Separable Reflexive Banach Space Research Articles

An analysis on time optimal feedback control for Caputo fractional neutral systems with delay

ABSTRACT This paper aims to provide a systematic approach to investigate the optimal feedback control systems governed by the fractional neutral evolution system with fixed delay in the sense of Caputo fractional derivatives in separable reflexive Banach spaces. Initially, we study the existence result of mild solutions for the system by applying fixed point theorem for a condensing map due to Martelli. The existence of feasible pairs is obtained by using the Filippov theorem and the Cesari property. Furthermore, the existence of optimal feedback control and time optimal control are studied for the corresponding control problem. Finally, an example is given to validate our findings.

Upper semicontinuous convex-valued selectors of a Nemytskii operator with nonconvex values and evolution inclusions with maximal monotone operators

On the space of continuous functions, we consider a multivalued superposition operator, a Nemytskii operator valued in the space of p-integrable functions with 1≤p<∞. The Nemytskii operator is generated by a multivalued mapping defined on the direct product of a segment of the real line and a separable reflexive Banach space and having closed values in this space. At each point of the phase variable, the mapping either has a closed graph and is convex-valued or it is lower semicontinuous in some neighborhood of this point. In recent years, such mappings are called mappings with mixed semicontinuity properties. We prove that the restriction of the Nemytskii operator onto any compact subset of the space of continuous functions with the sup-norm has a multivalued selector with convex closed values that is upper semicontinuous in the weak topology of the space of p-integrable functions. This result generalizes some well-known results of the author valid in the case when the multivalued mapping generating the Nemytskii operator has compact values. A multivalued selector theorem is used to study an evolution inclusion in a separable Hilbert space with a time-dependent maximal monotone operator and a composite perturbation that is the sum of two multivalued mappings. The values of the first multivalued mapping are closed, bounded, not necessarily convex sets. The second multivalued mapping has closed values and possesses mixed semicontinuity properties. The existence of an absolutely continuous solution is proved. The proof is based on the theorem on an upper semicontinuous convex-valued selector and the author's theorem on selectors continuous in parameter and passing through fixed points of a parameter-dependent multivalued mapping with closed, nonconvex, decomposable values in the space of p-integrable functions as well as on the classical Ky Fan's fixed point theorem. Our approach is fairly simple, concise and clear as compared to other known approaches based on discretization.

Approximate Controllability of Non-autonomous Second Order Impulsive Functional Evolution Equations in Banach Spaces

This article investigates the approximate controllability of second order non-autonomous functional evolution equations involving non-instantaneous impulses and nonlocal conditions. First, we discuss the approximate controllability of second order linear system in detail, which lacks in the existing literature. Then, we derive sufficient conditions for approximate controllability of our system in separable reflexive Banach spaces via linear evolution operator, resolvent operator conditions, and Schauder’s fixed point theorem. Moreover, in this paper, we define proper identification of resolvent operator in Banach spaces. Finally, we provide two concrete examples to validate our results.

Approximate controllability of fractional order non-instantaneous impulsive functional evolution equations with state-dependent delay in Banach spaces

Abstract This paper deals with the control problems governed by fractional impulsive functional evolution equations with state-dependent delay involving Caputo fractional derivatives in Banach spaces. The main objective of this work is to formulate sufficient conditions for the approximate controllability of the considered system in separable reflexive Banach spaces. We have exploited the resolvent operator technique and Schauder’s fixed point theorem in the proofs to achieve this goal. The approximate controllability of linear system is discussed in detail, which lacks in the existing literature. Moreover, we point out some shortcomings of the existing works in the context of characterization of mild solution, phase space, and approximate controllability of fractional order impulsive systems in Banach spaces. Finally, we investigate the approximate controllability of the fractional order heat equation with non-instantaneous impulses and delay by using the developed results.

On Reflexivity and Point Spectrum

In this note, we prove a characterization of reflexive Banach spaces with unconditional basis and study the set theoretical complexity of certain sets associated to linear bounded operators acting on reflexive separable Banach spaces.

Optimal feedback control for a class of fractional evolution equations with history-dependent operators

In this paper, we will study optimal feedback control problems derived by a class of Riemann-Liouville fractional evolution equations with history-dependent operators in separable reflexive Banach spaces. We firstly introduce suitable hypotheses to prove the existence and uniqueness of mild solutions for this kind of Riemann-Liouville fractional evolution equations with history-dependent operators. Then, by introducing a feedback iterative technique and applying Filippov theorem, we show the existence of feasible pairs and optimal control pairs of the optimal feedback control systems with history-dependent operators. Finally, we give some applications to illustrate our main results.

Existence and approximate controllability of non-autonomous functional impulsive evolution inclusions in Banach spaces

In this paper, we are concerned with the approximate controllability results for a class of impulsive functional differential control systems involving time dependent operators in Banach spaces. First, we show the existence of a mild solution for non-autonomous functional impulsive evolution inclusions in separable reflexive Banach spaces with the help of the evolution family and a generalization of the Leray-Schauder fixed point theorem for multi-valued maps. In order to establish sufficient conditions for the approximate controllability of our problem, we first consider a linear-quadratic regulator problem and obtain the optimal control in the feedback form, which contains the resolvent operator consisting of duality mapping. With the help of this optimal control, we prove the approximate controllability of the linear system and hence derive sufficient conditions for the approximate controllability of our problem. Moreover, in this paper, we rectify several shortcomings of the related works available in the literature, namely, proper identification of resolvent operator in Banach spaces, characterization of phase space in the presence of impulsive effects, and lack of compactness of the operator h(⋅)↦∫0⋅U(⋅,s)h(s)ds:L1([0,T];Y)→C([0,T];Y), where Y is a Banach space and U(⋅,⋅) is the evolution family, etc. Finally, we provide a concrete example to illustrate the efficiency of our results.

Operators Birkhoff–James Orthogonal to Spaces of Operators

Let be the space of bounded linear operators between real Banach spaces X, Y. For a closed subspace we partially solve the operator version of Birkhoff–James orthogonality problem, if is orthogonal to when does there exist a unit vector x 0 such that and is orthogonal to Z? In order to achieve this we first develop a compact optimization for a Y-valued compact operator T, via a minimax formula for in terms of point-wise best approximations, that links local optimization and global optimization. Our result gives an operator version of a classical minimax formula of Light and Cheney, proved for continuous vector-valued functions. For any separable reflexive Banach space X and for is a L 1-predual space as well as a M-ideal in Y, we show that if is orthogonal to then there is a unit vector x 0 with and is orthogonal to Z. In the general case we also give some local conditions on T, when this can be achieved.

Control systems described by a class of fractional semilinear evolution hemivariational inequalities and their relaxation property

ABSTRACT We are concerned with control systems described by a class of semilinear fractional evolution hemivariational inequalities in separable reflexive Banach spaces. The constraint on the control is a nonconvex valued multivalued function which is lower semicontinuous with respect to the variable states. Along with the original system we also consider the system in which the constraint on the control is the upper semicontinuous convex valued regularization of the original constraint. We obtain the existence results of the control systems and the relaxation property among the solution sets of these systems. In the end, an example is provided to illustrate the application of the obtained theory.

Existence results for fractional impulsive delay feedback control systems with Caputo fractional derivatives

&lt;p style='text-indent:20px;'&gt;The goal of this paper is to provide systematic approaches to study the feedback control systems governed by fractional impulsive delay evolution equations involving Caputo fractional derivatives in separable reflexive Banach spaces. This work is a continuation of previous work. We firstly give an existence result of mild solutions for the equations by applying the Banach's fixed point theorem and the Leray-Schauder alternative fixed point theorem. Next, by using the Filippove theorem and the Cesari property, we obtain the existence result of feasible pairs for the feedback control system. Finally, some applications are given to illustrate our main results.&lt;/p&gt;

Uniformly factoring weakly compact operators and parametrised dualisation

Abstract This article deals with the problem of when, given a collection $\mathcal {C}$ of weakly compact operators between separable Banach spaces, there exists a separable reflexive Banach space Z with a Schauder basis so that every element in $\mathcal {C}$ factors through Z (or through a subspace of Z). In particular, we show that there exists a reflexive space Z with a Schauder basis so that for each separable Banach space X, each weakly compact operator from X to $L_1[0,1]$ factors through Z. We also prove the following descriptive set theoretical result: Let $\mathcal {L}$ be the standard Borel space of bounded operators between separable Banach spaces. We show that if $\mathcal {B}$ is a Borel subset of weakly compact operators between Banach spaces with separable duals, then for $A \in \mathcal {B}$ , the assignment $A \to A^*$ can be realised by a Borel map $\mathcal {B}\to \mathcal {L}$ .

Approximate controllability of the non-autonomous impulsive evolution equation with state-dependent delay in Banach spaces

In this paper, we consider the non-autonomous semilinear impulsive differential equations with state-dependent delay. The approximate controllability results of the first-order systems are obtained in a separable reflexive Banach space, which has a uniformly convex dual. In order to establish sufficient conditions of the approximate controllability of such a system, we have used the theory of linear evolution systems, properties of the resolvent operator and Schauder’s fixed point theorem. Finally, we provide two concrete examples to validate our results.

Approximate controllability of a non-autonomous evolution equation in Banach spaces

In this paper, we consider a class of non-autonomous nonlinear evolution equations in separable reflexive Banach spaces. First, we consider a linear problem and establish the approximate controllability results by finding a feedback control with the help of an optimal control problem. We then establish the approximate controllability results for a semilinear differential equation in Banach spaces using the theory of linear evolution systems, properties of resolvent operator and Schauder's fixed point theorem. Finally, we provide an example of a non-autonomous, nonlinear diffusion equation in Banach spaces to validate the results we obtained.

On Gâteaux differentiability of strongly cone paraconvex vector-valued mappings

ABSTRACTWe prove that a strongly cone paraconvex mapping defined on a normed space X and taking values in a reflexive separable Banach space Y is Gâteaux differentiable on a dense subset of X. We also discuss Fréchet differentiability in the case when X is an Asplund space. Our results are generalizations of Rolewicz's theorems (Theorem 3.1) from Rolewicz [Differentiability of strongly paraconvex vector-valued functions. Funct Approx. 2011;2:273–277].

Feedback control systems governed by evolution equations

ABSTRACTThe goal of this paper is to provide systematic approaches to study the feedback control systems governed by evolution equations in separable reflexive Banach spaces. We firstly give some existence results of mild solutions for the equations by applying the Banach's fixed point theorem and the Leray–Schauder alternative fixed point theorem with Lipschitz conditions and some types of boundedness conditions. Next, by using the Filippove theorem and the Cesari property, a new set of sufficient assumptions are formulated to guarantee the existence of feasible pairs for the feedback control systems. Some existence results for an optimal control problem are given. Finally, we apply our main result to obtain a controllability result for semilinear evolution equations and existence results for a class of differential variational inequalities and Clarke's subdifferential inclusions.

Existence results for impulsive feedback control systems

The goal of this paper is to provide systematic approaches to study the feedback control systems governed by impulsive evolution equations in separable reflexive Banach spaces. We firstly give some existence results of mild solutions for the equations by applying the Banach’s fixed point theorem and the Leray–Schauder alternative fixed point theorem with Lipschitz conditions and different types of boundedness conditions. Then, by using the Filippove theorem and the Cesari property, a new set of sufficient assumptions are formulated to guarantee the existence results of feasible pairs for the feedback control systems. Finally, we apply our main result to obtain a controllability result for impulsive evolution equations and two existence results for a class of impulsive differential variational inequalities and impulsive Clarke’s subdifferential inclusions.

Integral representation of random set-valued measures

We consider random set-valued measures with values in a separable Banach space. We prove two integral representation theorems using measurable multifunctions and set-valued integrals. The first theorem is valid for all separable Banach spaces, while the second holds for reflexive separable Banach spaces.

Cylindrical Martingale Problems Associated with Lévy Generators

We introduce and discuss Levy-type cylindrical martingale problems on separable reflexive Banach spaces. Our main observations are the following: Cylindrical martingale problems have a one-to-one relation to weak solutions of stochastic partial differential equations, and well-posed problems possess the strong Markov property and a Cameron–Martin–Girsanov-type formula holds. As applications, we derive existence and uniqueness results.

Zippin’s embedding theorem and amalgamations of classes of Banach spaces

It was proved by Dodos and Ferenczi that the classes of Banach spaces with a separable dual and of separable reflexive Banach spaces are strongly bounded. In this paper, we provide an isometric version of this result.

Amalgamations of classes of Banach spaces with a monotone basis

It was proved by Argyros and Dodos that, for many classes $ C $ of separable Banach spaces which share some property $ P $, there exists an isomorphically universal space that satisfies $ P $ as well. We introduce a variant of their amalgamation technique which provides an isometrically universal space in the case that $ C $ consists of spaces with a monotone Schauder basis. For example, we prove that if $ C $ is a set of separable Banach spaces which is analytic with respect to the Effros-Borel structure and every $ X \in C $ is reflexive and has a monotone Schauder basis, then there exists a separable reflexive Banach space that is isometrically universal for $ C $.