Geodesic
Optimal control problem for a hyperbolic system with delay on the boundary in the class of smooth control actions
Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Optimal Control, Tome 183 (2020), pp. 14-21

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In this paper, we examine the optimal control problem for a hyperbolic system with differential constraints on the boundary, taking into account the delay. Controls are selected from the class of smooth functions that satisfy pointwise constraints. Problems of this type arise, in particular, in modeling the processes of population dynamics. The approach proposed in this paper is based on the use of “internal variations” of the control, which preserves the smoothness of the control function and ensures the fulfillment of pointwise constraints. We obtain an estimate of the state increment, prove a necessary optimality condition, and develop a scheme of an iterative method.
Keywords: hyperbolic system, system with delay, necessary optimality condition, smooth control. 
@article{INTO_2020_183_a1,
     author = {A. V. Arguchintsev and V. P. Poplevko},
     title = {Optimal control problem for a hyperbolic system with delay on the boundary in the class of smooth control actions},
     journal = {Itogi nauki i tehniki. Sovremenna\^a matematika i e\"e prilo\v{z}eni\^a. Temati\v{c}eskie obzory},
     pages = {14--21},
     publisher = {mathdoc},
     volume = {183},
     year = {2020},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/INTO_2020_183_a1/}
}
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A. V. Arguchintsev; V. P. Poplevko. Optimal control problem for a hyperbolic system with delay on the boundary in the class of smooth control actions. Itogi nauki i tehniki. Sovremennaâ matematika i eë priloženiâ. Tematičeskie obzory, Differential Equations and Optimal Control, Tome 183 (2020), pp. 14-21. http://geodesic.mathdoc.fr/item/INTO_2020_183_a1/