Abstract
The optimization of a bilinear functional related to a linear state system with a modular control constraint is considered. Exact formulas for the functional increment are used to obtain sufficient conditions for the optimality of extremal controls that supplement the maximum principle. These conditions are represented in the form of inequalities and equalities for one-variable functions on a time interval. The optimization of a quadratic functional with the help of a matrix conjugate function is reduced to the bilinear case.
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References
A. V. Arguchintsev, V. A. Dykhta, and V. A. Srochko, “Optimal control: Nonlocal conditions, computational methods, and the variational principle of maximum,” Russ. Math. 53 (1), 1–35 (2009).
V. A. Srochko, Iterative Methods for Optimal Control Problems (Fizmatlit, Moscow, 2000) [in Russian].
V. A. Baturin and D. E. Urbanovich, Approximate Methods for Optimal Control Based on the Expansion Principle (Nauka, Novosibirsk, 1997) [in Russian].
A. S. Strekalovsky, Elements of Nonconvex Optimization (Nauka, Novosibirsk, 2003) [in Russian].
A. I. Tyatyushkin, Multimethod Technique for Optimization of Controlled Systems (Nauka, Novosibirsk, 2006) [in Russian].
N. V. Antipina and V. A. Dykhta, “Linear Lyapunov–Krotov functions and sufficient conditions for optimality in the form of the maximum principle,” Russ. Math. 46 (12), 9–20 (2002).
V. A. Dykhta, “Analysis of sufficient optimality conditions with a set of Lyapunov-type functions,” Tr. Inst. Mat. Mekh. Ural. Otd. Ross. Akad. Nauk 16 (5), 66–75 (2010).
M. S. Nikol’skii, “On the sufficiency of the Pontryagin maximum principle in some optimization problems,” Vestn. Mosk. Univ., Ser. 15: Vychisl. Mat. Kibern., No. 1, 35–43 (2005).
V. A. Srochko and V. G. Antonik, “Sufficient optimality conditions for extremal controls based on functional increment formulas,” Russ. Math. 58 (8), 78–83 (2014).
E. V. Aksenyushkina and V. A. Srochko, “Sufficient optimality conditions for a class of nonconvex control problems,” Comput. Math. Math. Phys. 55 (10), 1642–1652 (2015).
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Original Russian Text © V.G. Antonik, V.A. Srochko, 2016, published in Zhurnal Vychislitel’noi Matematiki i Matematicheskoi Fiziki, 2016, Vol. 56, No. 12, pp. 2054–2064.
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Antonik, V.G., Srochko, V.A. Optimality conditions of the maximum principle type in bilinear control problems. Comput. Math. and Math. Phys. 56, 2023–2034 (2016). https://doi.org/10.1134/S0965542516120022
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DOI: https://doi.org/10.1134/S0965542516120022