Receding horizon control of tumor growth based on optimal control

Receding horizon control (RH) is a powerful and well known technique used to embed feedback in the solution of a dynamic optimization problem. In most published approaches, RH control is associated to model predictive control and amounts to minimize a cost defined over an horizon that slides in time. The optimization is done with respect to a sequence of candidate values for the manipulated variable, of which only the first is used. When considering nonlinear control problems, if the candidate sequence of the manipulated variables is left free of any constraint related to the plant dynamics (apart from operational constraints), there is the danger that the numerical method used converges to a local minimum. In this paper, instead, the minimization is performed using a relaxation algorithm that approximates the solution of Pontryagin´s optimality conditions. This approach has the advantage of shaping the solution using the state and adjoint equations and, in addition, provides a natural approach to continuous RH problems. This algorithm is applied here to design therapies for tumor growth, modeled by the Gompertz model. A comparison of quadratic costs with costs that lead to sparse control signals, i. e. that are zero instead of assuming a small value, is also done.