QUARTERLY OF APPLIED MATHEMATICS VOLUME LXXI, NUMBER 2 JUNE 2013, PAGES 339–368 S 0033-569X(2012)01287-7 Article electronically published on October 22, 2012 SIMULTANEOUS TEMPERATURE AND FLUX CONTROLLABILITY FOR HEAT EQUATIONS WITH MEMORY By S. AVDONIN (Department of Mathematics, University of Tennessee at Chattanooga, Chattanooga, Tennessee 37403-2598, USA) and L. PANDOLFI (Dipartimento di Scienze Matematiche, Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy ) Abstract. It is known that, in the case of the heat equation with memory, tempera- ture can be controlled to an arbitrary square integrable target provided that the system evolves for a sufficiently long time. The control is the temperature on the boundary. In this paper we consider heat equations with memory (one-dimensional space variable) and we first show that when the control is square integrable, then the flux is square integrable too. Then we prove that both temperature and flux can be simultaneously controlled to a pair of independent targets, both square integrable. This solves a problem first raised by Renardy. The method of proof relies on moment theory, and one of the contributions of this paper is the identification of L-bases and Riesz bases especially suited to heat equations with memory, which so appear to be endowed with a very rich bases structure. 1. Introduction. The derivation of heat equations depends on conservation of en- ergy (and temperature being a measure of energy): ∂ ∂t e(x, t)= −∇ · q(x, t) , ∂ ∂t e(x, t)= γ ∂ ∂t θ(x, t) . (1.1) The quantity denoted q(x, t) is the (density of) heat flux at time t at the position x while e(x, t) and θ(x, t) denote energy and temperature. This is just a balance law. To Received July 27, 2011. 2010 Mathematics Subject Classification. Primary 76A10, 93C05, 47N70. Supported in part by the National Science Foundation, grant ARC 0724860. This paper was partly written while the first author visited the Dipartimento di Matematica, Politecnico di Torino, as a visiting professor supported by GNAMPA-INDAM. Supported in part by Italian MURST and by the project “Groupement de Recherche en Contrˆ ole des EDP entre la France et l’Italie (CONEDP)”. This paper fits into the research programs of GNAMPA- INDAM. E-mail address : s.avdonin@alaska.edu E-mail address : luciano.pandolfi@polito.it c 2012 Brown University 339 License or copyright restrictions may apply to redistribution; see https://www.ams.org/license/jour-dist-license.pdf