Switched system optimal control approach for drug administration in cancer chemotherapy

This paper considers a drug administration problem in cancer chemotherapy. Our main aim is to choose a drug delivery rate such that the final tumour population is minimized. For this purpose, the drug administration problem is modeled as an optimal control problem of switched systems (OCPSS) with state-dependent switching. Since most of existing numerical algorithms are designed for the OCPSS with time-dependent switching, these algorithms can not be directly used to solve the OCPSS when the subsystems switch under state-dependent and not time-dependent. To tackle this issue, the OCPSS is formulated as a non-convex nonlinear parameter selection problem (NNPSP) with simple bounds on the decision variables by using the relaxation technique, the control parameterization method, and the penalty function idea. To obtain a global optimal solution of this NNPSP, a penalty function-based continuous filled function (PFCFF) algorithm is proposed based on a novel flattening function (NFF), a novel continuous filled function (NCFF), and an adaptive strategy to obtain the initial points. Compared with the existing continuous filled function (CFF) algorithms, the PFCFF algorithm has the following four advantages: a lot of local optimal solutions can be eliminated by using the NFF, which implies that it easier to find the global optimal solution; no parameters need to be adjusted in the NCFF; there exist no the exponential and logarithmic terms in the NCFF, which can effectively avoid memory overflow during the process of numerical computing; and an adaptive strategy to generate the initial points is proposed for obtaining the local minimum of the NCFF. Finally, numerical results show that the PFCFF algorithm has good performance and can obtain a stable and robust performance when considering the small noise perturbations in initial system state.