ω, thereexists an arbitrarily small smooth deformation u, such that the system isapproximately controllable in the new domain Ω+u underthis constraint. We reduce our control problem to a nonstandard uniquenessproblem. This uniqueness property is shown to hold generically with respectto the domain.