arXiv subjects

Lev Lokutsievskiy

Publications and source records attributed to Lev Lokutsievskiy.

9 recordsLinked to original sources

arXiv2026

Isoperimetric problem for quasi-Finsler metrics and shapes of least aerodynamic resistance

We consider the following model in the framework of Newtonian aerodynamics: a 2D convex body moves forward and slowly oscillates in a rarefied medium on the plane. The law of oscillations is given. The problem is to find a body of fixed area that has the smallest resistance. We solve this problem by reducing it to an isoperimetric problem for quasi-Finsler metrics. Further, we introduce two criteria of smallness of oscillations. We show that the optimal body has singularity at the front (back) part of its boundary iff the former (latter) criterion is satisfied. The rest of the boundary is smooth. Finally, we find exact optimal shapes in the case of uniform oscillations and construct several shapes explicitly.

math.OC↗

arXiv2024

Strict Lyapunov functions and energy decay in Hamiltonian chains with degenerate damping

We consider a Hamiltonian chain of rotators (in general nonlinear) in which the first rotator is damped. Being motivated by problems of nonequilibrium statistical mechanics of crystals, we construct a strict Lyapunov function that allows us to find a lower bound for the total energy dissipation rate when the energy and time are large. Our construction is explicit and its analysis is rather straightforward. We rely on a method going back to Matrosov, Malisoff and Mazenc, which we review in our paper. The method is rather universal and we show that it is applicable to a chain of oscillators as well.

math.DS↗

arXiv2023

Derivatives of Sub-Riemannian Geodesics are $L_p$-Hölder Continuous

This article is devoted to the long-standing problem on the smoothness of sub-Riemannian geodesics. We prove that the derivatives of sub-Riemannian geodesics are always $L_p$-Hölder continuous. Additionally, this result has several interesting implications. These include (i) the decay of Fourier coefficients on abnormal controls, (ii) the rate at which they can be approximated by smooth functions, (iii) a generalization of the Poincaré inequality, and (iv) a compact embedding of the set of shortest paths into the space of Bessel potentials.

math.DG↗

arXiv2022

Asymptotic control theory for a closed string II

We develop an asymptotical control theory for one of the simplest distributed (infinite dimensional) oscillating systems, namely, for a closed string under a bounded load applied to a single distinguished point. We find exact classes of string states that admit complete damping and an asymptotically exact value of the required time. By using approximate reachable sets instead of exact ones, we design a feedback control, which turns out to be asymptotically optimal. The main results are an exact algebraic formula for asymptotic shape of the reachable sets, asymptotically optimal time of motion, and an asymptotically optimal control thus constructed.

math.OC↗

arXiv2021

Reachable sets for two-level open quantum systems driven by coherent and incoherent controls

In this work, we study controllability in the set of all density matrices for a two-level open quantum system driven by coherent and incoherent controls. In [A. Pechen, Phys. Rev. A 84, 042106 (2011)] an approximate controllability, i.e., controllability with some precision, was shown for generic $N$-level open quantum systems driven by coherent and incoherent controls. However, the explicit formulation of this property, including the behavior of this precision as a function of transition frequencies and decoherence rates of the system, was not known. The present work provides a rigorous analytical study of reachable sets for two-level open quantum systems. First, it is shown that for $N=2$ the presence of incoherent control does not affect the reachable set (while incoherent control may affect the time necessary to reach particular state). Second, the reachable set in the Bloch ball is described and it is shown that already just for one coherent control any point in the Bloch ball can be achieved with precision $δ\sim γ/ω$, where $γ$ is the decoherence rate and $ω$ is the transition frequency. Typical values are $δ\lesssim10^{-3}$ that implies high accuracy of achieving any density matrix. Moreover, we show that most points in the Bloch ball can be exactly reached, except of two lacunae of size $\simδ$. For two coherent controls, the system is shown to be completely controllable in the set of all density matrices. Third, the reachable set as a function of the final time is found and shown to exhibit a non-trivial structure.

quant-ph↗

arXiv2021

Non-optimality of conical parts for Newton's problem of minimal resistance in the class of convex bodies and the limiting case of infinite height

We consider Newton's problem of minimal resistance, in particular we address the problem arising in the limit if the height goes to infinity. We establish existence of solutions and lack radial symmetry of solutions. Moreover, we show that certain conical parts contained in the boundary of a convex body inhibit the optimality in the classical Newton's problem with finite height. This result is applied to certain bodies considered in the literature, which are conjectured to be optimal for the classical Newton's problem, and we show that they are not.

math.OC↗

arXiv2015

Typicalness of chaotic fractal behaviour of integral vortexes in Hamiltonian systems with discontinuous right hand side

We consider a linear-quadratic deterministic optimal control problem where the control takes 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 finite time, so-called chattering. We prove the presence of an entirely new phenomenon, namely the chaotic behaviour of bounded pieces of optimal trajectories. We find the hyperbolic domains in the neighbourhood of a homoclinic point and estimate the corresponding contraction-extension coefficients. This gives us the possibility to calculate the entropy and the Hausdorff dimension of the non-wandering 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 behaviour is generic for piece-wise smooth Hamiltonian systems in the vicinity of a junction of three discontinuity hyper-surface strata.

math.DS↗

arXiv2011

Universal spaces for finite group actions on spaces of type $K(π,1)$

In this paper author proposes a construction of a universal space of type $K(π,1)$ such that any action (up to homotopy conjugation) of a given finite group $G$ on spaces of the same homotopy type is presented on the constructed space. Moreover, any action of $G$ on any space of type $K(π,1)$ is covered by some action of $G$ on this universal space.

math.AT↗

arXiv2010

Homotopy classification of finite group actions on aspherical spaces

The author proposes a method for investigating actions of finite groups on aspherical spaces. Complete homotopy classification of free actions of finite groups on aspherical spaces is obtained. Also there are some results about non-free actions. For example a relation between the cohomology of finite groups and the lattice structure of its subgroups is obtained by the proposed method. This relation is formulated in terms of spectral sequences.

math.GN↗