Carleman discretization of impulsive systems: application to the optimal control problem of anti-angiogenic tumor therapies

Abstract

Impulsive systems model continuous-time frameworks with control actions occurring at discrete time instants. Among the others, such models assume relevance in medical situations, where the physical system under control evolves continuously in time, whilst the control therapy is instantaneously administered, e.g. by means of intravenous injections. This note proposes a discretization algorithm for an impulsive system, whose methods relies on the Carleman embedding techinique. The discretization times are given by the impulsive control action and do not require to have a fixed discretization period. On the ground of the resulting discrete-time system (which can be computed with arbitrary level of accuracy) we propose an optimal control algorithm on a finite horizon. Simulations are carried out on a model exploited for anti-angiogenic tumor therapies and show the effectiveness of the theoretical results.

Optimal Impulsive Control With Application to Antiangiogenic Tumor Therapy

“…T is the sequence of control impulses and x is the chosen reference level for the state of the system at the end of the control application (e.g., at the end of the period of drug administration therapy in case of medical/healthcare frameworks). Such a problem has been addressed in [7], by means of the Carleman-based discretization of the original continuous-time system at the time instants τ k ∈ T related to the impulsive control. According to [7], the discretized version of ( 1) is:…”

“…Such a problem has been addressed in [7], by means of the Carleman-based discretization of the original continuous-time system at the time instants τ k ∈ T related to the impulsive control. According to [7], the discretized version of ( 1) is:…”

“…The Carleman embedding provides the exact solution to the discretization problem, whose finite-dimensional implementation is based on a suitable truncation of the involved series expansion (7).…”

“…Indeed, both these approaches have been followed in [7], under the assumption of complete knowledge of the initial state of the system. Unfortunately, such assumption is not realistic, especially in biomedical applications where only few compartments are accessible for measurements.…”

“…By using these two distinct tools it is relatively easy to extend the proposed methodology to other scenarios, like continuous-time measurements with impulsive control, or discrete-time measurements with continuous or piece-wise continuous infusion therapy. The use of a state observer allows to overcome the drawback of [7], where preliminary results on impulsive control where presented according to a complete knowledge of the state of the system.…”