Dynamic programming approach to the numerical solution of optimal control with paradigm by a mathematical model for drug therapies of HIV/AIDS

Parameter estimation in nonlinear mixed effect models based on ordinary differential equations: an optimal control approach

“…ODE models are standard in population dynamics, epidemiology, virology, pharmacokinetics, or genetic regulation networks analysis due to their ability to describe the main mechanisms of interaction between different biological components of complex systems, their evolution in time and to provide reasonable approximations of stochastic dynamics Perelson et al (1996); Lavielle and Mentré (2007); Wakefield and Racine-Poon (1995); Andraud et al (2012); Pasin et al (2019); M. Lavielle and Mentre (2011); Le et al (2015); Engl et al (2009); Wu et al (2014). Evidence of the relevance of ODEs resides for example in their joint use with control theory methods for the purpose of optimal treatment design Guo and Sun (2012); Agusto and Adekunle (2014); Zhang and Xu (2016); Pasin et al (2018); Villain et al (2019). In cases of experimental designs involving a large number of subjects and limited number of individual measurements, non-linear mixed-effect models may be more relevant than subject-by-subject model to gather information from the whole population while allowing between-individual variability.…”

A feedback design for numerical solution to optimal control problems based on Hamilton-Jacobi-Bellman equation

“…Many unnecessary disturbances and noise signals from outside the system can be rejected by using feedback ( [4]). Furthermore, even from the numerical solution point of view, for the two-point boundary problem obtained by the necessary condition of optimality, the recognized shooting method has to overcome the difficulty of "guess" for the initial data to start the iterative numerical process ( [14,27]), which is generally not an easy job.…”

“…It makes some effective algorithms for low dimensional problems, thus initially confined to only "toy" problems, can be generalized to deal with much more complicated practical optimal control problems. The upwind finite-difference scheme is such a well-adapted algorithm and has been successfully applied to many examples ( [13,14,15]). And most of all, its convergence has been rigorously proven in [28].…”

“…We can unfortunately select only some more relevant research to mention here. For two practical optimal control problems, [14,15] obtain the optimal feedback controls by repeatedly calling the algorithm to the HJB equation. In [2,10,18], the HJB equation is firstly solved to gain the value function.…”

“…Next we revisit the upwind finite-difference scheme to the HJB equation (2.5). Actually, it is such a well-adapted algorithm and has been successfully applied to many examples ( [14,13,15]). And most of all, its convergence has been rigorously proven in [28].…”