Alexander Tolstonogov

• Institute for System Dynamics and Control Theory, Russian Academy of Sciences

В сепарабельных гильбертовых пространствах рассматриваются связанные между собой эволюционное включение и процесс выметания (sweeping process). Эволюционное включение описывается с помощью максимально монотонных операторов, зависящих от времени, и переменных состояния как включения, так и процесса выметания. Оно содержит многозначное возмущение с з...

We study the minimization problem for an integral functional on the solutions to a coupled system. The system consists of an evolutionary inclusion in a separable Hilbert space with maximal monotone operators and an ordinary differential equation in a separable Banach space containing the control. The control constraint is a multivalued mapping wit...

The existence of an absolutely continuous solution of a differential inclusion whose right-hand side contains a time- and state-dependent maximal monotone operator and a nonconvex perturbation is proved in a Hilbert space. The proofs are based on our comparison theorems for inclusions with maximal monotone operators and a fixed point theorem for mu...

We consider a sweeping process with a triple perturbation defined on a separable Hilbert space. The values of the moving set are time- and state-dependent prox-regular sets. The perturbation is given by the sum of three multivalued mappings having different semicontinuity properties with respect to the state variable. The first mapping with closed...

In this paper we study the existence and properties of solutions for a discontinuous sweeping process involving prox-regular sets in a Hilbert spaces. The variation of the moving set is controlled by a positive Radon measure and the perturbation is the sum of two multivalued mappings. The values of the first one are closed, bounded, not necessarily...

An evolution inclusion with time-dependent family of maximal monotone operators in considered in a separable Hilbert space. If the elements with minimum norm of the family of maximal monotone operators satisfy certain growth conditions, then the domains of definition of this family are closed convex sets. Hence the sweeping process is well defined,...

A differential inclusion with a time-dependent maximal monotone operator and a perturbation is studied in a separable Hilbert space. The perturbation is the sum of a time-dependent single-valued operator and a multivalued mapping with closed nonconvex values. A particular feature of the single-valued operator is that its sum its with the identity o...

We introduce a series of distances between maximal monotone operators and study their properties. As applications, we consider the existence of solutions to evolutionary inclusions with maximal monotone operators.

В сепарабельном гильбертовом пространстве изучается эволюционное включение с зависящим от времени семейством максимально монотонных операторов. Если элементы минимальной нормы семейства максимально монотонных операторов удовлетворяют условиям роста, то области определения семейства максимально монотонных операторов являются замкнутыми выпуклыми мно...

We consider a sequence of superposition operators (Nemytskii operators) from the space of square-integrable functions on a line segment to a separable Hilbert space. Each term of the sequence is generated by a time-dependent family of maximal monotone operators in the Hilbert space. Under sufficiently general assumptions we show that every superpos...

We consider some questions on G-convergence of a sequence of Nemytskii maximal monotone operators defined on the space of square integrable functions acting from a real interval to a separable Hilbert space. Every Nemytskii operator is generated by a time dependent family of maximal monotone operators. The values of these maximal monotone operators...

A convex sweeping process is considered in a separable Hilbert space. The majority of works on sweeping processes use the Hausdorff distance to describe the movement of the convex set generating the process. However, for unbounded sets the use of the Hausdorff distance does not always guarantee the fulfilment of conditions under which a solution ex...

Continuing the first part of this work, we study an implicit evolution inclusion with time-dependent maximal monotone operator in a separable Hilbert space. This inclusion involves some perturbation given by a multivalued history-dependent operator. We establish the existence of a solution. The solution is unique provided that the values of the mul...

This paper addresses an evolution inclusion of subdifferential type with a multivalued perturbation. The values of the latter are closed, not necessarily convex sets. Our inclusion is implicit in the sense that the velocity enters it implicitly: the subdifferential is evaluated not at the state, but at a function depending both on the state and the...

A differential inclusion whose right-hand side is the sum of two multivalued mappings is considered in a separable Banach space. The values of one mapping are closed, bounded, and not necessarily convex sets. This mapping is measurable in the time variable, is Lipschitz in the phase variable, and satisfies the traditional growth condition. The valu...

В сепарабельном гильбертовом пространстве рассматривается семейство максимально монотонных операторов с областями определения, зависящими на отрезке числовой прямой от времени. Рассматривается также пространство интегрируемых с квадратом функций, определенных на этом отрезке, со значениями в указанном гильбертовом пространстве. Исходя из семейства...

On the space of continuous functions from a line segment to a reflexive Banach space, we consider some operator whose values are closed convex subsets of the space. If the values are singletons, the operator becomes a well-known single-valued history-dependent operator. We study the properties of the operator, prove a fixed-point theorem analogous...

An evolution inclusion with the right-hand side containing a time-dependent maximal monotone operator and a multivalued mapping with closed nonconvex values is studied in a separable Hilbert space. The dependence of the maximal monotone operator on time is described with the help of the distance between maximal monotone operators in the sense of Vl...

A measurable sweeping process with a composed perturbation is considered in a separable Hilbert space. The values of the moving set generating the sweeping process are closed, convex sets. The retraction of the sweeping process is bounded by a positive Radon measure. The perturbation is the sum of two multivalued mappings. The values of the first o...

Рассматривается задача минимизации интегрального функционала на решениях управляемой системы, описываемой нелинейным дифференциальным уравнением в сепарабельном банаховом пространстве и вариационным неравенством. Это вариационное неравенство определяет гистерезисный оператор, входом которого является траектория управляемой системы, а выход содержит...

В сепарабельном банаховом пространстве рассматривается дифференциальное включение, правая часть которого является суммой двух многозначных отображений. Значениями первого являются замкнутые ограниченные не обязательно выпуклые множества, и оно является липшицевым по фазовой переменной. Значениями второго отображения являются замкнутые множества, и...

We study the multivalued mappings on a closed interval of the real line whose values are polyhedra in a separable Hilbert space. The polyhedron space is endowed with the metric of the Mosco convergence of sequences of closed convex sets. A polyhedron is defined as the intersection of finitely many closed half-spaces. The equations of the correspond...

В статье рассматриваются правила вычисления плотностей борелевских мер, абсолютно непрерывных относительно положительной неатомической меры Радона. Борелевские меры порождены сложными функциями, которые зависят от определенных на отрезке непрерывных функций ограниченной вариации. Изучаются вопросы абсолютной непрерывности борелевских мер, порожденн...

Rules for calculating the densities of Borel measures which are absolutely continuous with respect to a positive nonatomic Radon measure are considered. The Borel measures are generated by composite functions which depend on continuous functions of bounded variation defined on an interval. Questions related to the absolute continuity of Borel measu...

Достаточным условием существования абсолютно непрерывного решения процесса выметания является абсолютная непрерывность в определенном смысле многозначного отображения, порождающего процесс выметания. Это свойство описывается в терминах расстояния по Хаусдорфу между значениями многозначного отображения. Однако существуют многозначные отображения, дл...

We consider a space of continuous set-valued mappings defined on a locally compact space T with a countable base. The values of these mappings are closed (not necessarily bounded) sets in a metric space (X, d(·)) whose closed balls are compact. The space (X, d(·)) is locally compact and separable. Let Y be a countable dense set in X. The distance ρ...

An evolution inclusion of subdifferential type with time dependent subdifferentials and a multivalued perturbation is considered in a separable Hilbert space. The perturbation has closed unbounded values. We also consider the inclusion with the values of the perturbation being convexified (convexified inclusion). The notion of a regular solution fo...

A polyhedral sweeping process with a multivalued perturbation whose values are nonconvex unbounded sets is studied in a separable Hilbert space. Polyhedral sweeping processes do not satisfy the traditional assumptions used to prove existence theorems for convex sweeping processes. We consider the polyhedral sweeping process as an evolution inclusio...

In a separable Banach space we consider a differential inclusion whose values are nonconvex, closed, but not necessarily bounded sets. Along with the original inclusion, we consider the inclusion with convexified right-hand side. We prove existence theorems and establish relations between solutions to the original and convexified differential inclu...

The present paper is concerned with a nonlinear partial differential control system arising in phase transition modeling. The control constraint is given by a state-dependent nonconvex-valued multivalued mapping. We study the problem of minimization of a given integral cost functional over solutions to the above system. The cost integrand is noncon...

An evolution inclusion is considered in a separable Hilbert space. The right-hand side of the inclusion contains the subdifferential of a time-dependent proper convex lower semicontinuous function and a multivalued perturbation with nonempty closed, not necessarily, bound-ed values. Along with this inclusion we consider the inclusion with the pertu...

An evolution inclusion with the right-hand side containing the difference of subdifferentials of proper convex lower semicontinuous functions and a multivalued perturbation whose values are nonconvex closed sets is considered in a separable Hilbert space. In addition to the original inclusion, we consider an inclusion with convexified perturbation...

The paper studies an evolution inclusion in a separable Hilbert space whose right-hand side contains the subdifferential of a proper convex lower semicontinuous function of time and a set-valued perturbation. Together with this inclusion, an inclusion with convexified perturbation values is considered. The existence and density of the solution set...

We consider the problem of minimization of an integral functional with a nonconvex with respect to the control integrand. We minimize our functional over the solution set of a control system with mixed nonconvex control constraint. The right-hand side of the system contains the difference of the subdifferentials of two proper convex, lower semicont...

A control system with internal and external controls is considered in a finite dimensional space. The internal control is a multivalued function of time with convex closed possibly unbounded values, while the external control is represented by a measurable function of time. The internal control acts on the system by normal cones at the points of co...

A differential inclusion in which the values of the right-hand side are nonconvex closed possibly unbounded sets is considered in a finite-dimensional space. Existence theorems for solutions and a relaxation theorem are proved. Relaxation theorems for a differential inclusion with bounded right-hand side, as a rule, are proved under the Lipschitz c...

The Nemytskii operator generated by a multivalued mapping whose values are compacts from a Banach space is considered. In every point this mapping is either upper semicontinuous and has convex values or it is lower semicontinuous in a neighborhood of the point. We prove that the multivalued Nemytskii operator has a multivalued selector with convex...

We consider a nonlinear partial differential control system describing phase transitions taking account of hysteresis effects. The control constraint is given by a multivalued mapping with nonconvex closed bounded values in a finite dimensional space depending on the phase variables. Existence of solutions and topological properties of the set of a...

A convex sweeping process with unbounded nonconvex perturbation term of a rather general form is investigated in a separable Hilbert space. We consider our sweeping process as an evolution inclusion with subdifferential operators. This allows to prove the theorem on existence of solutions under conditions generalizing the usual assumptions for conv...

We study the variational stability of an optimal control problem for a Volterra-type nonlinear functional-operator equation. This means that for this optimal control problem (P ɛ ) with a parameter ɛ we study how its minimum value min(P ɛ ) and its set of minimizers argmin(P ɛ ) depend on ɛ. We illustrate the use of the variational stability theore...

We consider a control system described by two nonlinear related equations. The first equation expresses connection between the input and output of the hysteresis operator, whereas the second one is the diffusion equation. The control constraint is expressed by a multivalued mapping of a phase variable with closed nonconvex values in a finitedimensi...

We consider the problem of minimization of an integral functional with nonconvex with respect to the control integrand. We minimize our functional over the solution set of a control system described by two ordinary differential equations subject to a control constraint given by a multivalued mapping with closed nonconvex values. The coefficients of...

The problem of minimization of an integral functional with an integrand that is nonconvex with respect to the control is considered. We minimize our functional over the solution set of a nonlinear evolution control system with a time-dependent subdifferential operator in a Hilbert space. The control constraint is given by a nonconvex closed bounded...

We consider a control system described by two ordinary nonlinear differential equations subject to a control constraint given by a multivalued mapping with closed nonconvex values, which depends on the phase variables. One of the equations contains the subdifferential of the indicator function of a closed convex set depending on the unknown phase v...

This paper is concerned with the problem of minimizing an integral functional with control-nonconvex integrand over the class of solutions of a control system in a Hilbert space subject to a control constraint given by a phase-dependent multivalued map with closed nonconvex values. The integrand, the subdifferential operators, the perturbation term...

We consider a control system described by an evolution equation with control constraint which is a multivalued mapping of a phase variable with closed nonconvex values. One of the evolution operators of the system is the subdifferential of a time-dependent proper, convex, and lower semicontinuous function. The other operator, acting on the derivati...

Questions relating to the Mosco convergence of integral functionals defined on the space of square integrable functions taking values in a Hilbert space are investigated. The integrands of these functionals are time-dependent proper, convex, lower semicontinuous functions on the Hilbert space. The results obtained are applied to the analysis of the...

In a separable Hilbert space, we consider a control system with a subdifferential operator and a non-linear perturbation of monotonic type. The control is subject to a restriction that is a multi-valued map depending on the phase variables with closed non-convex values in a reflexive separable Banach space. The subdifferential operator, the perturb...

We present some extreme continuous selector theorems, synthesizing the author's results; namely, we study existence and properties of continuous selectors from the set of extreme points of multifunctions with closed convex decomposable values in the space of Bochner integrable functions.

An analogue of the classical theorem of Bogolyubov with non-convex constraint is proved. The constraint is the solution set of a differential inclusion with non-convex lower semicontinuous right-hand side. As an application we study the interrelation between the solutions of the problem of minimizing an integral functional with non-convex integrand...

A control system governed by a non-linear first-order evolution equation with mixed non-convex control constraints is examined. The system depends on parameters entering all its data, including the non-linear evolution operator and the control constraints. The system with convexified control constraints is also considered. The general concept of -c...

The problem is considered of minimizing an integral functional with integrand that is not convex in the control, on solutions of a control system described by a first-order non-linear evolution equation with mixed non-convex constraints on the control. A relaxation problem is treated along with the original problem. Under appropriate assumptions it...

A class of multivalued maps with non-convex non-closed decomposable values is distinguished, and theorems are proved on the existence of continuous selections for such maps. This class contains multivalued maps whose values are extreme points of continuous multivalued maps with closed convex decomposable values in a Banach space of Bochner-integrab...

In a separable Hilbert space we consider a control system with evolution operators that are subdifferentials of a proper convex lower semicontinuous function depending on time. The constraint on the control is given by a multivalued function with non-convex values that is lower semicontinuous with respect to the variable states. Along with the orig...

We prove an analogue of Bogolyubov's theorem with constraints in the form of a controlled second-order evolution system. The main assertion of this theorem deals with relations between the values of an integral functional that is non-convex with respect to control on the solutions of a controlled system with non-convex constraints on the control an...

In this article a differential inclusion is considered, where the mapping takes values in the family of all nonempty compact convex subsets of a Banach space, is upper semicontinuous with respect to for almost every , and has a strongly measurable selection for every . Under certain compactness conditions on proofs are given for a theorem on the ex...

We prove a theorem on the existence of solutions of the minimization problem for the integral functional on solutions of the controlled system described by the Goursat-Darboux equation, which is linear with respect to the phase variables and their derivatives, with constraints on the control, the phase variables, and their first partial derivatives...

This article gives an approach to the study of both differential inclusions and ordinary differential equations in a Banach space X. The central point concerns the question of the existence and properties of the solution set of a differential inclusion whose right-hand side has the weak Scorza Dragoni property.Bibliography: 37 titles.

A continuous version of a theorem of Lyapunov on convexity for measures with values in a Banach space is proved, and then used to obtain two results on the existence of a common continuous selection of finitely many multivalued mappings with values in a space of Bochner-integrable functions. These results are applied to the investigation of propert...

We consider the minimization problem of an integral functional in a separable Hilbert space with integrand not convex in the control defined on solutions of the control system described by nonlinear evolutionary equations with mixed nonconvex constraints. The evolutionary operator of the system is the subdifferential of a proper, convex, lower semi...

In a separable Hilbert space we consider an evolution inclusion with a multivalued perturbation and the evolution operators that are the compositions of a linear operator and the subdifferentials of a time-dependent proper convex lower semicontinuous function. Alongside the initial inclusion, we consider a sequence of approximating evolution inclus...

We prove an analogue of the classical Bogolyubov theorem, with a nonconvex constraint. In the case we consider, the constraint is the solution set of a Cauchy problem for a differential inclusion with a nonconvex right-hand side satisfying a Lipschitz condition. Our approach is based on a relaxation argument, as in the Filippov-Wazewski theorem.

An existence result is proved for set differential equation when the function involved is upper semicontinuous and the result is very general. Then, the connection between the solutions of fuzzy differential equation and the set differential equation that is generated from it, is studied.

In a separable Hilbert space we consider an evolution inclusion with a multivalued perturbation and evolution operators that are subdifferentials of a proper convex lower semicontinuous function depending on time. Along with the original inclusion, we consider a sequence of approximating evolution inclusions with the same perturbation and the evolu...

We continue the research of the first part of the article. We mainly study codensity for the set of admissible “trajectory-control” pairs of a system with nonconvex constraints in the set of admissible “trajectory-control” pairs of the system with convexified constraints. We state necessary and sufficient conditions for the set of admissible “traje...

We consider a control system described by a nonlinear second order evolution equation defined on an evolution triple of Banach spaces (Gelfand triple) with a mixed multivalued control constraint whose values are nonconvex closed sets. Alongside the original system we consider a system with the following control constraints: a constraint whose value...

We establish the existence of extreme solutions for a class of nonlinear second-order evolution inclusions with a nonconvex right-hand side defined on an evolution triple of Banach spaces. Then we show that extreme solutions which belong to the solution set of the original system are in fact dense and codense in the solution set of a system with a...

A characterization of strongly exposed points of a decomposable bounded closed convex set G Ì Lp (T,X)\Gamma \subset L_p (T,X) , where 1 \leqslant p < ¥1 \leqslant p < \infty , in terms of strongly exposed points of values of the set-valued representation F:T ® 2XF:T \to 2^X of G\Gamma is given. As a corollary, necessary conditions characte...

An existence theorem is proved for an optimal control problem with constraints that is described by a first-order linear evolution equation, without the usual convexity assumptions relating to the control. The result is obtained by using properties of a multivalued integral and quasi-concavity conditions relating to the state variable. An example o...

We consider a control system described by a non-linear first-order evolution equation on an evolution triple of Banach spaces (a "Gelfand triple") with a mixed multivalued control constraint whose values are non-convex closed sets in the control space. Besides the original system, we consider systems with the following control constraints: the cons...

Preface. 1. Multi-Valued Differential Equation Generated by a Differential Inclusion. 2. Differential Inclusions. Existence of Solutions. 3. Properties of Solutions. 4. Integral Funnel of the Differential Inclusion. 5. Inclusions with Non-Compact Right Hand Side. Appendices. References. Index. Symbols.

In this Chapter a multi-valued differential equation generated by a differential inclusion, is introduced in a semi-linear space of all convex compact sets of an initial Banach space. The solution of this equation is a multi-function of time having convex compact sets as its values. Questions of the existence of both local and global solutions of t...

In this Chapter differential inclusions with non-convex, non-compact right hand side are considered. Questions of the existence and properties of Caratheodory type of solution sets are studied.

In this Chapter we study interrelationships between a set of all, of some type or other, solutions of a differential inclusion with non-convex right hand side and a set of all, of the same type of solutions of a differential inclusions with convexified right hand side. It is shown that each, of some type or other, solution of the differential inclu...

In this Chapter questions of the existence of classical, regular, and Caratheodory type of solutions of a differential inclusion with non-convex right hand side are considered. The solutions of the differential inclusion are sought as continuous selectors of a solution of a multi-valued differential equation generated by a differential inclusion, a...