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Math. Comput. Appl. 2018, 23(3), 45; https://doi.org/10.3390/mca23030045

Optimal Strategies for Psoriasis Treatment

1
TWU Department of Mathematics and Computer Science, Texas Woman’s University, Denton, TX 76204, USA
2
MSU Faculty of Computational Mathematics and Cybernetics, Lomonosov Moscow State University, Moscow 119992, Russia
*
Author to whom correspondence should be addressed.
Received: 1 August 2018 / Revised: 31 August 2018 / Accepted: 31 August 2018 / Published: 4 September 2018
(This article belongs to the Special Issue Optimization in Control Applications)
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Abstract

Within a given time interval we consider a nonlinear system of differential equations describing psoriasis treatment. Its phase variables define the concentrations of T-lymphocytes, keratinocytes and dendritic cells. Two scalar bounded controls are introduced into this system to reflect medication dosages aimed at suppressing interactions between T-lymphocytes and keratinocytes, and between T-lymphocytes and dendritic cells. For such a controlled system, a minimization problem of the concentration of keratinocytes at the terminal time is considered. For its analysis, the Pontryagin maximum principle is applied. As a result of this analysis, the properties of the optimal controls and their possible types are established. It is shown that each of these controls is either a bang-bang type on the entire time interval or (in addition to bang-bang type) contains a singular arc. The obtained analytical results are confirmed by numerical calculations using the software “BOCOP-2.0.5”. Their detailed analysis and the corresponding conclusions are presented. View Full-Text
Keywords: psoriasis; nonlinear control system; optimal control; Pontryagin maximum principle; switching function; Lie brackets; singular arc; chattering control psoriasis; nonlinear control system; optimal control; Pontryagin maximum principle; switching function; Lie brackets; singular arc; chattering control
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Grigorieva, E.; Khailov, E. Optimal Strategies for Psoriasis Treatment. Math. Comput. Appl. 2018, 23, 45.

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