The paper deals with an optimal control problem by a system of semilinear hyperbolic equations with boundary differential conditions with delay. This problem is considered for smooth controls. Because this requirement it is impossible to prove optimality conditions of Pontryagin maximum principle type and classic optimality conditions of gradient type. Problems of this kind arise when modeling the dynamics of non-interacting age-structured populations. Independent variables in this case are the age of the individuals and the time during which the process is considered. The functions of the process state describe the age-related population density. The goal of the control problem may be to achieve the specified population densities at the end of the process.The problem of identifying the functional parameters of models can also be considered as the optimal control problem with a quadratic cost functional. For the problem we obtain a non-classic necessary optimality condition which is based on using a special control variation that provides smoothness of controls. An iterative method for improving admissible controls is developed. An illustrative example demonstrates the effectiveness of the proposed approach.

Alexander Arguchintsev, Dr. Sci. (Phys.– Math.), Prof., Irkutsk State University, 1, K. Marks St., Irkutsk, 664003, Russian Federation, tel.: (3952)99-84-40, e-mail: arguch@math.isu.ru

Vasilisa Poplevko, Cand. Sci. (Phys.– Math.), Assoc. Prof., Irkutsk State University, 1, st. K.Marks, Irkutsk, 664003, Russian Federation, tel.:(3952)52-12-82, email: vasilisa@math.isu.ru

Arguchintsev A.V., Poplevko V.P. An Optimal Control Problem by a Hyperbolic System with Boundary Delay. The Bulletin of Irkutsk State University. Series Mathematics, 2021, vol. 35, pp. 3-17. https://doi.org/10.26516/1997-7670.2021.35.3

1. Arguchintsev A.V. Optimalnoe upravlenie giperbolicheskimi sistemami [Optimal Control of Hyperbolic Systems]. Moscow, Fizmatlit Publ., 2007, 186 p.
2. Arguchintsev A., Poplevko V. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control and Optimization, 2018, vol. 8, no. 2, pp. 193–202. https://doi.org/10.3934/naco.2018011
3. Brogan W. Optimal Control Theory Applied to Systems Described by Partial Differential Equations. Advances in Control Systems, 1968, vol. 6, pp. 221–316.
4. Colombo G., Henrion R., Hoang N.D. and Mordukhovich B. S. Optimal control of sweeping processes over polyhedral controlled sets. J. Diff. Eqs, 2016, vol. 260,pp. 3397-3447. https://doi.org/10.1016/j.jde.2015.10.039
5. Faggian S., Gozzi F. On the dynamic programming approach for optimal control problems of PDE’s with age structure. Mathematical Population Studies, 2010, vol. 11, no. 3–4, pp. 233–270. https://doi.org/10.1080/08898480490513625
6. Gabasov R., Churakova S. The existence of optimal controls in systems with lag. Differencialnye Uravnenija, 1967, vol. 3, no. 12, pp. 2067–2080. (in Russian)
7. Mordukhovich B. S. Optimal control of Lipschitzian and discontinuous differential inclusions with various applications. Proc. Inst. Math. Mech. Azer. Acad. Sci., 2019, vol. 45, pp. 52–74.
8. Mordukhovich B. S. Optimal control of differential inclusions, II: Sweeping. The Bulletin of Irkutsk State University. Series Mathematics, 2020, vol. 31, pp. 62–77. https://doi.org/10.26516/1997-7670.2020.31.62
9. Sadek I. Optimal control of time-delay systems with distributed parameters. J Optim Theory Appl., 1990, vol. 67, pp. 567-585.https://doi.org/10.1007/BF00939650
10. Teo K. Existence of optimal controls for systems governed by second order linear parabolic partial delay-differential equations with first boundary conditions. J. Austral. Math. Soc., 1979, vol. 21 (Series B), pp. 21–36.
11. Teo K. L. Optimal control of systems governed by time-delayed, second-order, linear, parabolic partial differential equations with a first boundary condition. J. Optim. Theory Appl., 1979, vol. 29, pp. 437–481.https://doi.org/10.1007/BF00933144
12. Zabello L. On the theory of necessary conditions for optimality with delay and a derivative of the control. Differential Equations, 1989, vol. 25, pp. 371–379. (in Russian)