8ed66b8b-9698-446a-969d-da119098eafa20210322034807141wseamdt@crossref.orgMDT DepositWSEAS TRANSACTIONS ON SYSTEMS1109-277710.37394/23202http://wseas.org/wseas/cms.action?id=4067220202022020201910.37394/23202.2020.19http://wseas.org/wseas/cms.action?id=23194ImadRezzougLaboratory of Dynamic Systems and Control, Department of Mathematics and Computer Science, Larbi Ben M’hidi University Oum El Bouaghi, AlgeriaOussaiefTaki-EddineOEB University, AlgeriaThe objective of this work is to study the approximate controllability with constraints on the control of the heat equation. To demonstrate the approximate controllability, it suffices to use Hahn Banach theorem and notice that the orthogonal of the set of reachable states at the moment t=0 is reduced to zero. Finally we adapted the method of duality (primal-dual) of Fenchel-Rockafellar to characterize the optimal control224202022420201419https://www.wseas.org/multimedia/journals/systems/2020/a065102-059.pdf10.37394/23202.2020.19.3http://www.wseas.org/multimedia/journals/systems/2020/a065102-059.pdfA. EL JAI , A. J. PRITCHARD (1986).Cap-teurs et actionneurs dans l’analyse des syst`emesdistribu ́es. Masson. R.M.A 3. Paris.A. PAZY (1983).Semigroups of linear op-erators and applications to partial differentialequation. Applied Maths. Sciences 44, Springer-Verlag. New-York.B. AMEL (2013). ́Etude des syst`emes hyper-boliques`a donn ́ees manquantes, Th`ese de doc-torat, Universit ́e de mentouri, Constantine.B. ALI (2000).Contrˆolabilit ́e r ́egionale, Ciblefronti`ere et contrˆolabilit ́e du gradient dansles syst`emes distribues. Th`ese de doctorat,L’universit ́e Moulay Ismail.E. H. ZERRIK (1993).Analyse r ́egionale dessyst`emes distribu ́es. Th`ese doctorat. Universit ́e.Mohammed V. Rabat, Maroc.I. EKLAND, AND R. TEMAM (1976).Con-vex analysis and variational problems. North-Holland, Amsterdam.J. L. LIONS (1992).Sentinelles pour lessyst`emes distribu ́es`a donn ́ees incompl`etes.RMA21. Paris.J. L. LIONS (1968).contrˆole optimal dessyst`emes gouvern ́es par des ́equations auxd ́eriv ́ees partielles, Dunod, Gauthier-Villars,Paris.L. AFIFI, A. EL JAI, E. ZERRIK (2008).Syst`emes Dynamiques. Universit ́e de Perpignan.R.IMAD(2014). ́Etudeth ́eoriqueetnum ́eriquedesprobl`emesd’identificationdes syst`emes gouvern ́es par des ́equationsaux d ́eriv ́ees partielles, Th`ese de doctorat,Universit ́e de Oum El Bouaghi, Alg ́erie.R. IMAD (2009).Identification d’une partiede la fronti`ere inconnue d’une membrane, Th`esede magister, Universit ́e de Oum El Bouaghi,Alg ́erie.10.1215/kjm/1250776858S. MIZOHATA (1958).Unicit ́e du prolonge-ment des solutions pour quelques op ́erateursdiff ́erentiels paraboliques, Mem. Coll. Sc. Univ.Kyoto, S ́erie A, Vol 31, pp 219-239.T. CHAHNAZ ZAKIA (2013).D ́etection dela pollution et identification des d ́efauts en ́elasticit ́e lin ́eaire, Th`ese de doctorat, Universit ́ed’Oran.