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Том 221, № 1 (2017)
- Год: 2017
- Статей: 2
- URL: https://ogarev-online.ru/1072-3374/issue/view/14808
Article
Typicality of Chaotic Fractal Behavior of Integral Vortices in Hamiltonian Systems with Discontinuous Right Hand Side
Аннотация
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.
1-136
Optimal Synthesis in the Control Problem of an n-Link Inverted Pendulum with a Moving Base
Аннотация
In this paper, we consider the problem of stabilization of an n-link inverted pendulum on a movable base (cart). A cart is allowed to move along the horizontal axis. A force applied to the cart is considered as a control. The problem is to minimize the mean square deviation of the pendulum from the vertical line. For the linearized model, we show that, for small deviations from the upper unstable equilibrium position, the optimal regime contains trajectories with more and more frequent switchings. Namely, the optimal trajectories with infinite number of switchings are shown to attain, in finite time, the singular surface and then continue these motion with singular control over the singular surface, approaching the origin in an infinite time. It is shown that the costructed solutions are globally optimal.
137-153