Optimal control for a nonlinear stochastic PDE model of cancer growth

ABSTRACTABSTRACTWe study an optimal control problem for a stochastic model of tumour growth with drug application. This model consists of three stochastic hyperbolic equations describing the evolution of tumour cells. It also includes two stochastic parabolic equations describing the diffusions of nutrient and drug concentrations. Since all systems are subject to many uncertainties, we have added stochastic terms to the deterministic model to consider the random perturbations. Then, we have added control variables to the model according to the medical concepts to control the concentrations of drug and nutrient. In the optimal control problem, we have defined the stochastic and deterministic cost functions and we have proved the problems have unique optimal controls. For deriving the necessary conditions for optimal control variables, the stochastic adjoint equations are derived. We have proved the stochastic model of tumour growth and the stochastic adjoint equations have unique solutions. For proving the theoretical results, we have used a change of variable which changes the stochastic model and adjoint equations (a.s.) to deterministic equations. Then we have employed the techniques used for deterministic ones to prove the existence and uniqueness of optimal control.KEYWORDS: Stochastic optimal controlstochastic parabolic-hyperbolic equationEkeland variational principlemulticellular tumour spheroid modelfree boundary problemMATHEMATICS SUBJECT CLASSIFICATIONS 2010: 49J5549J2049J1549K4549K2049K15 AcknowledgementsThe authors are very grateful to the editor and the referees for their valuable comments and suggestions which improved the original submission of this paper.Disclosure statementNo potential conflict of interest was reported by the author(s).Data availability statementNo datasets were generated or analysed during the current study.Additional informationFundingTorres was supported by the Portuguese Foundation for Science and Technology (FCT - Fundação para a Ciência e a Tecnologia) through CIDMA, reference UIDB/04106/2020.