Stepwise Solutions for Optimal Control Problems

Yıl 2016, Cilt: 13 Sayı: 2, - , 01.11.2016

Mehdi Afshar Farshad Merrikh-bayat Mohammad Reza Razvan

Öz

A new stepwise method for solving optimal control problems is introduced. The main motivation
for developing this new approach is the limitation of the continuous-time Pontryagin Maximum Principle
(PMP) where all control functions must be continuous. However, in many real-world applications such as
drug injection or resource allocation problems, it is not practical to continuously change the control. In
dealing with these problems it is strictly preferred to change the control only at certain moments of time and
keep it constant otherwise. Clearly, in this case the resulting stepwise solution cannot be calculated optimally
using PMP since it is not continuous anymore. The other advantage of stepwise solutions is that they can
be obtained much easier compared to the PMP approach when the system has complex dynamics or the cost
function is more complicated. Some numerical examples are solved by using both the classical PMP and the
proposed stepwise method and the results are compared, which prove the high performance of the proposed
method.

Anahtar Kelimeler

Optimal control theory, Heuristic and meta-heuristic optimization, Nonlinear dynamics

Kaynakça

• [1] S. P. Sethi, and G. L. Thompson, Optimal Control Theory: Applications to Management Science and Economics, Kluwer, Boston, 2nd edition, (2000).
• [2] S. Lenhart, J. T. Workman, Optimal Control Applied to Biological Models, Chapman and Hall, London, (2007).
• [3] K. Renee Fister, J.C. Panetta, Optimal Control Applied to Competing Chemotherapeutic Cell-kill Strategies, Chapman and Hall, London, (2007).
• [4] J.M. Hyman, J. Li, Differential Susceptibility and Infectivity Epidemic Models, Mathematical Bioscience and Engineering, 3(1), (2006), 89–100.
• [5] M. Afshar, M.R. Razvan. Optimal control of the Differential Infectivity Models, International Journal of Applied and Computational Mathematics, (2015), 1–15.
• [6] D. Kirschner, S. Lenhart, S. SerBin, Optimal Control of the Chemotherapy of HIV, Journal of Mathematical Biology, 35, (2007), 775–792.
• [7] K. Fister, J. Donnelly, Immunotherapy: An Optimal Control Theory Approach, Mathematical Biosciences and Engineering, 2(3), (20059; 499–510.
• [8] H.R. Thieme, Mathematics in Population Biology, Princeton University Press, Princeton, (2003).
• [9] M. McAsey, L. Mou, W. Han, Convergence of the Forward-backward Sweep Method in Optimal Control, Computational Optimization and Applications, 53(1), (2012), 207–226.