Controllability on infinite time horizon for first and second order functional differential inclusions in Banach spaces (original) (raw)
An existence result for impulsive functional differential inclusions in Banach spaces
Discussiones Mathematicae. Differential Inclusions, Control and Optimization, 2004
We use the topological degree theory for condensing multimaps to present an existence result for impulsive semilinear functional differential inclusions in Banach spaces, moreover under some additional assumptions we prove the compactness of the solution set.
On semilinear differential inclusions in Banach spaces with nondensely defined operators
Journal of Fixed Point Theory and Applications, 2011
We consider a semilinear differential inclusion in a Banach space assuming that its linear part is a nondensely defined Hille-Yosida operator whereas Carathèodory-type multivalued nonlinearity satisfies a regularity condition expressed in terms of the Hausdorff measure of noncompactness. We apply the theory of integrated semigroups and the fixed point theory of condensing multivalued maps to obtain local and global existence results and to prove the continuous dependence of the solutions set on initial data. An application to an optimization problem for a feedback control system is given.
Existence results for functional semilinear differential inclusions in Fréchet spaces
Mathematical and Computer Modelling, 2008
In this paper, a recent Frigon nonlinear alternative for contractive multivalued maps in Fréchet spaces, combined with semigroup theory, is used to investigate the existence of integral solutions for first order semilinear functional differential inclusions. An application to a control problem is studied. We assume that the linear part of the differential inclusion is a nondensely defined operator and satisfies the Hille-Yosida condition.
Banach fixed-point theorem in semilinear controllability problems – a survey
The main aim of this article is to review the existing state of art concerning the complete controllability of semilinear dynamical systems. The study focus on obtaining the sufficient conditions for the complete controllability for various systems using the Banach fixed-point theorem. We describe the results for stochastic semilinear functional integro-differential system, stochastic partial differential equations with finite delays, semilinear functional equations, a stochastic semilinear system, a impulsive stochastic integro-differential system, semilinear stochastic impulsive systems, an impulsive neutral functional evolution integro-differential system and a nonlinear stochastic neutral impulsive system. Finally, two examples are presented.
A Tikhonov type theorem for abstract parabolic differential inclusions in Banach spaces
We consider a class of singularly perturbed systems of semilinear parabolic differential inclusions in infinite dimensional spaces. For such a class we prove a Tikhonov-type theorem for a suitably defined subset of the set of all solutions for ε ≥ 0, where ε is the perturbation parameter. Specifically, assuming the existence of a Lipschitz selector of the involved multivalued maps we can define a nonempty subset Z L (ε) of the solution set of the singularly perturbed system. This subset is the set of the Hölder continuous solutions defined in [0, d], d > 0 † Research supported by the project "Qualitative analysis and control of dynamical systems" at the University of Siena and FRBR grant 99-01-00333.
Controllability for systems governed by semilinear evolution inclusions without compactness
Nonlinear Differential Equations and Applications NoDEA, 2014
In this paper we study the controllability for a class of semilinear differential inclusions in Banach spaces. Since we assume the regularity of the nonlinear part with respect to the weak topology, we do not require the compactness of the evolution operator generated by the linear part. As well we are not posing any conditions on the multivalued nonlinearity expressed in terms of measures of noncompactness. We are considering the usual assumption on the controllability of the associated linear problem. Notice that, in infinite dimensional spaces, the above mentioned compactness of the evolution operator and linear controllability condition are in contradiction with each other. We suppose that the nonlinear term has convex, closed, bounded values and a weakly sequentially closed graph when restricted to its second argument. This regularity setting allows us to solve controllability problem under various growth conditions. As application, a controllability result for hyperbolic integro-differential equations and inclusions is obtained. In particular, we consider controllability of a system arising in a model of nonlocal spatial population dispersal and a system governed by the second order one-dimensional telegraph equation.
Discussiones Mathematicae. Differential Inclusions, Control and Optimization, 2011
We study a controllability problem for a system governed by a semilinear functional differential inclusion in a Banach space in the presence of impulse effects and delay. Assuming a regularity of the multivalued non-linearity in terms of the Hausdorff measure of noncompactness we do not require the compactness of the evolution operator generated by the linear part of inclusion. We find existence results for mild solutions of this problem under various growth conditions on the nonlinear part 40 I. Benedetti, V. Obukhovskii and P. Zecca and on the jump functions. As example, we consider the controllability of an impulsive system governed by a wave equation with delayed feedback.