  We 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 that 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.
