We consider a phenotype-structured model of evolutionary dynamics in a population of cancer cells exposed to the action of a cytotoxic drug. The model consists of a nonlocal parabolic equation governing the evolution of the cell population density function. We develop a novel method for constructing exact solutions to the model equation, which allows for a systematic investigation of the way in which the size and the phenotypic composition of the cell population change in response to variations of the drug dose and other evolutionary parameters. Moreover, we address numerical optimal control for a calibrated version of the model based on biological data from the existing literature, in order to identify the drug delivery schedule that makes it possible to minimise either the population size at the end of the treatment or the average population size during the course of treatment. The results obtained challenge the notion that traditional high-dose therapy represents a 'one-fits-all solution' in anticancer therapy by showing that the continuous administration of a relatively low dose of the cytotoxic drug performs more closely to the optimal dosing regimen to minimise the average size of the cancer cell population during the course of treatment. * BDH acknowledges support from the Australian Research Council (DP140100339). LA and TL gratefully acknowledge support of the project PICS-CNRS no. 07688 and the French "ANR blanche" project Kibord: ANR-13-BS01-0004.
In this paper we use a reference trajectory computed by a model predictive method to shrink the computational domain where we set the Hamilton-Jacobi Bellman (HJB) equation. Via a reduced-order approach based on proper orthogonal decomposition(POD), this procedure allows for an efficient computation of feedback laws for systems driven by parabolic equations. Some numerical examples illustrate the successful realization of the proposed strategy. G. Fabrini gratefully acknowledges support by the German Science Fund DFG grant Reduced-Order Methods for Nonlinear Model Predictive Control.
In the present paper, a multiobjective optimal control problem governed by a linear parabolic advection-diffusion-reaction equation is considered. The optimal controls are computed by applying model predictive control (MPC), which is a method for controlling dynamical systems over long or infinite time horizons by successively computing optimal controls over a moving finite time horizon. Numerical experiments illustrate that the proposed solution approach can be successfully applied although some of the assumptions which are necessary to conduct the theoretical analysis cannot be guaranteed for the studied tests.
International audienceWe study the approximation of optimal control problems via the solution of a Hamilton-Jacobi equation in a tube around a reference trajectory which is first obtained solving a Model Predictive Control problem. The coupling between the two methods is introduced to improve the initial local solution and to reduce the computational complexity of the Dynamic Programming algorithm. We present some features of the method and show some results obtained via this technique showing that it can produce an improvement with respect to the two uncoupled methods
Often a dynamical system is characterized by one or more parameters describing physical features of the problem or geometrical configurations of the computational domain. As a consequence, by assuming that the system is controllable, a range of optimal controls exists corresponding to different parameter values. The goal of the proposed approach is to avoid the computation of a control function for any instance of the parameters. The greedy controllability consists in the selection of the most representative values of the parameter set that allows a rapid approximation of the control function for any desired new parameter value, ensuring that the system is steered to the target within a certain accuracy. By proposing the reduced basis (RB) method in this framework, we are able to consider linear parametrized partial differential equations (PDEs) in our setting. The computational costs are drastically reduced and the efficiency of the greedy controllability approach is significantly improved. As a numerical example a heat equation with convection is studied to illustrate our proposed RB greedy controllability strategy.
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