Andrei Dmitruk - Academia.edu (original) (raw)

Papers by Andrei Dmitruk

Research paper thumbnail of Lyusternik theorem and regularity of collections of sets

The paper is devoted to a revision of the metric regularity property for mappings between metric ... more The paper is devoted to a revision of the metric regularity property for mappings between metric or Banach spaces. Some new concepts are introduced: uniform metric regularity and metric multi-regularity for mappings into product spaces, when each component is perturbed independently. Regularity criteria are established based on a nonlocal version of Lyusternik-Graves theorem due to Milyutin. The criteria are applied to systems of generalized equations producing some "error bound" type estimates.

Research paper thumbnail of Necessary conditions for a weak minimum in optimal control problems with integral equations on a variable time interval

Discrete and Continuous Dynamical Systems, 2015

Research paper thumbnail of Conditions quadratiques pour des extrémales bang-singulières

This paper deals with optimal control problems for systems affine in the control variable. We con... more This paper deals with optimal control problems for systems affine in the control variable. We consider nonnegativity constraints on the control, and finitely many equality and inequality constraints on the final state. First, we obtain second order necessary optimality conditions. Secondly, we derive a second order sufficient condition for the scalar control case.

Research paper thumbnail of Lyusternik’s theorem and extremum theory

Research paper thumbnail of Solution by means of the maximum principle of a problem of energetically optimal control of train motion

Research paper thumbnail of A condition of Legendre type for optimal control problems, linear in control

For singular extremals in the above class of problems conditions of a Pontryagin minimum include ... more For singular extremals in the above class of problems conditions of a Pontryagin minimum include a new pointwise condition, i.e. a condition of Legendre type. It involves not only the second, but also the third variation of Lagrange function, and requires to solve a nontrivial auxiliary optimal control problem. Some cases of its exact solution are presented.

Research paper thumbnail of On the proof of Pontryagin's maximum principle by means of needle variations

We propose a proof of the maximum principle for the general Pontryagin type optimal control probl... more We propose a proof of the maximum principle for the general Pontryagin type optimal control problem, based on packages of needle variations. The optimal control problem is first reduced to a family of smooth finite-dimensional problems, the arguments of which are the widths of the needles in each packet, then, for each of these problems, the standard Lagrange multipliers rule is applied, and finally, the obtained family of necessary conditions is "compressed" in one universal optimality condition by using the concept of centered family of compacta.

Research paper thumbnail of A simplified Goddard problem in the presence of a nonlinear media resistance and a bounded thrust

We consider a problem of maximization of the distance traveled by a material point in the presenc... more We consider a problem of maximization of the distance traveled by a material point in the presence of a nonlinear friction under a bounded thrust and fuel expenditure. Using the maximum principle we obtain the form of optimal control and establish conditions under which it contains a singular subarc. This problem seems to be the simplest one having a mechanical sense in which singular subarcs appear in a nontrivial way.

Research paper thumbnail of Classification of extremals in a simplified Goddard model on the maximal height of rocket flight

We consider a problem on maximizing the height of vertical flight of a material point ("mete... more We consider a problem on maximizing the height of vertical flight of a material point ("meteorological rocket") in the presence of a nonlinear friction and a constant flat gravity field under a bounded thrust and fuel expenditure. The original Goddard problem is simplified by removing the dependence on the rocket mass from the equations of motion. Using the maximum principle we find all possible types of Pontryagin extremals and classify them w.r.t. problem parameters. Since the velocity of the point can be negative, we obtain some new types of extremals with two or three switching points, which optimality should be further investigated.

Research paper thumbnail of Necessary Conditions for a Weak Minimum in Optimal Control Problems with Integral Equations Subject to State and Mixed Constraints

Research paper thumbnail of Solution of the problem of the energetically optimal control of the motion of a train by the maximum principle

Research paper thumbnail of The Hybrid Maximum Principle is a consequence of Pontryagin Maximum Principle

Systems & Control Letters, 2008

We give a simple proof of the Maximum Principle for smooth hybrid control systems by reducing the... more We give a simple proof of the Maximum Principle for smooth hybrid control systems by reducing the hybrid problem to an optimal control problem of Pontryagin type and then by using the classical Pontryagin Maximum Principle.

Research paper thumbnail of О равномерной сходимости решений управляемой системы интегральных уравнений типа Вольтерра, линейной по управлению

Математические заметки, 2014

Том 96 выпуск 3 сентябрь 2014 УДК 517.968.22 О равномерной сходимости решений управляемой системы... more Том 96 выпуск 3 сентябрь 2014 УДК 517.968.22 О равномерной сходимости решений управляемой системы интегральных уравнений типа Вольтерра, линейной по управлению Ю. И. Белоглазов, А. В. Дмитрук

Research paper thumbnail of On a trolley-like problem in the presence of a nonlinear friction and a bounded fuel expenditure

2013 9th Asian Control Conference (ASCC), 2013

We consider a problem of maximization of the distance traveled by a material point in the presenc... more We consider a problem of maximization of the distance traveled by a material point in the presence of a nonlinear friction under a bounded thrust and fuel expenditure. Using the maximum principle we obtain the form of optimal control and establish conditions under which it contains a singular subarc. This problem seems to be the simplest one having a mechanical sense in which singular subarcs appear in a nontrivial way.

Research paper thumbnail of Алексей Алексеевич Милютин (некролог)

Успехи математических наук, 2002

Research paper thumbnail of Quadratic order conditions for bang-singular extremals

Numerical Algebra, Control and Optimization, 2012

This paper deals with optimal control problems for systems affine in the control variable. We hav... more This paper deals with optimal control problems for systems affine in the control variable. We have nonnegativity constraints on the control, and finitely many equality and inequality constraints on the final state. First, we obtain second order necessary optimality conditions. Secondly, we get a second order sufficient condition for the scalar control case. The results use in an essential way the Goh transformation.

Research paper thumbnail of Lyusternik's Theorem and the Theory of Extrema

Russian Mathematical Surveys, 1980

Research paper thumbnail of Aleksei Alekseevich Milyutin (obituary)

Russian Mathematical Surveys, 2002

Research paper thumbnail of Теорема существования в задаче оптимального управления на бесконечном интервале времени

Математические заметки, 2005

Research paper thumbnail of Existence Theorem in the Optimal Control Problem on an Infinite Time Interval

Mathematical Notes, 2005

We consider an optimal control problem on an infinite time interval. The system is linear in the ... more We consider an optimal control problem on an infinite time interval. The system is linear in the control, the cost functional is convex in the control, and the control set is convex and compact. We propose a new condition on the behavior of the cost at infinity, which is weaker than the previously known conditions, and prove the existence theorem under this condition. We consider several special cases and propose a general abstract scheme.

Research paper thumbnail of Lyusternik theorem and regularity of collections of sets

The paper is devoted to a revision of the metric regularity property for mappings between metric ... more The paper is devoted to a revision of the metric regularity property for mappings between metric or Banach spaces. Some new concepts are introduced: uniform metric regularity and metric multi-regularity for mappings into product spaces, when each component is perturbed independently. Regularity criteria are established based on a nonlocal version of Lyusternik-Graves theorem due to Milyutin. The criteria are applied to systems of generalized equations producing some "error bound" type estimates.

Research paper thumbnail of Necessary conditions for a weak minimum in optimal control problems with integral equations on a variable time interval

Discrete and Continuous Dynamical Systems, 2015

Research paper thumbnail of Conditions quadratiques pour des extrémales bang-singulières

This paper deals with optimal control problems for systems affine in the control variable. We con... more This paper deals with optimal control problems for systems affine in the control variable. We consider nonnegativity constraints on the control, and finitely many equality and inequality constraints on the final state. First, we obtain second order necessary optimality conditions. Secondly, we derive a second order sufficient condition for the scalar control case.

Research paper thumbnail of Lyusternik’s theorem and extremum theory

Research paper thumbnail of Solution by means of the maximum principle of a problem of energetically optimal control of train motion

Research paper thumbnail of A condition of Legendre type for optimal control problems, linear in control

For singular extremals in the above class of problems conditions of a Pontryagin minimum include ... more For singular extremals in the above class of problems conditions of a Pontryagin minimum include a new pointwise condition, i.e. a condition of Legendre type. It involves not only the second, but also the third variation of Lagrange function, and requires to solve a nontrivial auxiliary optimal control problem. Some cases of its exact solution are presented.

Research paper thumbnail of On the proof of Pontryagin's maximum principle by means of needle variations

We propose a proof of the maximum principle for the general Pontryagin type optimal control probl... more We propose a proof of the maximum principle for the general Pontryagin type optimal control problem, based on packages of needle variations. The optimal control problem is first reduced to a family of smooth finite-dimensional problems, the arguments of which are the widths of the needles in each packet, then, for each of these problems, the standard Lagrange multipliers rule is applied, and finally, the obtained family of necessary conditions is "compressed" in one universal optimality condition by using the concept of centered family of compacta.

Research paper thumbnail of A simplified Goddard problem in the presence of a nonlinear media resistance and a bounded thrust

We consider a problem of maximization of the distance traveled by a material point in the presenc... more We consider a problem of maximization of the distance traveled by a material point in the presence of a nonlinear friction under a bounded thrust and fuel expenditure. Using the maximum principle we obtain the form of optimal control and establish conditions under which it contains a singular subarc. This problem seems to be the simplest one having a mechanical sense in which singular subarcs appear in a nontrivial way.

Research paper thumbnail of Classification of extremals in a simplified Goddard model on the maximal height of rocket flight

We consider a problem on maximizing the height of vertical flight of a material point ("mete... more We consider a problem on maximizing the height of vertical flight of a material point ("meteorological rocket") in the presence of a nonlinear friction and a constant flat gravity field under a bounded thrust and fuel expenditure. The original Goddard problem is simplified by removing the dependence on the rocket mass from the equations of motion. Using the maximum principle we find all possible types of Pontryagin extremals and classify them w.r.t. problem parameters. Since the velocity of the point can be negative, we obtain some new types of extremals with two or three switching points, which optimality should be further investigated.

Research paper thumbnail of Necessary Conditions for a Weak Minimum in Optimal Control Problems with Integral Equations Subject to State and Mixed Constraints

Research paper thumbnail of Solution of the problem of the energetically optimal control of the motion of a train by the maximum principle

Research paper thumbnail of The Hybrid Maximum Principle is a consequence of Pontryagin Maximum Principle

Systems & Control Letters, 2008

We give a simple proof of the Maximum Principle for smooth hybrid control systems by reducing the... more We give a simple proof of the Maximum Principle for smooth hybrid control systems by reducing the hybrid problem to an optimal control problem of Pontryagin type and then by using the classical Pontryagin Maximum Principle.

Research paper thumbnail of О равномерной сходимости решений управляемой системы интегральных уравнений типа Вольтерра, линейной по управлению

Математические заметки, 2014

Том 96 выпуск 3 сентябрь 2014 УДК 517.968.22 О равномерной сходимости решений управляемой системы... more Том 96 выпуск 3 сентябрь 2014 УДК 517.968.22 О равномерной сходимости решений управляемой системы интегральных уравнений типа Вольтерра, линейной по управлению Ю. И. Белоглазов, А. В. Дмитрук

Research paper thumbnail of On a trolley-like problem in the presence of a nonlinear friction and a bounded fuel expenditure

2013 9th Asian Control Conference (ASCC), 2013

We consider a problem of maximization of the distance traveled by a material point in the presenc... more We consider a problem of maximization of the distance traveled by a material point in the presence of a nonlinear friction under a bounded thrust and fuel expenditure. Using the maximum principle we obtain the form of optimal control and establish conditions under which it contains a singular subarc. This problem seems to be the simplest one having a mechanical sense in which singular subarcs appear in a nontrivial way.

Research paper thumbnail of Алексей Алексеевич Милютин (некролог)

Успехи математических наук, 2002

Research paper thumbnail of Quadratic order conditions for bang-singular extremals

Numerical Algebra, Control and Optimization, 2012

This paper deals with optimal control problems for systems affine in the control variable. We hav... more This paper deals with optimal control problems for systems affine in the control variable. We have nonnegativity constraints on the control, and finitely many equality and inequality constraints on the final state. First, we obtain second order necessary optimality conditions. Secondly, we get a second order sufficient condition for the scalar control case. The results use in an essential way the Goh transformation.

Research paper thumbnail of Lyusternik's Theorem and the Theory of Extrema

Russian Mathematical Surveys, 1980

Research paper thumbnail of Aleksei Alekseevich Milyutin (obituary)

Russian Mathematical Surveys, 2002

Research paper thumbnail of Теорема существования в задаче оптимального управления на бесконечном интервале времени

Математические заметки, 2005

Research paper thumbnail of Existence Theorem in the Optimal Control Problem on an Infinite Time Interval

Mathematical Notes, 2005

We consider an optimal control problem on an infinite time interval. The system is linear in the ... more We consider an optimal control problem on an infinite time interval. The system is linear in the control, the cost functional is convex in the control, and the control set is convex and compact. We propose a new condition on the behavior of the cost at infinity, which is weaker than the previously known conditions, and prove the existence theorem under this condition. We consider several special cases and propose a general abstract scheme.