Local Controllability of Functional Integrodifferential Equations in Banach Space

In this paper, we establish a set of sufficient conditions for the local controllability of functional integrodifferential equations in Banach space. The results are obtained by using the methods of analytic semigroups, fractional powers of operators, and a fixed-point argument. These results generalize previous results on bounded linear operators to unbounded linear operators in which the equation involves a nonlinear delay term. An application to a partial integrodifferential equation is given.     

On controllability of fractional impulsive neutral infinite delay evolution integrodifferential systems in Banach spaces

In this work, controllability of fractional impulsive neutral evolution integrodifferential systems in a Banach space has been addressed. Sufficient conditions for the controllability are established using fractional calculus, resolvent operators and Krasnoselskii’s fixed point theorem.     

Controllability of Nonlinear Neutral Evolution Integrodifferential Systems with Infinite Delay

Sufficient conditions are derived for the controllability of nonlinear neutral evolution integrodifferential systems with infinite delay in a Banach space. The results are obtained by using the resolvent operators and the Schaefer fixed-point theorem. An example is given to illustrate the results.     

On stabilization of solutions to incomplete second-order integrodifferential equations

We consider abstract incomplete linear second-order integrodifferential equations in a Hilbert space. Operator coefficients of the equations are unbounded selfadjoint nonnegative operators. These equations arise naturally in viscoelasticity and hydroelasticity. We prove a theorem on asymptotic stability of strong solutions of the equations.     

Controllability of nonlinear neutral evolution integrodifferential systems

Sufficient conditions for controllability of nonlinear neutral evolution integrodifferential systems in a Banach space are established. The results are obtained by using the resolvent operators and the Schaefer fixed-point theorem. An application to partial integrodifferential equation is given.     

Approximate Controllability for a Class of Abstract Second-Order Functional Evolution Equations

Results providing sufficient conditions for the approximate controllability of a class of second-order abstract functional evolution equations governed by the generator of a strongly continuous cosine family of linear operators, together with nonlocal initial conditions, are developed. This work extends results included in recent work by Naito, Park, Zhou, and others. The abstract theory is then applied to a parabolic partial integrodifferential equation.     

Existence results of second-order impulsive neutral functional integrodifferential inclusions with unbounded delay in Banach spaces

This paper investigates a class of second-order impulsive neutral functional integrodifferential inclusions with unbounded delay in Banach spaces. The existence of mild solutions of these inclusions is determined by using the theory of cosine families of bounded linear operators and a recent fixed point theorem for multivalued maps due to Dhage.     

Regularity of solutions of evolution integrodifferential equations with deviating argument

In this paper we prove the existence of mild and classical solutions of delay integrodifferential equation and delay integrodifferential evolution equations with nonlocal condition in Banach spaces. The regularity solutions of integrodifferential evolution equations interconnected with viscoelastic material is derived to guarantee the stabilization. The results are established by using the resolvent operators and the fixed point principles. Finally, an example is given to show the potential of the proposed techniques.     

Controllability Results for Second Order Neutral Impulsive Integrodifferential Systems

In this paper, we address the problem of controllability for abstract second order neutral integrodifferential systems with impulsive conditions in Banach spaces. Furthermore, we discuss the controllability of second-order neutral integrodifferential systems in finite dimensional spaces. The results are obtained by using strongly continuous cosine families of operators and the Banach fixed point theorem.     

State space theory of linear time-invariant systems with delays in state,control, and observation variables,II

We study a coupled mathematical system which provides a good model for important families of linear time-invariant hereditary systems: delay-differential equations, integrodifferential equations, Volterra-Stieltjes integral equations, functional differential equations of retarded and neutral types, etc. Appropriate states are constructed and associated semigroups and abstract differential equations are obtained. We emphasize the structural operator approach as in Delfour and Manitius. Control operators are added to the coupled mathematical system allowing delays in the control variables. Again structural operators are introduced to define the state and obtain abstract differential equations without delays in the control variable as in the work of Vinter and Kwong. Finally observation operators are added which allow for delays in the observation variable. Again a new state and a state equations are constructed in such a way that no delay appear in the new observation operator as in the recent work of D. Salamon.     

SELFADJOINT EXTENSIONS OF VOLTERRA STIELTJES INTEGRODIFFERENTIAL OPERATORS

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Existence and regularity of solutions for neutral partial functional integrodifferential equations

In this paper, we study a class of neutral partial functional integrodifferential equations with finite delay by using the theory of resolvent operators. We give some sufficient conditions ensuring the existence, uniqueness and regularity of solutions. As an application, we also consider a diffusive neutral partial functional integrodifferential equation.     

Well-defined solvability and spectral analysis of abstract hyperbolic integrodifferential equations

The authors study integrodifferential equations in Hilbert space. The coefficients of the equations are unbounded and the principal part is an abstract hyperbolic equation perturbed by terms with Volterra integral operators. Such equations can be regarded as an abstract generalization of the Gurtin–Pipkin integrodifferential equation that describes heat transfer in materials with memory and has a number of other applications. Well-defined solvability of initial boundary value problems for such equations is established in weighted Sobolev spaces on the positive semi-axis. The authors examine spectral problems for operator-valued functions representing the symbols of the said equations and study the spectrum of the abstract Gurtin–Pipkin integrodifferential equation.     

Spectral analysis and correct solvability of abstract integrodifferential equations arising in thermophysics and acoustics

In the present paper, we study integrodifferential equations with unbounded operator coefficients in Hilbert spaces. The principal part of the equation is an abstract hyperbolic equation perturbed by summands with Volterra integral operators. These equations represent an abstract form of the Gurtin–Pipkin integrodifferential equation describing the process of heat conduction in media with memory and the process of sound conduction in viscoelastic media and arise in averaging problems in perforated media (the Darcy law). The correct solvability of initial-boundary problems for the specified equations is established in weighted Sobolev spaces on a positive semiaxis. Spectral problems for operator-functions are analyzed. Such functions are symbols of these equations. The spectrum of the abstract integrodifferential Gurtin–Pipkin equation is investigated.     

Controllability of Neutral Evolution Integrodifferential Systems with State Dependent Delay

In this paper, we address the problem of controllability of neutral evolution integrodifferential systems with state dependent delay in a Banach space. The results are obtained by using the theory of resolvent operators.     

Exact null controllability of semilinear integrodifferential systems in Hilbert spaces

Sufficient conditions for exact null controllability of the semilinear integrodifferential systems in Hilbert spaces are obtained. It is shown that under some natural conditions exact null controllability of the semilinear integrodifferential system is implied by the exact null controllability of the corresponding linear system with additive term. An application to partial integrodifferential equations is given.     

Boundary controllability of Sobolev-type abstract nonlinear integrodifferential systems

Sufficient conditions are established for boundary controllability of various classes of Sobolev-type nonlinear systems including integrodifferential systems in Banach spaces. The results are obtained using the strongly continuous semigroup of operators and the Banach contraction principle. Examples are provided to illustrate the theory.     

Controllability of fractional-order impulsive neutral functional infinite delay integrodifferential systems in Banach spaces

In this work the controllability of fractional impulsive neutral functional integrodifferential systems in a Banach space has been addressed. Sufficient conditions for the controllability are established using fractional calculus, a semigroup of operators and Krasnoselskii’s fixed point theorem.     

The Cauchy Problem for a Class of Coupled Systems Containing a Convolution Operator

Some results on the invariant regions ,existence and uniqueness of solutions to a class of integrodifferential systems are established. Applying these results to integrodifferential systems with a small parameter ε > 0, we obtain,in particular, some estimates of solutions uniform in ε > 0.     

Controllability of fractional impulsive neutral integrodifferential systems with a nonlocal Cauchy condition in Banach spaces

In this work the controllability of fractional impulsive neutral functional integrodifferential systems with a nonlocal Cauchy condition in a Banach space has been addressed. Sufficient conditions for the controllability are established using fractional powers of operators and the Banach contraction mapping theorem.