Abstract:
A mathematical model of psoriasis treatment is considered on a given time interval. The model consists of three nonlinear differential equations that describe the relationship between the concentrations of T-lymphocytes, keratinocytes, and dendritic cells. The model also includes a bounded control defining the drug dose to suppress the interaction between T-lymphocytes and keratinocytes. For this model, a problem of minimizing the concentration of keratinocytes at the end point of a given time interval is stated. The Pontryagin maximum principle is applied to the analysis of this optimal control problem. For certain relations between the parameters of the model, this principle is used for studying a possible existence of a third-order singular arc of optimal control. Namely, the corresponding necessary optimality condition is verified, and formulas for the optimal solutions of differential equations on this arc are obtained. Finally, a connection of a control on such an arc with nonsingular bang-bang arcs of optimal control is investigated.
The work of the first author was supported by the Russian Foundation for Basic Research and by the Department of Science and Technology of the Government of India, project no. 18-51-45003 IND_a.
Citation:
E. N. Khailov, E. V. Grigorieva, “On a Third-Order Singular Arc of Optimal Control in a Minimization Problem for a Mathematical Model of Psoriasis Treatment”, Optimal control and differential equations, Collected papers. On the occasion of the 110th anniversary of the birth of Academician Lev Semenovich Pontryagin, Trudy Mat. Inst. Steklova, 304, Steklov Math. Inst. RAS, Moscow, 2019, 298–308; Proc. Steklov Inst. Math., 304 (2019), 281–291
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\by E.~N.~Khailov, E.~V.~Grigorieva
\paper On a Third-Order Singular Arc of Optimal Control in a Minimization Problem for a Mathematical Model of Psoriasis Treatment
\inbook Optimal control and differential equations
\bookinfo Collected papers. On the occasion of the 110th anniversary of the birth of Academician Lev Semenovich Pontryagin
\serial Trudy Mat. Inst. Steklova
\yr 2019
\vol 304
\pages 298--308
\publ Steklov Math. Inst. RAS
\publaddr Moscow
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\jour Proc. Steklov Inst. Math.
\yr 2019
\vol 304
\pages 281--291
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https://doi.org/10.4213/tm3973
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This publication is cited in the following 3 articles:
E. N. Khailov, “Bang-Bang Property and Singular Regimens of Optimal Control in one Mathematical Model of Treating Psoriasis”, MoscowUniv.Comput.Math.Cybern., 49:1 (2025), 73
E. N. Khailov, E. V. Grigorieva, “Connecting a Third-Order Singular Arc with Nonsingular Arcs of Optimal Control in a Minimization Problem for a Psoriasis Treatment Model”, Proc. Steklov Inst. Math., 315 (2021), 257–269
P. K. Roy, A. K. Roy, E. N. Khailov, F. Al Basir, E. V. Grigorieva, “A model of the optimal immunotherapy of psoriasis by introducing il-10 and il-22 inhibitors”, J. Biol. Syst., 28:3 (2020), 609–639
Citation in format AMSBIB
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