Typicality of chaotic fractal behavior of integral vortices in Hamiltonian systems with discontinuous right hand side

M. I. Zelikin ; L. V. Lokutsievskii ; R. Hildebrand

Contemporary Mathematics. Fundamental Directions, Optimal control, Tome 56 (2015), pp. 5-128

Résumé

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.