Singular controls and chattering arcs in optimal control problems arising in biomedicine

Abstract: We consider an optimal control problem of the Mayertype for a single-input, control affine, nonlinear system in small dimension. In this paper, we analyze effects that a modeling extension has on the optimality of singular controls when the control is replaced with the output of a first-order, time-invariant linear system driven by a new control. This analysis is motivated by an optimal control problem for a novel cancer treatment method, tumor anti-angiogenesis, when such a linear differential equation, which represents the pharmacokinetics of the therapeutic agent, is added to the model. We show that formulas that define a singular control of order 1 and its associated singular arc carry over verbatim under this model extension, albeit with a different interpretation. But the intrinsic order of the singular control increases to 2. As a consequence, optimal concatenation sequences with the singular control change and the possibility of optimal chattering arcs arises.