Asymptotic representation of a solution to a singular perturbation linear time-optimal problem

A time-optimal control problem is considered for a linear system with fast and slow variables and smooth geometric constraints on the control. An asymptotic expansion of the optimal time up to the second order of smallness is constructed and validated. © 2013 Pleiades Publishing, Ltd

Asymptotic expansion of solution to singularly perturbed optimal control problem with convex integral quality functional with terminal part depending on slow and fast variables

We consider an optimal control problem with a convex integral quality functional for a linear system with fast and slow variables in the class of piecewise continuous controls with smooth constraints on the control, where x ∈ Rn, y ∈ Rm, u ∈ Rr; Aij and Bi, i, j = 1, 2, are constant matrices of corresponding dimension, and the functions ϕ1(·), ϕ2(·) are continuously differentiable in Rn, Rm, strictly convex, and cofinite in the sense of the convex analysis. In the general case, for such problem, the Pontryagin maximum principle is a necessary and sufficient optimality condition and there exist unique vectors le and pe determining an optimal control by the formula, The main difference of our problem from the previous papers is that the terminal part of quality functional depends on the slow and fast variables and the controlled system is a more general form. We prove that in the case of a finite number of control change points, a power asymptotic expansion can be constructed for the initial vector of dual state, which determines the type of the optimal control. © 2019 Danilin A.R., Shaburov A.A

Halo Excitation of 6^6He in Inelastic and Charge-Exchange Reactions

Four-body distorted wave theory appropriate for nucleon-nucleus reactions leading to 3-body continuum excitations of two-neutron Borromean halo nuclei is developed. The peculiarities of the halo bound state and 3-body continuum are fully taken into account by using the method of hyperspherical harmonics. The procedure is applied for A=6 test-bench nuclei; thus we report detailed studies of inclusive cross sections for inelastic $^6$He(p,p')$^6$He$^*$ and charge-exchange $^6$Li(n,p)$^6$He$^*$ reactions at nucleon energy 50 MeV. The theoretical low-energy spectra exhibit two resonance-like structures. The first (narrow) is the excitation of the well-known $2^+$ three-body resonance. The second (broad) bump is a composition of overlapping soft modes of multipolarities $1^-, 2^+, 1^+, 0^+$ whose relative weights depend on transferred momentum and reaction type. Inelastic scattering is the most selective tool for studying the soft dipole excitation mode.Comment: Submitted to Phys. Rev. C., 11 figures using eps

Асимптотическое разложение решения одной сингулярно возмущенной задачи оптимального управления с интегральным выпуклым критерием качества, терминальная часть которого аддитивно зависит от медленных и быстрых переменных

The paper deals with the problem of optimal control with a Boltz–type quality index over a finite time interval for a linear steady–state control system in the class of piecewise continuous controls with smooth control constraints. In particular, we study the problem of controlling the motion of a system of small mass points under the action of a bounded force. The terminal part of the convex integral quality index additively depends on slow and fast variables, and the integral term is a strictly convex function of control variable. If the system is completely controllable, then the Pontryagin maximum principle is a necessary and sufficient condition for optimality. The main difference between this study and previous works is that the equation contains the zero matrix of fast variables and, thus, the results of A. B. Vasilieva on the asymptotic of the fundamental matrix of a control system are not valid. However, the linear steady–state system satisfies the condition of complete controllability. The article shows that problems of optimal control with a convex integral quality index are more regular than time–optimal problems. © 2020 Izvestiya Instituta Matematiki i Informatiki Udmurtskogo Gosudarstvennogo Universiteta. All rights reserved

Asymptotics of the solution to a singularly perturbed timeoptimal control problem with two small parameters

The paper continues the author's previous studies. We consider a time-optimal control problem for a singularly perturbed linear autonomous system with two independent small parameters and smooth geometric constraints on the control in the form of a ball (Formula Presented) The main difference of this case from the systems with fast and slow variables studied earlier is that here the matrix J at the fast variables is the second-order Jordan block with zero eigenvalue and, thus, does not satisfy the standard asymptotic stability condition. Continuing the research, we consider initial conditions depending on the second small parameter μ. We derive and justify a complete asymptotic expansion in the sense of Erdelyi of the optimal time and optimal control with respect to the asymptotic sequence "("k + μk), 0 < < 1. Keywords: Optimal control, time-optimal control problem, asymptotic expansion, singularly perturbed problem, small parameter. © 2019 Trudy Instituta Matematiki i Mekhaniki UrO RAN. All rights reserved

Asymptotics of the Solution to a Singularly Perturbed Time-Optimal Control Problem with Two Small Parameters

The paper continues the authors’ previous studies. We consider a time-optimal control problem for a singularly perturbed linear autonomous system with two independent small parameters and smooth geometric constraints on the control in the form of a ball cases formulae-sequence superscript R2 superscript R2 missing-subexpressionmissing-subexpressionsuperscript 3 formulae-sequencenorm 1formulae-sequence0 much-less-than 1missing-subexpressionmissing-subexpressionformulae-sequence 0subscript 0 superscriptsubscript 01superscript 3 0subscript 0missing-subexpressionmissing-subexpressionmissing-subexpressionformulae-sequence subscript 0formulae-sequence subscript 0 subscript missing-subexpressionmissing-subexpressionmissing-subexpression where 0100 The main difference of this case from the systems with fast and slow variables studied earlier is that here the matrix at the fast variables is the second-order Jordan block with zero eigenvalue and, thus, does not satisfy the standard asymptotic stability condition. Continuing the research, we consider initial conditions depending on the second small parameter. We derive and justify a complete asymptotic expansion in the sense of Erdelyi of the optimal time and optimal control with respect to the asymptotic sequence superscript superscript superscript, 01. © 2020, Pleiades Publishing, Ltd.This work was partially supported by the Russian Academic Excellence Project (agreement no. 02.A03.21.0006 of August 27, 2013, between the Ministry of Education and Science of the Russian Federation and Ural Federal University)

Computations of Three-Body Continuum Spectra

We formulate a method to solve the coordinate space Faddeev equations for positive energies. The method employs hyperspherical coordinates and analytical expressions for the effective potentials at large distances. Realistic computations of the parameters of the resonances and the strength functions are carried out for the Borromean halo nucleus 6He (n+n+alpha) for J = 0+, 0-, 1+, 1-, 2+,2-. PACS numbers: 21.45.+v, 11.80.Jy, 31.15.Ja, 21.60.GxComment: 10 pages, 3 postscript figures, LaTeX, epsf.sty, corrected misprints in the caption of Fig.