Optimal Control of Adaptive Cancer Therapy:Ceiling Policies, Resistant-Lineage Risk, and

We study adaptive cancer therapy as an optimal control problem for a two-clone competitive model in which drug-sensitive cells are killed by treatment while drug-resistant cells are not, and the two compete for a shared resource.We obtain three families of results, each with complete proofs.First, for the objective of maximizing the time until the resistant burden reaches a prescribed threshold, subject to a hard ceiling on total tumor burden, we prove that the unique optimal policy is a ``ceiling-hold'' policy: withhold treatment until the total burden reaches the ceiling, then apply exactly the dose that holds the burden constant. The proof rests on a cooperative comparison principle for the pair (total burden, resistant burden), and the optimal time is given in closed form.Second, for a stochastic lineage model in which resistant lineages are seeded by mutation and evolve as independent time-inhomogeneous birth--death processes, we derive an exact formula for the probability that resistance is present at a horizon, prove existence of an optimal open-loop schedule, compute the first variation of the risk through an explicit adjoint system, and deduce structural properties: maximal dosing is optimal absent competitive suppression, and a treatment holiday is optimal near the horizon whenever an explicit sign condition holds.Third, we characterize exactly when the clonal composition can be recovered from total burden alone: the initial state is identifiable if and only if the dose differs from a single critical value on a set of positive measure; we give a reconstruction formula and a Lipschitz stability estimate, and show that the ceiling-hold policy is always observable.Numerical experiments are included as illustrations only.This paper is an update of Youn and Chatterjea (2025).