Abstract
We consider three optimal control problems (for linear, bilinear, and quadratic functionals) for a special bilinear system with a matrix of rank 1. For the first problem, we obtain two versions of conditions for initial data of the system and functional such that the maximum principle becomes a sufficient optimality condition. In this case, the problem becomes very simple: the optimal control is determined in the integration process of the phase or adjoint system (one Cauchy problem). Next, the optimization problem for a bilinear functional is considered. Sufficient optimality conditions for the boundary controls without switching points are obtained. These conditions are represented as inequalities for functions of one variable (time). The optimal control problem with the quadratic functional is reduced to the bilinear case on the basis of special increment formula.
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References
P. D. Alessandro, “Bilinearity and sensitivity in macroeconomy,” Lect. Notes Econ. Math. Syst., 111, 170–199 (1975).
V. G. Antonik and V. A. Srochko, “Optimality conditions of the maximum principle type in bilinear control problems,” Zh. Vychisl. Mat. Mat. Fiz., 56, No. 12, 2054–2064 (2016).
E. N. Khailov, “On extremal controls of a homogeneous bilinear system which is controlled in the positive orthant,” Tr. Mat. Inst. Steklova, 220, 217–235 (1998).
G. Oster, “Bilinear models in ecology,” Lect. Notes Econ. Math. Syst., 162, 260–271 (1978).
A. S. Perelson, “Applications of optimal control theory to immunology,” Lect. Notes Econ. Math. Syst., 162, 272–287 (1978).
V. Srochko, V. Antonik, and E. Aksenyushkina, “Sufficient optimality conditions for extremal controls based on functional increment formulas,” Num. Alg. Control Optim., 7, No. 2, 191–199 (2017).
A. Swierniak, “Some control problems for simplest differential models of proliferation cycle,” Appl. Math. Comput. Sci., 4, No. 2, 223–232 (1994).
A. Swierniak, “Cell cycle as an object of control,” J. Biol. Syst., 3, No. 1, 41–54 (1995).
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Translated from Itogi Nauki i Tekhniki, Seriya Sovremennaya Matematika i Ee Prilozheniya. Tematicheskie Obzory, Vol. 183, Differential Equations and Optimal Control, 2020.
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Srochko, V.A., Antonik, V.G. & Aksenyushkina, E.V. Optimal Control Problems for Bilinear Systems of Special Structure. J Math Sci 279, 701–709 (2024). https://doi.org/10.1007/s10958-024-07049-5
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DOI: https://doi.org/10.1007/s10958-024-07049-5