APPROXIMATE CONTROLLABILITY OF INFINITE DIMENSIONAL SEMILINEAR CONTROL SYSTEMS

2023-06-30T11:57:55Z

Title: APPROXIMATE CONTROLLABILITY OF INFINITE DIMENSIONAL SEMILINEAR CONTROL SYSTEMS Authors: HAQ, ABDUL Abstract: The present research work deals with the existence of solutions and approximate controllability of deterministic semilinear integer order systems with control delays and fractional order systems without delay. To derive the existence and controllability results, various techniques have been applied along with the semigroup, cosine and sine families, fractional calculus, fractional cosine family, fractional resolvent, xed point theory. Some examples are provided for the illustration of the obtained results. Some introductory matter along with literature survey on controllability of nonlinear and linear control systems of fractional and integer orders are given in Chapter 1. Basic concepts and de nitions of control theory, semigroup theory, cosine family, fractional calculus, fractional cosine family and nonlinear functional analysis which are utilized in forthcoming chapters, are given in Chapter 2. In Chapter 3, the existence of mild solutions of rst-order retarded semilinear system with control delay is proved under the locally Lipschitz continuity of nonlinear function and a xed point theorem. Then the approximate controllability of semilinear system is proved provided that the associated linear system without delay is approximately controllable. Controllability results are obtained by using the method of steps and semigroup theory. The results of this chapter are illustrated with controlled heat equation.

APPROXIMATE CONTROLLABILITY OF SEMILINEAR DELAY CONTROL SYSTEMS

2023-06-23T12:14:12Z

Title: APPROXIMATE CONTROLLABILITY OF SEMILINEAR DELAY CONTROL SYSTEMS Authors: Shukla, Anurag Abstract: Controllability is an important area in the study of control systems. The present work deals with the approximate controllability of deterministic and stochastic semilinear delayed first order systems and fractional order systems in Banach spaces. In chapter 1, a general introduction about the control theory is given. A brief account of the related work by various authors in this direction is presented. Chapter 2, contains basic concepts and definitions of control theory and nonlinear functional analysis that will be used in subsequent chapters. In chapter 3, we studied the approximate controllability of semilinear system with state delay. Instead of a CO-semigroup associated with the mild solution of the system, we use the so-called fundamental solution. Controllability results are obtained by using sequential approach and the operator semigroup theory. In chapter 4, we discuss the approximate controllability of retarded semilinear stochastic system with nonlocal conditions. Using the infinite dimensional controllability operator the control function for the system is constructed. By using this control function, Banach fixed point theorem and stochastic analysis, some results for proposed problems in Hubert space are presented. The objective of this chapter is to study the approximate controllability of semilinear fractional stochastic control system with delay. Sufficient conditions are obtained by separating the given fractional semilinear stochastic system into two systems viz, a fractional linear stochastic system and a fractional semilinear deterministic system. To prove our results Schauder fixed point theorem has been applied. 1 11 Chapter 6 contains two sections. In the first section we studied the approximate controllability of fractional order semilinear system of order c E (1, 2] in Hubert spaces. The results of first section are obtained by using Schauder's fixed point theorem. In the second section we studied approximate controllability of fractional order semilinear delay system of order o E (1, 21. The results of second section are obtained by using the theory of strongly continuous a-order cosine family and Gronwall's Inequality. In chapter 7, we studied the approximate controllability of semilinear fractional control system of order a E (1,2] with infinite delay. The results are obtained with the help of strongly continuous a-order cosine family and sequence method. In chapter 8, we studied the approximate controllability of fractional semilinear stochastic system of order (1, 2] in L spaces. A set of sufficient conditions is obtained using the theory of strongly continuous a-order cosine family, Banach fixed point theorem and stochastic analysis techniques.

STABILITY OF NON-ISOTHERMAL POISEUILLE FLOW IN VERTICAL ANNULUS FILLED WITH POROUS MEDIUM

2023-06-23T12:15:19Z

Title: STABILITY OF NON-ISOTHERMAL POISEUILLE FLOW IN VERTICAL ANNULUS FILLED WITH POROUS MEDIUM Authors: Bhowmik, Moumita Abstract: The non-isothermal Poiseuille flow in porous media has been a subject of intense research for over four decades. This type of flow in pipe/channel/annulus is used in many industrial situations such as extraction of bio-fuel [51], packed-bed chemical reactors [2], oil recovery process [63], solid-matrix heat exchangers and cooling of nuclear plants [54], etc. However, most of the available studies are done in vertical channel or pipe. The results of channel • or pipe can not be used to predict the flow mechanism in annular geometry. Therefore, to • understand the flow configuration in annular geometry formed by two concentric cylinders a step has been taken in the present thesis. Both linear and nonlinear theories have been used to examine the stability mechanism of the flow. The objective of this study is to investigate the effect of gap between the two concentric cylinders on the above flow for different permeable medium as well as non-isothermal resources. The annulus is filled with a homogeneous and isotropic porous medium. An external pressure gradient and a buoyancy force (due to temperature difference) drive the fully developed water flow in the annular region. The inner wall temperature of the annulus increases linearly with the axial coordinate from an upstream reference temperature and the outer wall is adiabatic. In the limit of fully developed flow, this simulates a constant heat flux condition on the inner cylinder. Note that depending on the sign of Rayleigh number, the fully developed flow may be stably stratified (i.e., the buoyancy force acts in the direction of forced flow) or unstably stratified (i.e., buoyancy force in the negative direction of forced flow). The linear stability of the above flow for both stably stratified and unstably stratified cases are analyzed in this thesis. Following the previous efforts of Yao & Rogers [123], the weakly 11 nonlinear stability of non-isothermal Poiseuille flow in vertical annulus filled with porous medium is developed. The present thesis is compiled in six chapters and the chapter wise description is given below. Chapter 1 is an introductory and contains some basic definitions, preliminaries of the flow in porous medium, brief description of hydrodynamic stability theory, work done by various authors in the field of linear and nonlinear stability analysis of Poiseuille flow, and justification regarding the model, which has been adopted for this problem. Chapter 2 addresses the basic flow characteristic of the non-isothermal Poiseuille flow in vertical annulus filled with porous medium. Both stably stratified and unstably stratified situations are considered for this study. The non-Darcy-Brinkman-Forchheimer model is used. The governing equations are solved analytically for a special case: form drag equal to zero and numerically by Chebyshev spectral collocation method. Along with the other controlling parameters, a special attention is given to understand the effect of curvature parameter (C) of the annulus on the flow configuration as well as heat transfer rate. The numerical experiments show that reducing the value of C enhances the maximum magnitude of the velocity along with heat transfer rate in the system. The impact of C (C> 10) on the flow profile as well as heat transfer rate is negligible. Furthermore, the analysis shows that the tendency of appearance of back flow, point of inflection and flow separation (in case of unstably stratified flow) in the flow profile is highly sensitive to C. Apart from this, for a small increase in Ra, a drastic change (up side down) in the flow profile can also be seen. The appearance of flow separation shifted from the vicinity of the inner wall to the outer wall. Hence, to shed more light on this phenomenon and to find the appropriate non-isothermal parameter space as a function of gap between the two concentric cylinders, in which the flow will remain as fully developed, stability analysis is needed. Chapter 3 contains the linear stability of the above Poiseuille flow for stably stratified case. For a given annulus, the stability of the basic flow is controlled by different parameters such as Reynolds number (Re), Rayleigh number (Ra), Darcy number (Da), Prandtl number (Pr), heat capacity ratio (a), viscosity ratio (A), porosity (E), and modified Forchheimer 111 number (F'). Since curvature parameter (C) plays a vital role to describe the size of the annulus, therefore impact of C on the transition mechanism of basic flow for relatively high permeable medium is considered in this chapter. To avoid numerous parametric study we have fixed the value of some of the parameters such as A, Pr, and a. at 1, 7 and 1, respectively. The disturbance momentum and energy equations are numerically solved by spectral collocation method. We have also analyzed the energy budget spectrum at critical point. The linear stability results show that increasing C as well as decreasing Da stabilizes the basic flow. 1-lowever, beyond C = 10 the impact of curvature parameter on the stabilization of the basic flow is almost negligible. From the energy analysis at critical level it is observed that the thermal-buoyant instability is the only mode of instability. Furthermore, the analysis of linear stability shows that although the impact of form drag upto a threshold value is negligible on instability but its contribution in energy dissipation - is significant. In Chapter 4, we have investigated the stability of stably stratified non-isothermal Poiseuille flow of water in vertical porous-medium annulus using weakly nonlinear stability theory, with particular emphasis on the impact of gap between the two vertical axisymmetric cylinders. For a comparative study, we have considered three different values (1 0, 0.6, 10) of C for three different values (10_I, 10_2, 10) of Da. The flow in the annulus is governed by the volume-averaged forms of the Naiver-Stokes and continuity equations derived by [117]. To carry out the weakly nonlinear analysis, we started by analyzing the range of Ra, beyond the critical point, in which the growth rate varies linearly using perturbation series solution approach. From this analysis it has been found that for high permeable medium the linear relationship between growth rate and Ra holds good for very small neighborhood of critical (bifurcation) point, however for low permeable medium it is relatively large. This gives an impression that the nonlinear interaction is not effective for low permeable medium, which is also supported by finite amplitude analysis. The finite amplitude analysis predicts both the supercritical as well as subcritical bifurcation at and in the vicinity of bifurcation point, which are also investigated by nonlinear energy spectrum. The analysis lv of the nonlinear energy spectrum for the disturbance reveals that in case of Da = IO 2 or C = 10 an instability that is supercritical for some wavenumber may be supercritical or subcritical at other nearby wavenumber. The equilibrium amplitude increases on decreasing the media permeability as well as reducing the gap between inner and outer cylinders. In the limiting case (i.e., at C = 10) the fundamental disturbance of stably stratified nonisothermal Poiseuille flow (SSNPF) of water in vertical channel filled with porous medium will have minimum amplitude. The influence of nonlinear interaction of different superimposed waves on some physical aspects: heat transfer, friction coefficient, nonlinear energy spectrum, and steady secondary flow is also investigated. Investigation related to impact of superimposed waves on the pattern of secondary flow, based on linear stability theory gives an impression that cells of flow pattern are just shifted. This is the consequence of negligible modification in the buoyant production of disturbance kinetic energy and significant modification in the rate of the viscous dissipation of disturbance energy for the considered set of parameters. In Chapter 5, the instability mechanism of the above flow is analyzed for unstably stratified case. Linear stability analysis predicts first azimuthal mode as the least stable mode in the entire range of C for Da = 10_I and 10. For Da = 10-2 first azimuthal mode is also the least stable mode except for 0.02 < C < 0.1 where zero azimuthal mode is the least stable mode. However, for Da = 10-2 (except for 0.02

A STUDY ON EXISTENCE OF SOLUTION AND CONTROLLABILITY OF DELAY DIFFERENTIAL SYSTEMS

2023-06-23T12:16:40Z

Title: A STUDY ON EXISTENCE OF SOLUTION AND CONTROLLABILITY OF DELAY DIFFERENTIAL SYSTEMS Authors: Das, Sanjukta Abstract: Controllability of distributed parameter systems, essentially of dymiamnical systems governe(l by partial differential equations, has evolved into a widely researched topic in less than t11r0e decades. Despite generating a (hstmctive identity and philosophy as a part of the theory of dynamical systems, this research field has played a significant role in the advancement of the extensive theory of partia.l differential equations. In last few decades, control theory has contributed enormously to study of realistic problems of elasticity such as thcrrnoelasticity, acroelasticity, problems depicting interactions between fluids and elastic structures and real world problems of fluid dynamics, to name but a few. Such real world problems present new mathematical challenges. For instance, the mathematical foundations of basic theoretical issues have to be enriched, along with the development of conceptual insights significant to the (lesigners and the practitioners. This poses novel challenges that need to be addressed. lii our present work we focuss on the existence, uniqueness and controllablity of nonlinear functional differential equations. We use theory of sernigroup, cosine family, measure of noncompactness and fixed point theorems to ol)tain the results. The results can be applied to a class of functional differential equations, appearing in the mathematical models of several physical phenomena to which the prototype of partial differential equations modeling the phenomena., belongs. rfll(s layout of the thesis, containing 10 chapters, is as follows. Chapter 1 is introductory in nature. The delay differential equations and their applications are discussed. The objective of work done, current status of the field and layout of the t11e5is is also presented in this chapter. Chapter 2 illustrates some basic properties of semigroup theory, cosine family, measure of noncompactness, controllability, fractional and stochastic differential equations. In chapter 3 we study a functional differential equation with deviating argument and finite delay to establish that it is approximately controllable. The results of this chapter are published as 'Approximate Controllability of a Funct. ioimal l)ilferential Equation with Deviated Argument' in Nonlinear Dynamics and Systems Theory, Imifor Math, volume 14, no. 3, (2014), 265-277. In chapter 4 existence of mild solution of a second order partial neutral (hffcreutial equation with state dependent delay and non-instantaneous impulses is investigated. We use Ilausdoril measure of nonconipactness and Darbo Sadovskii fixed point theorem to prove the existence. The results of this chapter are published as 'Existence of Solution for a Second-Order Neutral Differential Equation with State Dependent Delay and Non-instantaneous Impulses' in International JournaI of Nonlinear Science, World Scientific, volume 18, no.2, (2014). 145-155. Chapter 5 consists of two parts. The first part deals with the existence of mild solution of an instantaneous impulsive second order differential equation with state dependent delay. In second part non-instantaneous impulsive conditions are studied. We introduce new non-instantaneous impulses with fixed delays. The results of this chapter are in revision as 'Existence of Solution of Impulsive Second-Order Neutral Integro-Differential Equation with State Delay' in Journal of Integral Equations and Applications. In chapter 6 we establish the existence and uniqueness of mild solution and the approximate controllability of a second order neutral partial differential equation with state dependent delay. The conditions for approximate controllability are investigated for the distributed second order neutral differential system with respect to the approximate controllability of the corresponding linear system in a Ihilbert space. The results of this chapter are published as 'Approximate Controllability of a Seeond Order Neutral Differential Equation with State Dependent Delay' in Differential Equations and I)ynamical Systems, Springer, DOI 10.1007/.s12591 - 014 - 0218 - 6, (2014). Chapter 7 is divided in two parts. In the first, part we study a second order neutral differential equation with state dependent delay and non-instantaneous impulses. The existence and uniqueness of the mild solution are investigated via Flausdorif measure of norl-cOlnl)actlless and Darbo Sadovskii fixed point theorem. In the second part the conditions for approximate controllability are investigated for the neutral second order system under the assumption that the corresponding linear system is approximately controllable. A simple range condition is used to prove Hi approximate controllability. The results of this chapter are published as 'Existence of Solution and Approximate Controllability for Neutral Differential Equation with State Dependent Delay' in Internatiorial Journal of Partial Differential Equations, Hindawi, volume 2014 (2014), Article ID 787092, 12 pages. In chapter 8 we study a fractional neutral differential equation with deviating argument to establish the existence and uniqueness of mild solution. The approximate controllability of a class of fractional neutral differential equation with deviating argumdnt is discussed by assuming a simple range condition. The results of this chapter arc published as 'Approximate Controllability of a Fractional Neutral System with Deviated Argument in Banach Space' in Differential Equations and Dynamical Systems, Springer, DOI : 10.1007/812591 - 015 —0237— y, (2015). In chapter 9 the approximate controllability of an impulsive fractional stochastic neutral integro-differential equation with deviating argument and infinite delay is studied. The control parameter is also included inside the nonlinear term. Only Schauder fixed point theoremim and a few fundamental hypotheses are used to prove our result. The results of this chapter are published as 'Approximate controllability of an unpulsive neutral fractional stochastic differential equation with deviated argument and infinite delay' in Nonlinear Studies, volume 22, no. 1, 1-16, (2015), CSP - Cambridge, UK; 1&S - Florida, USA. In chapter 10 the existence, uniqueness and convergence of approximate solutions of a stochastic fractional differential equation with deviating argument is established. Analytic semigroup theory is used along with fixed point approach. Then we investigate Faedo-Galerkin approximation of solution and establish some convergence results. The results of this chapter are accepted for publication as 'Approximations of Solutions of a Fractional Stochastic Differential Equations with Deviated Argument' in Journal of Fractional Calculus and Applications in 2015.

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APPROXIMATE CONTROLLABILITY OF INFINITE DIMENSIONAL SEMILINEAR CONTROL SYSTEMS

APPROXIMATE CONTROLLABILITY OF SEMILINEAR DELAY CONTROL SYSTEMS

STABILITY OF NON-ISOTHERMAL POISEUILLE FLOW IN VERTICAL ANNULUS FILLED WITH POROUS MEDIUM

A STUDY ON EXISTENCE OF SOLUTION AND CONTROLLABILITY OF DELAY DIFFERENTIAL SYSTEMS