Abstract In t,his work, we are interested iri tlie optimal coritrol of tlie inrluctiori heatiiig pro- cess. This problerri has beeri posed by our iri<strong>du</strong>strial <strong>par</strong>trier Alcari withiri the frarrie- work of ari iritlustrial process of manufacture of aluniiriiurri <strong>par</strong>ts, called tliixoforrnirig. Duririg the rriaiiufacture, one Iieats by iriciuctiori an aliirrii~iiiirri alloy bill<strong>et</strong> uritil a <strong>par</strong>- ticular serni-solid state, called thixotropic state, before injcctirig it, uritler pressure, iri a mould. Tlie quality of the <strong>par</strong>t obtairied clepends directly on ttie qiiality of the heatirig. The bill<strong>et</strong> rriust, have t,he rriost. liorriogerieous possihle t.erriperat,itre corresporidirig to apprmirriately 50 % of solid fraction to guararitee the coriforrriity of the niouldirig. Thus, it is very sigriificaiit to be able to optirriize the coritrol <strong>par</strong>arri<strong>et</strong>ers of the iri- <strong>du</strong>ctiori heatirig iri order to achieve this goal. Otlier <strong>par</strong>arri<strong>et</strong>ers can also be takeri irito accouiit to guararitee ttie qualit,y of tlie heat,irig. The coiitrol variables are the frequency aiid tlie terisiori (or power) siipplirtl to tlie iritlrictor. More precisely, we seek a power wliose arriplitude varies in tirrie. We use riurnerical analysis techniques to solve the optirrial coritrol problern. The whole proceedirig plienonieria, <strong>du</strong>ring the heating, are described by non-liriear <strong>par</strong>tial derivative equatioris. We work withiii a two-dirrierisiorial frarriework. We use t,wo riiirrierical rnodels allowirig t,o obtairi the rriagu<strong>et</strong>ic field h arid the entlialpy u iti talie piece. From physical corisitlerations, we huild ari objective functiori J ctiaracteristic of the control wliicli orle warits to carry out. By a perturbatioris rri<strong>et</strong>hod, we tlieri seek to calculate the gradierit of tlie cost fuiictiori followirig the coritrol <strong>par</strong>ani<strong>et</strong>ers. This tecliriiqiie bririgs 11s to tlie calciilatiori of two associated acljoirit problerris, orle for electrorriagri<strong>et</strong>isrri, the other for thermies. These problerns heirig solv<strong>et</strong>l, it is pomible to evaliiate the gradierit of the ot>.ject,ive fuiictiori. Tlie value of coiitrol variables, the gratlierit and the furictiori J are theri trarisrriitted to a descerit algoritlirri based or1 a quasi-Newton wliich gives us the riew s<strong>et</strong> of <strong>par</strong>ani<strong>et</strong>ers. Theri, we reit,erate iiritil satisfyirig a stop criteriori deperidirig on tlie required precisioii. Firially, we obtairi the optirrial <strong>par</strong>arri<strong>et</strong>ers. For solvirig these prohlenis, we use a corribiriatiori of a firiite diff'ererice ni<strong>et</strong>liotl for the the discr<strong>et</strong>ixation arid of a finite elemerits ni<strong>et</strong>hoti for the space discr<strong>et</strong>iïiation. The so develop<strong>et</strong>l rri<strong>et</strong>hod was irriple~rieiited iri a already existiiig two-diiiierisioilal <strong>simulation</strong> code for iri<strong>du</strong>ctioii heatirig. We preserit riiirrierical tests, optirriixatiori results and a corn<strong>par</strong>isori with experirrierital results. We show like this the effectiveriess of our m<strong>et</strong>hod for t,tie optimal cont,rol of in<strong>du</strong>ctioii Iieating. We firiisli our study t>y the
corri<strong>par</strong>isori of the results obtairied by th adjoi~it state ni<strong>et</strong>lrod witlr tliose obtaitieci by autorrlatic differeritiatiori. Il'tius, we tiighligtit the atlvaritages arid the disadvaritages of this last one.