Abstract
In this paper, we consider linear-quadratic deterministic optimal control problems where the controls take values in a two-dimensional simplex. The phase portrait of the optimal synthesis contains second-order singular extremals and exhibits modes of infinite accumulations of switchings in a finite time, so-called chattering. We prove the presence of an entirely new phenomenon, namely, the chaotic behavior of bounded pieces of optimal trajectories. We find the hyperbolic domains in the neighborhood of a homoclinic point and estimate the corresponding contraction-extension coefficients. This gives us a possibility of calculating the entropy and the Hausdorff dimension of the nonwandering set, which appears to have a Cantor-like structure as in Smale’s horseshoe. The dynamics of the system is described by a topological Markov chain. In the second part it is shown that this behavior is generic for piecewise smooth Hamiltonian systems in the vicinity of a junction of three discontinuity hyper-surface strata.
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Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 56, Dynamical Systems and Optimal Control, 2015.
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Zelikin, M.I., Lokutsievskii, L.V. & Hildebrand, R. Typicality of Chaotic Fractal Behavior of Integral Vortices in Hamiltonian Systems with Discontinuous Right Hand Side. J Math Sci 221, 1–136 (2017). https://doi.org/10.1007/s10958-017-3221-y
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DOI: https://doi.org/10.1007/s10958-017-3221-y