arXiv subjects

Sergey Nikitin

Publications and source records attributed to Sergey Nikitin.

9 recordsLinked to original sources

arXiv2026

Diffusions with resetting boundaries: ergodic control and renewal optimization

We study ergodic optimization problems for one-dimensional diffusions with resetting boundaries, namely diffusions that are instantaneously reset according to prescribed distributions upon reaching designated boundary points. For controlled diffusions with resetting boundaries, we consider optimization of the ergodic cost over a broad class of admissible real-time controls. We derive a Hamilton--Jacobi--Bellman equation with non-local boundary conditions and establish a verification theorem for optimal stationary Markov controls. In the case of one resetting and one reflecting boundary, we also prove an existence result under weak integrability assumptions. We then study renewal diffusions. Using their regenerative structure, we derive explicit formulas for the ergodic cost. These formulas are applied to optimization problems over families of drift coefficients, yielding explicit characterizations of optimal drift parameters in several examples. Finally, we study an infinite-penalty regime in which boundary hitting is prohibited and characterize the corresponding optimal ergodic cost.

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arXiv2016

Leap Gradient Algorithm

The paper proposes a new algorithm for solving global univariate optimization problems. The algorithm does not require convexity of the target function. For a broad variety of target functions after performing (if necessary) several evolutionary leaps the algorithm naturally becomes the standard descent (or ascent) procedure near the global extremum. Moreover, it leads us to an efficient numerical method for calculating the global extrema of univariate real analytic functions.

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arXiv2007

Generalized persistency of excitation

The paper presents the generalized persistency of excitation conditions. They are not only valid for much broader range of applications than their classical counterparts but also elegantly prove the validity of the latter. The novelty and the significance of the approach presented in this publication is due to employing the time averaging technique.

math.GM↗

arXiv2007

Pontryagin's principle of stabilization

The paper presents necessary and sufficient conditions for a nonlinear system to be stabilized by a feedback. The conditions are based on the ideas related to the well-known Pontryagin's maximum principle. That allows us to formulate the results in terms that are valid for continuous, discontinuous, stationary and time-dependent feedbacks.

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arXiv2006

On 3-manifolds

It is well known that a three dimensional (closed, connected and compact) manifold is obtained by identifying boundary faces from a polyhedron P. The study of (\partial P)/~, the boundary \partial P with the polygonal faces identified in pairs leads us to the following conclusion: either a three dimensional manifold is homeomorphic to a sphere or to a polyhedron P with its boundary faces identified in pairs so that (\partial P)/~ is a finite number of internally flat complexes attached to each other along the edges of a finite graph that contains at least one closed circuit. Each of those internally flat complexes is obtained from a polygon where each side may be identified with more than one different sides. Moreover, Euler characteristic of (\partial P)/~ is equal to one and the fundamental group of (\partial P)/~ is not trivial.

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arXiv2005

On three dimensional stellar manifolds

It is well known that a three dimensional (closed, connected and compact) manifold is obtained by identifying boundary faces from a stellar ball a*S. The study of S/~, two dimensional stellar sphere S with 2-simplexes identified in pairs leads us to the following conclusion: either a three dimensional manifold is homeomorphic to a sphere or to a stellar ball a*S with its boundary 2-simplexes identified in pairs so that S/~ is a finite number of internally flat complexes attached to a finite graph that contains at least one closed circuit. Each of those internally flat complexes is obtained from a polygon where each side may be identified with one or more different other sides. Moreover, Euler characteristic of S/~ is equal to one and the fundamental group of S/~ is not trivial.

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arXiv2004

Coxeter groups of stellar manifolds

It is well known that a compact two dimensional surface is homeomorphic to a polygon with the edges identified in pairs. This paper not only presents a new proof of this statement but also generalizes it to any connected $n$-dimensional stellar manifold with a finite number of vertices. The study of an $n$-dimensional stellar sphere with the boundary faces identified in pairs leads us to Coxeter groups of stellar manifolds. Analysis of these Coxeter groups allows us to single out the class of so called flat stellar manifolds. Flat stellar manifolds are not only very common but also possess a range of simple properties that allow to prove some difficult long standing statements. In particular, there exists a simple proof of the Poincarè conjecture for flat manifolds.It is not possible to carry out this simple proof for a general stellar manifolds because of certain singularities that can be characterized in terms of Coxeter groups. The results and ideas presented in this paper not only open a new way that may lead to entirely combinatorial and algebraic proof of the Poincarè conjecture but also suggest a new technique for studying stellar manifolds in terms of properties of their Coxeter groups.

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arXiv2002

On a stellar structure for a stellar manifold

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arXiv2002

Proof of the Poincare' conjecture

This paper proves that any compact, closed, simply connected and connected three dimensional stellar manifold is stellar equivalent to the three dimensional sphere.

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