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Dmitry Khlopin

Publications and source records attributed to Dmitry Khlopin.

18 recordsLinked to original sources
arXiv2025

Pontryagin maximum principle for the deterministic mean field type optimal control problem via the Lagrangian approach

We study necessary optimality conditions for the deterministic mean field type free-endpoint optimal control problem. Our study relies on the Lagrangian approach that treats the mean field type control system as a crowd of infinitely many agents who are labeled by elements of some probability space. First, we derive the Pontryagin maximum principle in the Lagrangian form. Furthermore, we consider the Kantorovich and Eulerian formalizations which describe mean field type control systems via distributions on the set of trajectories and nonlocal continuity equation respectively. We prove that local minimizers in the Kantorovich or Eulerian formulations determine local minimizers within the Lagrangian approach. Using this, we deduce the Pontryagin maximum principle in the Kantorovich and Eulerian forms. To illustrate the general theory, we examine a model system of mean field type linear quadratic regulator. We show that the optimal strategy in this case is determined by a linear feedback.

math.OC↗
arXiv2022

Aluminum Cayley trees as scalable, broadband, multi-resonant optical antennas

An optical antenna can convert a propagative optical radiation into a localized excitation, and reciprocally. Although optical antennas can be readily created using resonant nanoparticles (metallic or dielectric) as elementary building blocks, the realization of antennas sustaining multiple resonances over a broad range of frequencies remains a challenging task. Here, we use aluminum self-similar, fractal-like structures as broadband optical antennas. Using electron energy loss spectroscopy, we experimentally evidence that a single aluminum Cayley tree, a simple self-similar structure, sustains multiple plasmonic resonances. The spectral position of these resonances is scalable over a broad spectral range spanning two decades, from ultraviolet to mid-infrared. Such multi-resonant structures are highly desirable for applications ranging from non-linear optics to light harvesting and photodetection, as well as surface-enhanced infrared absorption spectroscopy.

physics.optics↗
arXiv2018

One simple remark concerning the uniform value

The paper is devoted to dynamic games. We consider a general enough framework, which is not limited to e.g. differential games and could accommodate both discrete and continuous time. Assuming common dynamics, we study two game families with total payoffs that are defined either as the Cesàro average (long run average game family) or Abel average (discounting game family) of the running costs. We study a robust strategy that would provide a near-optimal total payoff for all sufficiently small discounts and for all sufficiently large planning horizons. Assuming merely the Dynamic Programming Principle, we prove the following Tauberian theorem: if a strategy is uniformly optimal for one of the families (when discount goes to zero for discounting games, when planning horizon goes to infinity in long run average games) and its value functions converge uniformly, then, for the other family, this strategy is also uniformly optimal and its value functions converge uniformly to the same limit.

math.OC↗
arXiv2017

Value Asymptotics in Dynamic Games on Large Horizons

This paper is concerned with two-person dynamic zero-sum games. Let games for some family have common dynamics, running costs and capabilities of players, and let these games differ in densities only. We show that the Dynamic Programming Principle directly leads to the General Tauberian Theorem---that the existence of a uniform limit of the value functions for uniform distribution or for exponential distribution implies that the value functions uniformly converge to the same limit for arbitrary distribution from large class. No assumptions on strategies are necessary. Applications to differential games and stochastic statement are considered.

math.OC↗
arXiv2017

Local field enhancement and thermoplasmonics in multimodal Aluminum structures

Aluminum nanostructures have recently been at the focus of numerous studies due to their properties including oxidation stability and surface plasmon resonances covering the ultraviolet and visible spectral windows. In this article, we reveal a new facet of this metal relevant for both plasmonics purpose and photo-thermal conversion. The field distribution of high order plasmonic resonances existing in two-dimensional Al structures is studied by nonlinear photoluminescence (nPL) microscopy in a spectral region where electronic interband transitions occur. The polarization sensitivity of the field intensity maps shows that the electric field concentration can be addressed and controlled on-demand. We use a numerical tool based on the Green dyadic method to analyze our results and to simulate the absorbed energy that is locally converted into heat. The polarization-dependent temperature increase of the Al structures is experimentally quantitatively measured, and is in an excellent agreement with theoretical predictions. Our work highlights Al as a promising candidate for designing thermal nanosources integrated in coplanar geometries for thermally assisted nanomanipulation or biophysical applications.

cond-mat.mes-hall↗
arXiv2017

On transversality condition for overtaking optimality in infinite horizon control problem

In this paper we investigate necessary conditions of optimality for infinite-horizon optimal control problems with overtaking optimality as an optimality criterion. For the case of local Lipschitz continuity of the payoff function, we construct a boundary condition on the co-state arc that is necessary for the optimality. We also show that, under additional assumptions on the payoff function's asymptotic behavior, the Pontryagin Maximum Principle with this condition becomes a complete system of relations, and this boundary condition points out the unique co-state arc through a Cauchy-type formula. An example is given to clarify the application of this formula as an explicit expression of the co-state arc. The cornerstone of this paper is the theorem on convergence of subdifferentials.

math.OC↗
arXiv2017

Tauberian theorem for value functions

For two-person dynamic zero-sum games (both discrete and continuous settings), we investigate the limit of value functions of finite horizon games with long run average cost as the time horizon tends to infinity and the limit of value functions of $λ$-discounted games as the discount tends to zero. We prove that the Dynamic Programming Principle for value functions directly leads to the Tauberian Theorem---that the existence of a uniform limit of the value functions for one of the families implies that the other one also uniformly converges to the same limit. No assumptions on strategies are necessary. To this end, we consider a mapping that takes each payoff to the corresponding value function and preserves the sub- and super- optimality principles (the Dynamic Programming Principle). With their aid, we obtain certain inequalities on asymptotics of sub- and super- solutions, which lead to the Tauberian Theorem. In particular, we consider the case of differential games without relying on the existence of the saddle point; a very simple stochastic game model is also considered.

math.OC↗
arXiv2016

On Uniform Tauberian Theorems for Dynamic Games

The paper is concerned with two-person dynamic zero-sum games. We investigate the limit of value functions of finite horizon games with long run average cost as the time horizon tends to infinity, and the limit of value functions of $λ$-discounted games as the discount tends to zero. Under quite weak assumptions on the game, we prove the Uniform Tauberian Theorem: existence a of uniform limit for one of the value functions implies the uniform convergence of the other one to the same limit. We also prove the analogs of the One-sided Tauberian Theorem, i.e., the inequalities on asymptotics of the lower and upper game. Special attention is devoted to the case of differential games. The key roles in the proof were played by Bellman's optimality principle and the closedness of strategies under concatenation.

math.OC↗
arXiv2016

On Lipschitz continuity of value functions for infinite horizon problem

We investigate conditions of optimality for an infinite horizon control problem and consider their correspondence with the value function. Assuming Lipschitz continuity of the value function, we prove that sensitivity relations plus the normal form version of the Pontryagin Maximum Principle is a necessary and sufficient condition for the optimality criteria that correspond to this value function. Different criteria of optimality under different asymptotic constraints may be used, including almost strong and classical optimality proposed by D.Bogusz. Special attention is devoted to weakly agreeable criteria. We also obtain the conditions on control system (like controllability) that guarantee the Lipschitz continuity of the value function, without any other asymptotic conditions besides finiteness of the value function. Some examples are discussed. In particular, it was shown that the same control, regarded as agreeable optimal and overtaking optimal control, can correspond to different (everywhere) value functions.

math.OC↗
arXiv2016

On an example for the Uniform Tauberian theorem in abstract control systems

The paper is devoted to the asymptotic behavior of value functions of abstract control problem with the long-time and discounted averages. The Uniform Tauberian Theorem for these problems states that the uniform convergence of value functions for long-time~averages (as the horizon tends to infinity) is equivalent to the uniform convergence of value functions for discounted averages (as the discount tends to zero), and that the limits are identical. According to Miquel Oliu-Barton and Guillaume Vigeral, this assertion holds if the set of all feasible processes is closed with respect to concatenation. In this paper, we refine this condition.

math.OC↗
arXiv2015

On Hamiltonian as limiting gradient in infinite horizon problem

Necessary conditions of optimality in the form of the Pontryagin Maximum Principle are derived for the Bolza-type discounted problem with free right end. The optimality is understood in the sense of the uniformly overtaking optimality. Such process is assumed to exist, and the corresponding payoff of the optimal process (expressed in the form of improper integral) is assumed to converge in the Riemann sense. No other assumptions on the asymptotic behaviour of trajectories or adjoint variables are required. In this paper, we prove that there exists a corresponding limiting solution of the Pontryagin Maximum Principle that satisfies the Michel transversality condition; in particular, the stationarity condition of the maximized Hamiltonian and the fact that the maximized Hamiltonian vanishes at infinity are proved. The connection of this condition with the limiting subdifferentials of payoff function along the optimal process at infinity is showed. The case of payoff without discount multiplier is also considered.

math.OC↗
arXiv2015

On asymptotic value for dynamic games with saddle point

The paper is concerned with two-person games with saddle point. We investigate the limits of value functions for long-time-average payoff, discounted average payoff, and the payoff that follows a probability density. Most of our assumptions restrict the dynamics of games. In particular, we assume the closedness of strategies under concatenation. It is also necessary for the value function to satisfy Bellman's optimality principle, even if in a weakened, asymptotic sense. We provide two results. The first one is a uniform Tauber result for games: if the value functions for long-time-average payoff converge uniformly, then there exists the uniform limit for probability densities from a sufficiently broad set; moreover, these limits coincide. The second one is the uniform Abel result: if a uniform limit for self-similar densities exists, then the uniform limit for long-time average payoff also exists, and they coincide.

math.OC↗
arXiv2014

Necessity of limiting co-state arc in Bolza-type infinite horizon problem

We investigate necessary conditions of optimality for the Bolza-type infinite horizon problem with free right end. The optimality is understood in the sense of weakly uniformly overtaking optimal control. No previous knowledge in the asymptotic behaviour of trajectories or adjoint variables is necessary. Following Seierstad's idea, we obtain the necessary boundary condition at infinity in the form of a transversality condition for the maximum principle. Those transversality conditions may be expressed in the integral form through an Aseev--Kryazhimskii-type formulae for co-state arcs. The connection between these formulae and limiting gradients of payoff function at infinity is identified; several conditions under which it is possible to explicitly specify the co-state arc through those Aseev--Kryazhimskii-type formulae are found. For infinite horizon problem of Bolza type, an example is given to clarify the use of the Aseev--Kryazhimskii formula as explicit expression of the co-state arc.

math.OC↗
arXiv2012

On necessary boundary conditions for strictly optimal control in infinite horizon control problems

In the paper we consider the infinite horizon control problems on the interval with free right-hand endpoint. We obtain the necessary conditions of strict optimality. The method of the proof actually follows the classic paper by Halkin, and the boundary condition for infinity that we construct in our paper is a stronger variety of the Seierstad condition. The complete system of relations of the maximum principle that was obtained in the paper allows us to write the expression for the adjoint variable in the form of improper integral that depends only on the developing trajectory. S.M. Aseev, A.V. Kryazhimskii, and V.M. Veliov obtained the similar condition as a necessary condition for certain classes of control problems. As we note in our paper, the obtained conditions of strict optimality lead us to a redefined system of relations for sufficiently broad class of control problems. An example is considered.

math.OC↗
arXiv2012

Necessity of vanishing shadow price in infinite horizon control problems

This paper investigates the necessary optimality conditions for uniformly overtaking optimal control on infinite horizon in the free end case. %with free right endpoint. In the papers of S.M.Aseev, A.V.Kryazhimskii, V.M.Veliov, K.O.Besov there was suggested the boundary condition for equations of the Pontryagin Maximum Principle. Each optimal process corresponds to a unique solution satisfying the boundary condition. Following A.Seierstad's idea, in this paper we prove a more general geometric variety of that boundary condition. We show that this condition is necessary for uniformly overtaking optimal control on infinite horizon in the free end case. A number of assumptions under which this condition selects a unique Lagrange multiplier is obtained. The results are applicable to general non-stationary systems and the optimal objective value is not necessarily finite. Some examples are discussed.

math.OC↗
arXiv2010

On extension for infinite horizon game of pursuit-evasion

The extension of a conflict control problem with infinite horizon is constructed. This extension is the projective limit of restricted games. Relations between "sensitivity to target set" and the existence of the optimal control are studied. Special attention is paid to the pursuit-evasion game with "joint control/relaxed control".

math.OC↗