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Abstract
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Thisstudyintroducestwo-andthree-dimensionaloptimalcontrolproblemscharacterisedbythe ψ-temperedfractionalderivativeandproposesanovelnumericalmethodologybasedondeep fractional-orderBernoullioptimisationtoachievetheirefficientsolution.Tothisend,theproblems underconsiderationarefirstreformulatedasequivalentvariationalproblems.Subsequently,adeep neuralnetworkemployingthefractional-orderBernoullifunctionsandsinhasactivationfunctionsis utilisedtoapproximatethestatevariable.Tofacilitatetheeffectiveimplementationoftheproposed method,weconstructseveral integraloperatorsofbothintegerandψ-temperedfractionalorders basedonthebasisfunctionsderivedfromthedeepneuralnetwork.Theseoperatorsarediscretised viatheFejérquadratureruleadaptedtotheψ-temperedfractionalcalculus,ensuringstabilityand high-precisionintegration.Theproposedapproachconvertsthe2D/3Dψ-temperedfractionalopti malcontrolproblemintoanalgebraicsystemusingdeepneuralnetworkswithintegraloperators andGauss–Legendreintegration,whichisefficientlysolvedviaNewton’smethod.Theproposed methodcombinessimpleimplementation,lowcomputationalcost,andhighaccuracy.Itseffective nessandrobustnessarevalidatedthroughseveralrepresentative2Dand3Dexamples,confirming themethod’sapplicabilitytocomplexfractionaloptimalcontrolproblems.
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