Analysis on Controllability Results for Wellposedness of Impulsive Functional Abstract Second-Order Differential Equation with State-Dependent Delay

The functional abstract second order impulsive differential equation with state dependent delay is studied in this paper. First, we consider a second order system and use a control to determine the controllability result. Then, using Sadovskii’s fixed point theorem, we get sufficient conditions for the controllability of the proposed system in a Banach space. The major goal of this study is to demonstrate the controllability of an abstract second-order impulsive differential system with a state dependent delay mechanism. The wellposed condition is then defined. Next, we studied whether the defined problem is wellposed. Finally, we apply our results to examine the controllability of the second order state dependent delay impulsive equation.

[1]  Ju H. Park,et al.  Existence and controllability results for second-order impulsive stochastic evolution systems with state-dependent delay , 2014, Appl. Math. Comput..

[2]  N. Sukavanam,et al.  Existence of solution and approximate controllability of a second-order neutral stochastic differential equation with state dependent delay , 2016 .

[3]  Jianhong Wu,et al.  Algebraic-delay differential systems: ⁰-extendable submanifolds and linearization , 2017 .

[4]  R. D. Driver,et al.  A neutral system with state-dependent delay☆ , 1984 .

[5]  C. Travis,et al.  Cosine families and abstract nonlinear second order differential equations , 1978 .

[6]  R. Sakthivel,et al.  Approximate Controllability of Second-Order Systems with State-Dependent Delay , 2008 .

[7]  Yong-Kui Chang,et al.  Existence for impulsive neutral integrodifferential inclusions with nonlocal initial conditions via fractional operators , 2009 .

[8]  Yongzhen Pei,et al.  Smoothness of semiflows for parabolic partial differential equations with state-dependent delay , 2016 .

[9]  S. I. Piskarev,et al.  Differential Equations in Banach Spaces II. Theory of Cosine Operator Functions , 2004 .

[10]  Marcus R. W. Martin,et al.  The escaping disaster: A problem related to state-dependent delays , 2004 .

[11]  D. N. Chalishajar,et al.  Controllability of Nonlocal Impulsive Functional Differential Equations with Measure of Noncompactness in Banach Spaces , 2021, Journal of Nonlinear Sciences and Applications.

[12]  K. Balachandran,et al.  CONTROLLABILITY OF SECOND-ORDER IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATIONS WITH STATE-DEPENDENT DELAY , 2011 .

[13]  Tibor Krisztin,et al.  Parabolic partial differential equations with discrete state-dependent delay: Classical solutions and solution manifold , 2014, 1412.0219.

[14]  I. Chueshov,et al.  Dynamics of second order in time evolution equations with state-dependent delay , 2013, 1310.2902.

[15]  Analytic solutions of a second-order functional differential equation with a state derivative dependent delay , 1999 .

[16]  Jianhong Wu,et al.  A non-local PDE model for population dynamics with state-selective delay: Local theory and global attractors , 2006 .

[17]  N. Mahmudov,et al.  On the Approximate Controllability of Second-Order Evolution Hemivariational Inequalities , 2020, Results in Mathematics.

[18]  C1 + α-strict solutions and wellposedness of abstract differential equations with state dependent delay , 2016 .

[19]  Alexander V. Rezounenko.  Partial differential equations with discrete and distributed state-dependent delays , 2007 .

[20]  F. Hartung.  On differentiability of solutions with respect to parameters in neutral differential equations with state-dependent delays , 2006, Annali di Matematica Pura ed Applicata.

[21]  J. Kisyński,et al.  On cosine operator functions and one-parameter groups of operators , 1972 .

[22]  Yong-Kui Chang,et al.  EXISTENCE RESULTS FOR A SECOND ORDER IMPULSIVE FUNCTIONAL DIFFERENTIAL EQUATION WITH STATE-DEPENDENT DELAY , 2009 .

[23]  Eduardo Hernández,et al.  A note on partial functional differential equations with state-dependent delay , 2006 .

[24]  EXISTENCE OF SOLUTIONS FOR A SECOND ORDER ABSTRACT FUNCTIONAL DIFFERENTIAL EQUATION WITH STATE-DEPENDENT DELAY , 2007 .

[25]  H. Henríquez,et al.  Existence of Global Solutions for a Class of Abstract Second-Order Nonlocal Cauchy Problem with Impulsive Conditions in Banach Spaces , 2018 .

[26]  Bheeman Radhakrishnan,et al.  Controllability of Neutral Evolution Integrodifferential Systems with State Dependent Delay , 2012, J. Optim. Theory Appl..

[27]  Omar Bazighifan,et al.  Philos-Type Oscillation Results for Third-Order Differential Equation with Mixed Neutral Terms , 2021, Mathematics.

[28]  Meili Li,et al.  Approximate controllability of second-order impulsive stochastic differential equations with state-dependent delay , 2018 .

[29]  H. M. Baskonus,et al.  Results on approximate controllability results for second‐order Sobolev‐type impulsive neutral differential evolution inclusions with infinite delay , 2020, Numerical Methods for Partial Differential Equations.

[30]  V. Vijayakumar,et al.  Controllability for a class of second-order evolution differential inclusions without compactness , 2019 .

[31]  A. Rezounenko.  A condition on delay for differential equations with discrete state-dependent delay , 2012 .

[32]  V Vijayakumar,et al.  Approximate controllability of second order nonlocal neutral differential evolution inclusions , 2020, IMA J. Math. Control. Inf..

[33]  H. I. Freedman,et al.  Analysis of a model representing stage-structured population growth with state-dependent time delay , 1992 .

[34]  Ferenc Hartung,et al.  On Differentiability of Solutions with Respect to Parameters in State-Dependent Delay Equations , 1997 .

[35]  Dimplekumar N. Chalishajar,et al.  Controllability of nonlocal neutral impulsive differential equations with measure of noncompactness , 2021 .

[36]  D. O’Regan,et al.  On second order differential equations with state-dependent delay , 2018 .

[37]  Yuri V. Rogovchenko.  Nonlinear Impulse Evolution Systems and Applications to Population Models , 1997 .

[38]  Jianhong Wu,et al.  Abstract algebraic-delay differential systems and age structured population dynamics , 2013 .

[39]  Existence of solutions for a second order abstract functional differential equation with state-dependent delay. , 2007 .

[40]  N. Y. Nadaf,et al.  Existence and controllability results for damped second order impulsive functional differential systems with state-dependent delay , 2014 .