CONTROLLABILITY RESULTS FOR NON-LINEAR NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS

 ABSTRACT

In this work, necessary and sufficient conditions are investigated and proved for the controllability of nonlinear functional neutral differential equations. The existence, form, and uniqueness of the optimal control of the linear systems are also derived. Global uniform asymptotic stability for nonlinear infinite neutral differential systems are investigated and proved and ultimately, the Shaefers’ fixed point theorem is used to forge a new and farreaching result for the existence of mild solutions of nonlinear neutral differential equations in Banach Spaces.

KEY WORDS:

Relative Controllability, Volterra Integro-Differential Equation, Optimal Control, Complete State, Unsymmetric Fubini Theorem, Neutral Systems, Linearization,  Exponential Estimate, , Stability in the Large, mild solution .

TABLE OF CONTENTS

 

 

Title                                                                                                                          i

Certification                                                                                                                                                                ii

Dedication                                                                                                                                                                  iii 

Acknowledgements                                                                                                                                                iv

Table of contents                                                                                                                                                      v

Abstract                                                                                                                                                                      vii     

 

 

CHAPTER 1:  GENERAL INTRODUCTION                                                       1

 

1.1   Background                                                                                                          1

1.2   Statement of the Problem/Objective                                                                    3

1.3   Scope of Study.                                                                                                    4

1.4    Definitions and Preliminaries                                                                                                    5

1.5    Functional Differential Equations                                                                             11

1.6  121.6   Difference between Retarded and Neutral Functional Differential Equations. 12

1.7   General Form of Functional Differential Equations.                                      12

1.8   General Solution Format or Variation of Parameters.		13
1.9   Volterra Integral Equation 1.10 Existence and Uniqueness of Optimal Control for linear neutral		13
Volerra Integro-Differential system. 1.11 Global Uniform Asymptotic Stability for nonlinear Infinite		14
Neutral differential systems 1.12 Existence of Mild Solution of Nonlinear Neutral Differential		16
Equations in Banach Spaces		19
CHAPTER 2:       REVIEW OF THE LITERATURE		22
2.1       Introduction		22
2.2       Controllability of Ordinary Differential Systems.		25
2.3      Computable Criterion for Controllability.		26
2.4      Controllability of Nonlinear Systems.		27
2.5      Relative Controllability of Nonlinear Neutral Systems.		27
2.6      Null Controllability of Functional Differential Systems		28
CHAPTER 3:  METHODOLOGY AND CONTROLLABILITY		31

3.1    Introduction                                                                                                       31

3.2    Description of System.                                                                                   32

3.3     Variation of constant formula                                                                     32

3.4. The following Methods/Techniques were used to obtain Results                       34

3.4.1 Relative Controllability Technique                                                                    34

3.4.2 The Energy Method of Alexander Mikhailovick Lyapunov(1829)                   34

3.5      Basic set functions and properties                                                                   35

3.6      Controllability Conditions or Controllability Standard                               35

3.6.1      Optimality Control                                                                                  35

37       Bang-Bang Principle                                                                                  36

3.8      Control theorem on nonlinear systems			36
3.9      Lyapunov Stability Results (theorem)			36
3.10    Existence Theorems			37
CHAPTER4:  MAIN RESULTS			38
4.1.   Relative Controllability Results 4.2. Optimality Conditions For The Linear Neutral Volterra			38
Intego-Differential Systems.			40
4.3. Existence of an Optimal Control.			41
4.4 The Form of Optimal Control.			43
4.5. Uniqueness of Optimal Control.			45
4.6. Global Uniform Asymptotic Stability for Nonlinear Infinite Neutral
Differential Systems.			46
4.6.1 Results on Stability on Neutral Systems. 4.6.2 Results on the Exponential Asymptotic Stability in the Large for			48
Nonlinear Neutral Systems.			48

4.7. Existence of Mild Solution of Nonlinear Neutral Differential Equations in

Banach Spaces.                                                                                                   55

4.9.1. Existence of Solutions by Fixed Point Technique.                                              56 CHAPTER5: SUMMARY, CONCLUSION, RECOMMENDATION

AND CONTRIBUTION TO KNOWLEDGE                                   61

5.1. Summary                                                                                                    61

5.2.   Conclusion                                                                                                  67

5.3. Recommendation                                                                                            68

5.4. Contribution to Knowledge                                                                        68

Reference.                                                                                             69

Appendix A.                                                                             74

CHAPTER 1

GENERAL INTRODUCTION

    1.1     Background

Controllability is one of the fundamental concepts in mathematical control theory. It is a qualitative property of dynamical control systems and is of particular importance to the control theorist. In the recent past, the theory of control of deterministic processes with several degrees of freedom appeared to have reached a satisfying stage of completeness. As interpreted by the theory of nonlinear ordinary differential equations, Iyai (2006) the fundamental problems of control theory have been mathematically posed and answered and hence the theory has reached a certain degree of stability and perfection. The authors as a result believed that a thorough and careful presentation of the current status of control theory would serve the useful purpose of offering a foundation on which later researches would be based. It is in this intent, that this work: “Controllability Results for Nonlinear Neutral Functional Differential Systems” is carried out. Our Objective therefore is to present an organized treatment of control theory that could be complete within the limitations set by the restrictions of deterministic problems identifiable in terms of functional differential equations. It is enough to mention here that two kinds of functional differential equations exist.

• The Retarded Functional Differential Equation given as

𝑥̇ = 𝑓(𝑡, 𝑥 )   ;   𝑥(𝑡 ) = 𝜙 = 𝑥                            (1. 1.1)   where ф is the initial function defined in the delay interval [-h,0] , h > 0.

• The Neutral Functional Differential Equation given as:

[𝐷(𝑡, 𝑥 )] = 𝑓(𝑡, 𝑥 )  ;   𝑥(𝑡 ) = 𝜙 = 𝑥                      (1.1. 2)             where D is a bounded linear operator

It is easily observed that, both equations (1.1.1) and (1.1.2) are characterized by delays. The motivation for this study stems from the fact that most realistic systems should encompass not only the present, but also the past state of the system. This is encountered in many areas of human activities. For a good grasp of the present, (t), some knowledge of the past, (t-h),  t ≥ 0, h > 0 , is very important.

In general, differential equations which include the present as well as the past state of any physical system is called a Delay Differential Equation (or Functional Differential Equations).

The Retarded Functional Differential Equations (RFDE) are characterized by delays in the state of the system.    An example is the system

𝑥(𝑡) = 𝑥(𝑡 − ℎ)  ,   ℎ > 0                                                   (1.1.3)

𝑂𝑛 𝑡ℎ𝑒 𝑜𝑡ℎ𝑒𝑟 ℎ𝑎𝑛𝑑, 𝑁𝑒𝑢𝑡𝑟𝑎𝑙 𝐹𝑢𝑛𝑐𝑡𝑖𝑜𝑛𝑎𝑙 𝐷𝑖𝑓𝑓𝑒𝑟𝑒𝑛𝑡𝑖𝑎𝑙 𝐸𝑞𝑢𝑎𝑡𝑖𝑜𝑛𝑠 (𝑁𝐹𝐷𝐸) are those

that have delays in the state as well as   in the derivatives. An example is the system                                   𝑥(𝑡) − 𝑐 𝑥(𝑡 − ℎ)       =    𝑏𝑥(𝑡 − ℎ)                                    (1.1.4)

𝑤ℎ𝑒𝑟𝑒 𝑐, 𝑏 , 𝑎𝑛𝑑 ℎ ( ℎ > 0)𝑎𝑟𝑒 𝑐𝑜𝑛𝑠𝑡𝑎𝑛𝑡𝑠.

Systematic study of controllability started over the years at the beginning of the sixties when the theory of controllability based on the description in the form of state space for both time –varying and time invariant linear control systems was carried out. Roughly speaking, controllability generally means that, it is possible to steer a dynamical control system from an initial state to a final state using the set of admissible controls. Optimal control means doing the same in the best conceivable way. There are many different definitions of controllability which strongly depend on the class of dynamical control systems. In recent years, various controllability problems for different types of nonlinear systems have been considered. However, it should be stressed that, most of the reported work in this direction has been mainly concerned with controllability for linear dimensional systems with constrained control and without delays (see Klamka (1991), Sun (1996), Underwood and Young (1979)). Later on delay differential equations came to limelight (see Nse (2007), Nse (2007)). A delayed equation on a linear system is one which affects the evolution of the system in an indirect manner.

If we consider the equation

𝑥̇  =  𝐴𝑥(𝑡) + 𝐵𝑢(𝑡)                                                            (1.1.5)

where A and B are nxn and mxn matrices, we see that the action of the control is direct in that the local behavior of the trajectory is affected only by the local behavior of the control u(t) at time t. It is known that, most natural applications give rise to mechanism of indirect actions where decisions in the control function are shifted, twisted or combined before affecting the evolution thus  comprising the delay u(t-h) represented by the system                                             𝑥̇  =  𝐴𝑥(𝑡) + 𝐵𝑢(𝑡 − ℎ)                                                 (1.1.6)

It is well known that the future state of realistic models in the Natural Science, Economics and Engineering depends not only on the present, but also on the past state and at times , even on the derivative of the  past state. There are simple examples from Biology (predatorprey, Lodka Volterra, Spread of Epidemics), from Economics (dynamics of capital growth of global economies) and from Engineering (mechanical and aero – space , aircraft stability, automatic steering using minimum fuel and effort, control of high speed, closed air circuit, wind tunnel computer and electric engineering , fluctuations of current in linear and nonlinear circuits, flip-flop circuits and lossless transmission lines).These examples are used to study the stability and time optimal control of  Functional Differential Systems. The results of this research effort are, therefore, intended to forge far-reaching solutions to these daily human endeavors.

1.2. Statement of the Problem / Objective.

Our principal objective in this work is to obtain Necessary and Sufficient Conditions for controllability, optimal control and stability for Neutral Functional Differential Systems. It is known from Onwuatu (1993) that, if a system is relatively controllable, then optimal control is unique and bang-bang. In the light of this, we shall consider the Neutral Volterra Integrodifferential Equation of the form

𝑥(𝑡) − ∫ 𝐶(𝑡, 𝑠)𝑥(𝑠)𝑑𝑠 − 𝑔(𝑡)

= A(t)x(t)  +    ∫ 𝐺(𝑡, 𝑠)𝑥(𝑠)𝑑𝑠 + ∫ 𝑑 𝐻(𝑡, 𝜃)𝑢(𝑡 + 𝜃)           (1.1.7)

with initial condition 𝑥(𝑡 ) = 𝑥 , where xє𝐸 is the state space and u є𝐸 is the        control function, H(t ,θ) is an nxm matrix continuous at t and of bounded variation in θ on

[-h,0] ; h>0 for each tє[t ₀,t₁] ; t₁ > t₀ . 𝑇ℎ𝑒 𝑛𝑥𝑛 𝑚𝑎𝑡𝑟𝑐𝑒𝑠 𝐴(𝑡), 𝐶(𝑡, 𝑠), 𝐺(𝑡, 𝑠) are continuous in their arguments. The n-vector function g is absolutely continuous.       The above system will be investigated for existence and uniqueness of optimal control by first of all considering the relative controllability of the system.

We shall then forge ahead to achieve solution near the origin to another Neutral

Functional Differential Equation of the form

[𝐷(𝑡, 𝑥 )] = 𝐿(𝑡, 𝑥 )𝑥 + 𝑓(𝑡, 𝑥 ) + ∫ 𝐴(𝑡)𝑥(𝑡 + 𝜃)𝑑𝜃                        (1.1.8).

𝑤ℎ𝑒𝑟𝑒 ,

𝐿(𝑡, 𝑥 ) = ∑      𝐴 𝑥(𝑡 − 𝑤 ) + ∫              𝐴(𝑡, 𝜃)𝑥(𝑡 + 𝜃)𝑑𝜃                                      (1.1.9) is a bounded linear operator, and f( 𝑡 , 𝑥 )is a perturbation function.  The nxn matrix functions A_k and A(t ,θ) are measurable  in (t, s) є ExE,  θ є [-∞,0).

The energy method of Alexander Mikhailovick Lyapunov (1829) which stipulates that, in a stable system, the total energy in the system will be a minimum at the equilibrium point will be used to establish results. This no doubt will pave the way for discussions on stability of various nonlinear functional equations.

The statements of the problem are thus formulated:

Suppose we are given a Neutral Functional Differential Equation as in equations (1.1.7) and it is required to move the solution 𝒙(𝒕) from an initial point  𝒙_𝟎  at time  𝒕_𝟎  to a terminal point 𝒙_𝟏  at time  𝒕_𝟏.The problem arises as to whether it is possible to carry out this task in finite time. As an illustration, we shall consider the system

𝑥̇   = −𝑎𝑥(𝑡)                                                              (1.2.0)

Clearly, the solution of the above system is

𝑥(𝑡) = A𝑒                                                             (1.2.1).

If we desire to drive this solution to the origin .that is, null controllability, we observe that, it cannot be achieved in finite time because 𝑥(𝑡)  tends to zero only when t tends to infinity. Since this cannot be achieved in finite time, we need to modify the system to be able to bring 𝒙(𝒕)  to 0 in finite time. The process of modification is called controllability which will answer the controllability problem.

The optimal control problem is formulated as follows: Having guaranteed controllability of the system in question is there an admissible control u* such that the solution  𝒙(𝒕, ф, 𝒖^∗) of the system hits a continuously moving target point in minimum time t*. Here u* is the optimal control and t* the minimum time. Once it is guaranteed that such a control exists, we shall show it is unique and bang-bang.

Finally, we shall ask the question:  Is the solution near the origin of system (1.1.8) going to remain quite close for all future times?  This is the stability problem which we are desirous to answer in the affirmative.

1.3.     Scope of Study.

Differential systems are generally important tools for harnessing different components into a single system and analyzing the inter-relationships that exists between them which otherwise might continue to remain independent of each other. Physical systems which express the present state of solutions are the most common system encountered in the theory of differential equations. The Scope of this work, therefore, is to go beyond these systems and address more realistic systems involving not only the present but also the past states of the system. This is because the latter permeates various aspects of life and has of late triggered interest in research.

Neutral differential equations arise in many areas of applied Mathematics and such equations have received much attention in recent years. For example, the mixed initial boundary hyperbolic partial differential equations which arise in the study of lossless transmission lines can be replaced by an associated neutral differential equation. This equivalence has been the basis of a number of investigations of the stability properties of distributed networks, (see Iyai (2006), Kwun (1991)).

It is in this light also that we intend to broaden the scope to involve systems of the Neutral type. This is because in recent years, it has emerged as independent branch of modern research due to its connection to many fields such as continuum mechanics, population dynamics, system theory, biology epidemics and chemical oscillations (see Balachandran,

Balasubramaniam and Dauer (1996), Burton (1983), Corduneanu (1985)).

1.4.   Definitions and Preliminaries

In Mathematical parlance, we consider a system of the form

𝑥̇ = 𝑓(𝑥, 𝑡, 𝑢)                                                            (1.2.2)                                               where x є  𝐸 and  u є 𝐸 .

This differs from the known familiar one-the usual first order ordinary differential equation (ODE) because of the presence of the time function u(t) in the right-hand side of (1.2.2). Many physical processes described by differential equations may have the time-dependence of the process influenced in some manner. This influence is generally referred to as steering or controlling of the process and in (1.2.2) the time function u denotes steering mechanism. Hence, u is often called the steering or control function. Thus, in Control Theory, the two dependent variables x and u are called the state variable and the control variable respectively.

In case of air spaceship, the state variable x may refer to the position and velocity of the spaceship, while the control function 𝑢 = ( 𝑢 , 𝑢 , … , 𝑢 ) may represent the controllable

individual thrusts of the engines.

Suppose now we desire to steer a process (system) from a state  𝑥 to  a state 𝑥 , two  very important questions arise:

(i).       Does a steering mechanism, a control function u that can be used to steer the process                     (system) from  𝑥 to  𝑥 in a finite time exist? This is the question of Controllability.

(ii)  Suppose the answer to question (i) above is affirmative, among all the possible steering functions that can be used to steer the process (system) from 𝑥 to 𝑥  , is there a best one, an optimal one? May be from the point of view of minimizing the travel time, fuel consumption, cost, side effects, or maximizing profit etc?   This is the question of Optimal Controllability. A solution of the system (1.2.2) depends on the function u(.) – the notation u (.) is used to denote a function defined on the interval [t_o ,t] . To reflect this dependence, we write a solution to system (1.2.2) as   𝜓(𝑥 , 𝑡 , 𝑢(. ), 𝑡 )

The Controllability question (i) may be put in mathematical parlance thus: Given  𝑥 , 𝑡 and  𝑥 , does there exist a time 𝑡 > 𝑡  and   a function u (.)  such that                                                𝜓(𝑥 , 𝑡 , 𝑢(. ), 𝑡 ) =  𝑥 .

If the answer to question (i) is affirmative, the system is said to be controllable, while the system is said to be optimally controllable if the answer to question (ii) is affirmative.      Let us consider the special case in which the right-hand side of system (1.2.2) is linear with constant coefficients. In such a case, we have

𝑥̇ = 𝐴(𝑡)𝑥 + 𝐵𝑢(𝑡)                                             (1.2.3).

where A is an nxn-matrix and B is an nxm-matrix. For such linear systems, the concept of complete or null-controllability makes sense.

Definition 1.4.1: (controllability)

The linear system (1.2.3) is said to be controllable if and only if for any initial state 𝑥  𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡 there exists steering function u (.) which steers the system from

𝑥 𝑎𝑡 𝑡  to   𝑥 𝑎𝑡  𝑡𝑖𝑚𝑒 𝑡 in finite time.

That is for any 𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑠𝑡𝑎𝑡𝑒  𝑥  ,𝑖𝑛𝑖𝑡𝑖𝑎𝑙 𝑡𝑖𝑚𝑒 𝑡  given, there exists  𝑡𝑖𝑚𝑒 𝑡  and u (.) such that     𝜓( 𝑥  ,𝑡 ,u (.) , 𝑡 ) = 𝑥

Definition 1.4.2: (null-controllability)

The linear system (1.2.3) is null-controllable if and only if for any initial state 𝑥  𝑎𝑡 𝑡    there exists steering function u (.) which steers the system from 𝑥 𝑎𝑡 𝑡𝑖𝑚𝑒 𝑡  to   𝑥 = 0  𝑎𝑡  𝑡

in finite time.

That is for any 𝑥  ,𝑡  given, there exists  𝑡  and u(.) such that     𝜓( 𝑥  ,𝑡  , u (.) , 𝑡 ) = 0. We state, without proof, the following very important result that provides criteria for determining the null-controllability for the system (1.2.3).

Definition 1.4.3: (Reachable Set)

Consider the system (1.2.3) given as

𝑥̇ = 𝐴(𝑡)𝑥 + 𝐵𝑢(𝑡)                                            (1.2.3)

Let the solution be 𝑥(𝑡)  such that

𝑥(𝑡) = 𝑋(𝑡)𝑋  (𝑡 )𝑥  + 𝑋(𝑡) ∫   𝑋 (𝑠)𝐵(𝑠) 𝑢(𝑠)𝑑𝑠,

𝑤ℎ𝑒𝑟𝑒 𝑋(𝑡) is a fundamental matrix and  𝑋(𝑡 ) = 𝑋(0) = 𝐼.

We define the reachable set as

𝑅(𝑡 , 𝑡 ) = ∫ 𝑋 (𝑠)𝐵(𝑠) 𝑢(𝑠)𝑑𝑠: 𝑢є𝑈 ,   where U is the set of admissible controls.

Definition 1.4.4: (Attainable Set)

Attainable set is the set of all possible solutions of a given control system. In the case of the system (1.2.3), for instance, it is given as

𝐴(𝑡 , 𝑡 ) =  𝑥(𝑡) = 𝑋(𝑡)𝑋    (𝑡 )𝑥 + 𝑋(𝑡) ∫   𝑋 (𝑠)𝐵(𝑠) 𝑢(𝑠)𝑑𝑠: 𝑢є𝑈 ,

𝐸𝑣𝑖𝑑𝑒𝑛𝑡𝑙𝑦 , 𝑅(𝑡 , 𝑡 )is a translation of attainable set through the origin 𝑥 , that is  𝐴(𝑡 , 𝑡 ) = { 𝑥(𝑡) =  𝑋(𝑡)𝑋 (𝑡 )𝑥 + 𝑋(𝑡) 𝑋 (𝑠)𝐵(𝑠) 𝑢(𝑠)𝑑𝑠: 𝑢є𝑈 }

= 𝑋(𝑡)[𝑋   (𝑡 )]

=  𝑥  + ∫     𝑋 (𝑠)𝐵(𝑠) 𝑢(𝑠)𝑑𝑠: 𝑢є𝑈

=  𝑥  +  𝑅(𝑡 , 𝑡 ), since 𝑋(𝑡)  is a fundamental matrix and  fundamental matrices are invertible .

Definition 1.4.5: (Properness)

The system(1.2.3) given as

𝑥̇ = 𝐴(𝑡)𝑥(𝑡 ) + 𝐵(𝑡)𝑢(𝑡), is proper on the interval [𝑡 , 𝑡 ]   𝑖𝑓 and only i𝑓    𝐶 𝑋 (𝑡)𝐵(𝑡) = 0, 𝑎. 𝑒 𝑜𝑛[𝑡 , 𝑡 ] , implies that 𝑐 = 0.

Here, the set function     𝑔(𝑡) = 𝐶 𝑋   (𝑡)𝐵(𝑡), is called the controllability index.

 Lyapunov Function

Consider the system

𝑥̇  = 𝑓(𝑥) , 𝑓(0) = 0                                                          (1.2.4)    where f ∶ D →  𝑅 is continuous, D is a subset of  𝑅 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑏𝑦                                      𝐷 = {𝑥є𝑅 :∥ 𝑥 ∥≤ 𝑟}.

The solutions of system(1.2.4) are uniquely stable for given initial data 𝑡 , | 𝑡 | < ∞and 𝑥є𝐷  Here, we shall be concerned with the stability of the trivial solution.

 

Definition 1.4.6: (positive definite function)

A function V: D→R is said to be positive definite if V varnishes only at the origin and 𝑉(𝑥) > 0 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ≠ 0.

Definition 1.4.7: (negative definite function)

A function V: D→R is said to be negative definite if V varnishes only at the origin and 𝑉(𝑥) < 0 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ≠ 0.

Definition 1.4.8: (positive semi – definite function)

A function V: D→R is said to be positive semi – definite function if V varnishes only at the origin and 𝑉(𝑥) ≥ 0 , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ≠ 0.

It is negative semi-definite if it varnishes only at the origin and 𝑉(𝑥) ≤ 0  , 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑥 ≠ 0.

Definition1.4.9: (Lyapunov Function)

Let V: D→R be continuously differentiable and positive definite on D. Let the derivative

(called Eulers’ derivative) of V along the solution path of the system (1.2.4) be defined by

̇

𝑉̇ (𝑥) =   𝑉(𝑥) = ∑  = ∑  𝑓(𝑥), then Vis called a Lyapunov function for the system (1.2.4)

Theorem 1.4.1.

The system (1.2.3) is completely controllable or null-controllable if and only if

𝑅𝑎𝑛𝑘(𝐵, 𝐴𝐵, 𝐴 𝐵, … , 𝐴 𝐵) = 𝑛                                            (1.2.5)

Example 1.4.1

Consider the motion of Satellite in a central gravitational field. Let the kinetic energy T and the potential energy P be given by

T =  m (𝑟̇ , 𝑟 𝜓̇ ), P =

where m is the mass of the satellite , if the product of the mass of the planet and the gravitational constant r  and ψ are polar coordinate of the satellite (which we idealized as a particle).  .

 

The equations of motion are then

 

𝑟̈  =      + r𝜓̇

 

(1.2.6)

𝜓̈  =        ^{̇ ̇}

 

 

Particular solution is given by the motion in a circular orbit with          .

r = R ,  𝜓̇ =  ω =

Deviations from this orbit are to be corrected by two rocket engines with thrust vectors in the direction   e_r   and  e_ψ   respectively. Thus, the equation of motion now becomes

𝑟̈  =        +  r𝜓̇ +  𝑢

 

𝜓^̈  =       ^{̇ ̇}                                                      (1.2.7)

 

where 𝑢 and 𝑢 are the radial and transverse (normal) components of the acceleration respectively (with respect to the circular orbit) as produced by the rocket engines.  Let us now introduce the following new variables:

 

𝑥 = r – R

 

𝑥 =        𝑥̇   = 𝑟̇

(1.2.8)

𝑥  = (𝜓 − 𝜔𝑡)𝑅

𝑥   =  𝑥̇   = ( 𝜓̇ − 𝜔)𝑅

 

which represent the (small) deviations from the ideal motion

 

Now,  𝑥̇  = 𝑥̈  = 𝑟̈= 𝑟̇𝜓 –    + 𝑢

 

= ( 𝑥 + R) ω²    –   _{( )} + 𝑢

 

= ( 𝑥 + R)ω²   −𝑅 ω² (1 +   )^-2  + 𝑢

 

=  ( 𝑥 + R)ω²  −𝑅 ω²[1 – 2       +  3(   )² – 4 (   )³  +  …]  +  𝑢

= 3ω² 𝑥 −3ω² 𝑥 ² R^-1 + 4ω² 𝑥   R^-2 −5ω² 𝑥 ⁴ R^-3 + …+ 𝑢 (using Binomial Theorem).

⇒                    𝑥̇    =  3ω² 𝑥 +  𝑢 +  h.o.t   ( higher order terms).                                (1.2.9)

 

𝑥̇ = 𝑥̈ = 𝜓̈𝑅 = (−2𝜔𝑟̇ ^̇ +   )𝑅  = −2𝜔𝑅𝑥 (𝑥 + 𝑅 )^-1 + 𝑢 𝑅(𝑥 + 𝑅 )^-1

=   −2𝜔𝑥 (1 +  )^-1    + 𝑢 (1 +   )^-1

^-1

=   ( −2𝜔𝑥 + 𝑢 ) (1 +       )

=   ( −2𝜔𝑥 + 𝑢 ){1 –(      )¹ + (     )² – (     )³ + … }

=   −2𝜔𝑥 +  𝑢    + h.o.t .                                                                                         (1.3.0)

Thus, the linearized system yields

.

𝑥̇ =  𝑥

𝑥̇  = 3𝜔 𝑥  + 𝑢                                                                               (1.3.1)

𝑥̇ =  𝑥

 

𝑥̇    =  −2𝜔𝑥 + 𝑢

 

with    𝑥̅    =   ( 𝑥  𝑥 𝑥  𝑥 ), and   𝑢   = (𝑢 𝑢 )

 

We have the equivalent system (1.2.3) with

 

 

0     1   0   0                                                                0         0

3ω₂ 0   0   0                                                                1         0

A = B =

0     0   0   1                        ;                                       0         0

0 -2ω   0   0                                                                0         1

 

which provides a first approximation to the controlled motion in a neighborhood of the reference orbit.

To determine the controllability of the system, we compute rank ( B, AB, A²B, A³B )  and show that it equals 4.

 

 

0  0  1   0   0       0    3ω²   0

Now, rank (B, AB, A²B, A³B)  =       1  0  0   0  3ω²    0     0       0             =     4 = n.

0  0  0   1 -2ω     0     0       0

0  1 -2ω 0 -6ω³   0     0       0

 

Thus, the system is completely controllable. This means that the satellite can be steered to “always” move in the circular orbit.

Suppose one of the rocket engines breaks down or is shut down for some reason; will the system remain completely controllable? Let us suppose the engine providing the radial thrust is shut down, then only the transverse (tangential) thrust is available, the   B^T  =  (0 0 0 1).

 

Computing, we have                          0   0   0   0

Rank(B,AB,A²B, A³B)  =  rank     0   0   0   0                 =  2  ≠ 4

• 1 0   0
• 0 0   0

The system, therefore, is no longer completely controllable.

Suppose now the engine providing the transverse thrust is shut down, then only the radial thrust is available. Then

B^T    =  ( 0 1 0 0 )

Computing, we have

 

0  1   0      3ω²  rank (B ,AB ,A²B,  A³B )    =        rank              1  0  3ω²   0                =   3  ≠  4.

0  0  -2ω   0

0 -2ω  0   -6ω³

The system is again no longer completely controllable. Thus, once one of the engines is shut down, the satellite can no longer be “constrained” to move along the circular (reference) orbit.

1.5 .Functional Differential Equations

Definition 1.5.1.

Functional differential equation is an equation (ordinary differential equation) where the derivative at various time instances t-h_i, (i = 0,1,2,…,n) is a relation to the state of the equation at equally various time instances. It is a more general type of differential equation. It comprises:

(a) .    Retarded equation and    (b).      Neutral equation.

Definition 1.5.2:  (Retarded Functional Differential Equation)

Retarded Functional differential equations are differential equations where the derivative of the state is expressed in terms of the state at various time instances   t-h_i,  (i =1, 2, 3,…, n)

Example 1.5.1

𝑥̇(𝑡)  = 2𝑥(𝑡) + 3𝑥(𝑡 − ℎ)      ,where  𝑥̇(𝑡) is a derivative of a state  𝑥(𝑡).

𝑥̇(𝑡)  = 1+ 𝑥(𝑡 − 1) + 𝑥(𝑡 − 2) + 𝑥(𝑡) .

Here,   1 = constant

(𝑡 − 1)= a day or a year ago. (I.e. earlier time)          (𝑡 − 2)= 2 days or 2 years ago.              t   = presently or present time.

Difinition1.5.3: (Neutral Functional Differential Equation)

Neutral Functional Differential Equation is the differential equation, where the derivative of the state at previous and present time instances is expressed in terms of time t.

Example 1.5.2

𝑥̇(𝑡) − 𝑥̇(𝑡 − 1)   =   𝑥(𝑡) + 2𝑥(𝑡 − 1)+ 3𝑥(𝑡 − 2)

1.6. Difference between Retarded Functional and Neutral Differential

Equations

Difference between the retarded functional differential equation and the neutral functional differential equation is that the retarded functional differential equation contains present time in its derivative, while the neutral functional differential equation contains both present and previous (past) time in its derivative. The retarded or the neutral functional differential equation is called Delay Equation.

Usually in the neutral functional differential equation, we define a continuous linear operator D to represent the derivative of the state at present, and earlier time instances. That is,   D(𝒕, 𝒙_𝒕) is called the functional difference operator,  where          (𝒕, 𝒙_𝒕)   D(𝑡, 𝑥 )  =  𝑥(𝑡)–𝑥(𝑡 − 1) .

We can now rewrite the equation of Example 1.5.2 in terms functional operator thus:

[𝐷(𝑡, 𝑥 )]     =      [𝑥(𝑡)–𝑥(𝑡 − 1) ]   =    𝑥(𝑡) + 2𝑥(𝑡 − 1)+ 3 𝑥(𝑡 − 2) .

=  𝑥̇(𝑡) − 𝑥̇(𝑡 − 1)   =   𝑥(𝑡) + 2𝑥(𝑡 − 1)+ 3 𝑥(𝑡 − 2) .

1.7.   General Form of Functional Differential Equation.

The initial conditions for general differential equation and functional differential equations are respectively given by:

𝑥̇   =   𝑓(𝑡, 𝑥)  ; 𝑥(𝑡 )      = 𝑥  ,                                                        (1.3.2)

Here, 𝑥 is a vector (i.e. initial point is a vector) and

𝑥̇      =  𝑓(𝑡, 𝑥 ) ; 𝑥(𝑡 ) = ф.                                                             (1.3.3)

Here the initial point φ is a function defined in the delay interval [-h, 0] , h > 0 ,

𝑥(𝑡 ) =   𝑥   = ф

Definition 1.7.1: (Retarded functional differential equation)

In Functional Differential Equation of the retarded type, whose general equation is given by

𝑥̇  = 𝑓(𝑡, 𝑥 )  ,

we prescribe an initial function  𝑥(𝑡 )  =  𝑥   = ф    ,  over the delay interval [-h, 0] ,  h > 0

Definition1.7.2: (Neutral Functional Differential Equations)

In functional differential equation of the neutral type, whose general equation is given by

[D(𝑡, 𝑥 )]       =      f (𝑡,𝑥 ), we also prescribe initial function  𝑥(𝑡 ) =  𝑥 =  ф  , over the delay interval  [-h, 0] ,  h > 0

Definition 1.7.3

Let 𝑥(𝑡)be a function defined over the interval ( -∞ ,t ) , the function 𝒙_𝒕  is defined over the delay interval [-h, 0] such that

𝑥 (s)  =  𝑥(𝑡 + 𝑠) ; s є [-h, 0] .

1.8. General Solution Format or Variation of Parameters.

Consider the delay equation (retarded type)

𝑥̇  =  𝑓(𝑡, 𝑥 ) ; 𝑥(𝑡 )  = 𝑥  = ф,

The solution format is given by direct integration with respect to time t to get,

𝑥(𝑡)  =  ф(0) +  ∫ 𝑓 𝑠, 𝑥 (𝑠) ds ,        t > 0                                  (1.3.4)

If it is the neutral functional differential equation given by

 

[D(𝑡, 𝑥 )]        =  𝑓(𝑡, 𝑥 ) ;  𝑥(𝑡 )  =   𝑥   = ф,    the solution format (variation of parameter) is given as

D (t,𝑥 )   =      ф (0) +  ∫ 𝑓(𝑠, 𝑥 (𝑠)  ) ds,        t > 0                             (1.3.5)

1.9.   Volterra Integral Equations

In applied mathematics, mathematical physics, radioactive transfer, theory of population and in engineering, most formulations are often presented in the forms of integral equations.

An integral equation is an equation in which the function to be determined appears under an integral sign.

In ordinary differential equations, integral equations can either be linear or non-linear. The linear integral equations were grouped into two, namely

 

(a).  Fredholm integral equations and  (b).Volterra integral equations.

Example 1.9.1 (Fredholm integral equations and/or forms)

∫ 𝐾(𝑥, 𝑦) ф(𝑦)𝑑𝑦 = 𝑓(𝑥)                                                        (i)                                             ф(𝑥) – λ ∫ 𝐾(𝑥, 𝑦) ф(𝑦)𝑑𝑦 = 𝑓(𝑥)                                       (ii)

𝛼(𝑥)ф(𝑥)  – λ∫ 𝐾(𝑥, 𝑦) ф(𝑦)𝑑𝑦 = 𝑓(𝑥)                                  (iii)

Equation (i) is called Fredholm integral equation of first kind.  While equation (ii) is the Fredholm integral equation of the second order and equation (iii) is the Fredholm integral equation of the third kind. In all the three forms of Fredholm integral equations

[(i) –    (iii)] , 𝐾(𝑥, 𝑦) is called the kernel and ф(𝑦) which appears under the integral sign is the dependent variable meant to be determined.  The integral from a- b may be infinite or may take the forms: (-∞,b] or [a,∞) or (-∞,∞).

Example 1.9.2: (Volterra integral equations and/or forms).

1. 𝒇𝒅𝒚

x

2. 𝒇(𝒙) =  ф(𝑥) – λK(x, y) ф(𝒚)𝒅𝒚

a

x

3. 𝒇(𝒙) =  𝛼(𝑥)ф(𝑥)   – λ K(x, y) ф(𝒚)𝒅𝒚

a

We observe that the upper limit in the above equations (1) – (3) is a variable x and not a constant as in the case of Fredholms’.

Definition 1.9.1: (Closed Operators)

An operator T: X → Y , where X, Y are linear spaces is said to be closed if for any sequence u_nєD(T) such that u_n → u  and Tu_n → v   , u є D(T) and Tu   =  v

1.10. Existence and Uniqueness of Optimal Control for Linear Neutral

Volterra Integro-Differential Systems

We recall here that our system of investigation is given by

𝑑

𝑥(𝑡) − 𝐶(𝑡, 𝑠)𝑥(𝑠)𝑑𝑠 − 𝑔(𝑡) = 𝐴(𝑡)𝑥(𝑡) + 𝐺(𝑡, 𝑠)𝑥(𝑠)𝑑𝑠

𝑑𝑡

• ∫ 𝑑 𝐻(𝑡, 𝜃)𝑢(𝑡 + 𝜃)                                  (1.3.6)

For purposes of clarity, we define the following terminologies as they relate to system (1.3.6)

With solution given by

x(t) = X(t ,0)[ x(0) – g(0) ] + g(t) − ∫ (  ) X(t, s)g(s)ds  

• ∫      dH ∫        X(t , s − θ)H(s − θ, θ)𝑢 (s)ds

• ∫    ∫     X(t , s − θ)dθH^∗( s − θ, θ) u(s) ds                                          (1.3.7)

Definition 1.10.1: (Complete state)

The complete state for system (1.3.6) is given by the set z (t) = {x, 𝑢 } .

Definition 1.10.2: (Relative Controllability)

The system (1.3.6) is said to be relatively controllable on [0,𝑡 ] if for every initial complete state z(0)  and 𝑥 є 𝐸 , there exists a control function u(t) defined on [0,𝑡 ]   such that the solution of system (1.3.6) satisfies 𝑥(𝑡 )  =  𝑥 .

Definition 1.10.3:     (Reachable Set)

The reachable set for the system (1.3.6) is given as

𝑅(𝑡 , 0) =    ∫ [∫ 𝑋(𝑡, 𝑠 − 𝜃)𝑑 𝐻^∗(𝑠 − 𝜃, 𝜃)𝑢(𝑠)]𝑑𝑠                       (1.3.8)

Definition 1.10.4:        (Attainable Set)

The attainable set for the system (1.3.6) is given as   , A(t, 0) = {x (t,  𝑥 , u) : u є U} where   ,   U =    { u є L₂( [0, t,] , E^m ) : │𝑢 │ ≤ 1 , j= 1,2,…., m}

Definition 1.10.5:     (Target Set)

The target set for system (1.3.6) denoted by G (𝑡 , 0) is given as

G(t₁, 0) = { x( 𝑡 ,  𝑥 , u) :  𝑡 ≥ т > 𝑡 for fixed т and u є U}

Definition 1.10.6:      (Controllability Grammian)

The controllability grammian for the system (1.3.6) is given as

𝑊(0, 𝑡) = ∫ 𝑧(𝑡, 𝑠)𝑧 (𝑡, 𝑠)𝑑𝑠.

𝑤ℎ𝑒𝑟𝑒   𝑧(𝑡, 𝑠) = ∫ 𝑋(𝑡, 𝑠 − 𝜃)𝑑 𝐻^∗(𝑠 − 𝜃, 𝜃)  and  т denotes matrix transpose.

Definition 1.10.7: (Relative Controllability)

The system (1.3.6) is relatively controllable on [ 0,  𝑡  ] if

𝐴(𝑡 , 0) ∩ 𝐺(𝑡 , 0) ≠ ∅; 𝑡 > 0

Definition 1.10.8: (Properness)

The system (1.3.6) is proper in  Eⁿon   [0,  𝑡 ] , if   span R( t, 0) = Eⁿ  that is if

𝑐 [∫     X(t, s − θ) d H^∗(s − θ, θ)] = 0       (almost everywhere.), 𝑡  > 0 ⇒ 𝑐 = 0; 𝑐𝜖𝐸

 

1.11. Global Uniform Asymptotic Stability for Nonlinear Infinite Neutral  Differential Systems

In stability theory, the desire is to achieve that solutions near the origin remain quite close for all future times. The desire to maintain a constant for the solutions of a system over time has given rise to different variants of stability. We have in the literature, uniform stability, asymptotic stability, exponential stability of neutral equations, (see Cheban (2000), Chukwu (1992), Chukwu (1981), Hale (1977), and Onwuatu (1994)).

The study of stability of neutral systems has given impetus to the task of investigating the stabilization of nonlinear neutral system. This is the ever growing interest by researchers in stability theory (see Eke (2000)).

The methods involved the computation of the eigen-values of certain matrices. With the computer now in vogue, the computations involved are less tedious.

The energy method of Alexander Mrkhailovick Lyapunov (1829), which revolves around the notion that in a stable system, the total energy in the system would be a minimum at the equilibrium point. The total energy is called the Lyapunov function, whose derivative along the solution path must be negative semi-definite for the solution of the system to have small upper bound, (see Hmanmed (1986)). The energy method has provided approval method for discussing stability of various non-linear functional equations. In his work, Chukwu (1992) extended the work in Cruz and Hale (1970) by monitoring nonlinear functional equations with positive definite Lyapunov functions where explicit solutions cannot be guaranteed. Hale (1977) has discussed extensively on the stability of nonlinear neutral equations using various methods, providing exponential estimates of solution of same thereby lending clarity of meaning to exponential stability ,which in Chukwu (1992) and Chukwu (1981) was extended to asymptotic exponential stability in the large for neutral systems. Stability of perturbations of linear neutral systems has received appreciable emphasis in Chukwu (1992) and Chukwu (1981).

In Onwuatu (1993) stability of infinite neutral systems is reported. We hope to extend the works in Onwuatu (1993) to systems of the form.

[𝐷(𝑡, 𝑥 )]    =  L(𝑡, 𝑥 )𝑥 + 𝑓(𝑡, 𝑥 ) + ∫ 𝐴(𝑡) 𝑥(𝑡 + 𝜃)𝑑 , 𝑥 = ф.                   (1.3.9)

(a nonlinear infinite neutral system)

Now, let n be a positive integer and E = (- ∞, ∞) be the real line.  Denote by 𝐸 the space of real n- tuples called the Euclidean space with norm denoted by │.│

If   J = [a ,b] is any interval of E, L₂ is the Lebesgue space of square integrable functions from J to 𝐸 written in full as  L₂([ a, b] , 𝐸 )  .

Let   h > 0 be a positive real number and let C ([-h,0] , 𝐸 )     be the Banach space of continuous function with the norm of uniform convergence defined by

║ ф║   = supф(s),     -h ≤ s ≤ 0, for   ф є C ([-h, 0] , 𝐸 ),

𝐼𝑓 𝑥 𝑖𝑠 𝑎 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑓𝑟𝑜𝑚[−ℎ, ∞) 𝑡𝑜 𝐸 , 𝑡ℎ𝑒𝑛  𝑥 𝑖𝑠 𝑎 𝑓𝑢𝑛𝑐𝑡𝑖𝑜𝑛 𝑑𝑒𝑓𝑖𝑛𝑒𝑑 𝑜𝑛 𝑡ℎ𝑒 𝑜𝑛𝑡ℎ𝑒𝑑𝑒𝑙𝑎𝑦 𝑖𝑛𝑡𝑒𝑟𝑣𝑎𝑙      𝑔𝑖𝑣𝑒𝑛 𝑎𝑠𝑥 (𝑠) = 𝑥(𝑡 + 𝑠); 𝑠𝜖[−ℎ, 0] , 𝑡𝜖[0, ∞).

Consider the nonlinear infinite neutral system.

[𝐷(𝑡, 𝑥 )] = 𝐿(𝑡, 𝑥 )𝑥 + ∫^∞ 𝐴(𝑡) 𝑥(𝑡 + 𝜃)𝑑 + 𝑓(𝑡, 𝑥 )                        (1.4.0)

𝑤ℎ𝑒𝑟𝑒𝐿(𝑡, 𝑥 ) = ∑    𝐴 𝑥(𝑡 − 𝑤 ) + ∫   𝐴(𝑡) 𝑥(𝑡 + 𝜃)𝑑

𝐿(𝑡, 𝑥 )𝑥 = ∫ 𝑑 η 𝑡, 𝑠, 𝑥(𝑡 + 𝑠) 𝑥(𝑡 + 𝜃)

η(t, s, ф, ψ) ≥ 0, for s ≥ 0 , and ф, ψϵC

η(t, s, ф, ψ) = η(t, s, ф, ψ) for  t < −ℎ.

η( t, s, ф, ψ) is a continuous matrix function of bounded variation in s є[−h ,0 ] ,  𝑣𝑎𝑟 η(t) ≤ m(t), m(t)ϵL .

L₁ is the space of integrable functions. Let Ω be an open subset of  ExC and D and L be bounded linear operators defined on ExC into Eⁿ.

|𝐿(𝑡, 𝑥 )𝑥(𝑡)| ≤ 𝑚(𝑡)‖𝑥 ‖, 𝑓𝑜𝑟 𝑎𝑙𝑙 𝑡𝜖𝐸, ψ(t) є 𝐂.

𝐷(𝑡, 𝑥 ) = 𝑥(𝑡)𝑔(𝑡, 𝑥 ),   𝑤ℎ𝑒𝑟𝑒

𝑔(𝑡, 𝑥 ) = ∑^∞ 𝐴 (𝑡) ф(t − w (t)) + ∫    𝐴(𝑡, 𝑠) ф(𝑠)𝑑𝑠 = ∫ 𝑑 𝐻(𝑡, 𝜃) ф(𝜃),

𝑤ℎ𝑒𝑟𝑒   0 ≤ 𝑤  ≤ ℎ 𝑎𝑛𝑑 | ∫    𝑑 𝐻(𝑡, 𝜃) ф(𝜃) | ≤ ℎ(𝛳)‖ф‖

𝐷(𝑡, 𝑥 ) 𝑖𝑠  𝒏𝒐𝒏 − 𝒂𝒕𝒐𝒎𝒊𝒄 𝒂𝒕 𝒛𝒆𝒓𝒐(differentiable and integrable at zero )

∫  |𝐴(𝑡, 𝑠)|𝑑𝑠 + ∑     |𝐴 (𝑡) | ≤ δ(є) , for all t , where  δ(є) → 0 .

 

f  is continuous and satisfies other smoothness conditions.

Consider  the  system (1.4.1) below:

[𝐷(𝑡, 𝑥 )] = 𝐿(𝑡, 𝑥 )𝑥 + ∫ 𝐴(𝑡, 𝜃)𝑥(𝑡 + 𝜃)𝑑 + 𝑓(𝑡, 𝑥 )                         (1.4.1).

(Circularity of the function from -∞ to 0, and from 0 to ∞)

We can linearize the system (1.4.1. ) as in 𝐂𝐡𝐮𝐤𝐰𝐮 (𝟏𝟗𝟗𝟐) by setting 𝑥 = 𝑧 ;

𝑎 specified function inside the function 𝐿(𝑡, 𝑥 )𝑥 𝑡𝑜 ℎ𝑎𝑣𝑒 L(t, z )𝑥 𝑤𝑖𝑡ℎ 𝑛𝑜 𝑙𝑜𝑠𝑠 𝑜𝑓 𝑔𝑒𝑛𝑒𝑟𝑎𝑙𝑖𝑡𝑦.

Thus , the system (1.4.1) becomes

𝐷(𝑡, 𝑥 ) = 𝐿(𝑡, 𝑧)𝑥 + ∫ 𝐴(𝑡, 𝜃)𝑥(𝑡 + 𝜃)𝑑 + 𝑓(𝑡, 𝑥 )                               (1.4.2)

Evidently,
𝐿(𝑡, 𝑧)𝑥 = ∑	𝐴 𝑥(𝑡 − 𝑤 ) + ∫	𝐴(𝑡, 𝜃)𝑥(𝑡 + 𝜃)𝑑 + ∫ 𝐴(𝑡, 𝜃)𝑥(𝑡 + 𝜃)𝑑    (1.4.3)
𝐿∗(𝑡, 𝑧) 𝑥 = ∑	𝐴 (𝑡 − 𝑤 ) + ∫	𝐴(𝑡, 𝜃)𝑥(𝑡 + 𝜃)𝑑                                                 (1.4.4)

𝑇ℎ𝑒 𝑟𝑒𝑝𝑟𝑒𝑠𝑒𝑛𝑡𝑎𝑡𝑖𝑜𝑛𝑠 𝐿, 𝐿^∗ are the same under the following assumptions

𝐿(𝑡, 𝑧)𝑥 = lim _→   ∑       𝐴 𝑥(𝑡 − 𝑤 ) + lim _{,   →}     ∫ 𝐴(𝑡, 𝜃)𝑥(𝑡, 𝜃)𝑑𝑠                (1.4.5)

We assume the limits exist, giving finite partial sum for the infinite series and the improper integrals. Thus the system

𝐿^∗(𝑡, 𝑧)𝑥 = ∑     𝐴 (𝑡 − 𝑤 ) + ∫   𝐴(𝑡, 𝜃)𝑥(𝑡 + 𝜃)𝑑 ,

is finite and well defined function.

In the light of the above,the system (1.4.1) reduces to

	[𝐷(𝑡, 𝑧)𝑥 ] = 𝐿(𝑡, 𝑧)𝑥	+ 𝑓(𝑡, 𝑥 ); 𝑥( 𝑡 ) = ф𝜖𝐶                                         (1.4.7)
	𝑤ℎ𝑒𝑟𝑒,     𝐿(𝑡, 𝑧)𝑥	=        𝐴 𝑥(𝑡 − 𝑤 ) +    𝐴(𝑡, 𝜃)𝑥(𝑡 + 𝜃)𝑑 ,

Integrating   (1.4.7), after linearizing, we have

𝑥(𝑡) = 𝑥(𝑡, 𝑡 , ф, 0) + ∫ 𝑋(𝑡, 𝑠)𝑓(𝑠, 𝑥 )𝑑𝑠 ,                                (1.4.8)

Where, X(t ,s) is the fundamental matrix of the homogenous part of the system  (1.4.7).

X (t, s) = I      (identity matrix); t = s

From the transformation in Hale (1977), there is a linear operator T such that

𝑋 (𝑠)ф = (𝑡, 𝑠)𝑋(𝜃); 𝛳𝜖[−ℎ 0] .                                                   (1.4.9)

𝑋(𝑡 + 𝜃, 𝑠) = 𝑇(𝑡, 𝑠)𝑋(𝜃)

For ө = 0, we have   𝑋(𝑡, 𝑠) = 𝑇(𝑡, 𝑠) = 𝑇(𝑡, 𝑠), where T is defined as follows:

(𝑖)       𝑇(𝑡, 𝑠)is an operator defined on C = C ([−h, 0] , E ), 𝑇(𝑡, 𝑠)is bounded for T є C.

(𝑖𝑖)     𝑇(0) = 𝐼 𝑎𝑛𝑑  𝑇 𝑖𝑠 𝑠𝑡𝑟𝑜𝑛𝑔𝑙𝑦 𝑐𝑜𝑛𝑡𝑖𝑛𝑢𝑜𝑢𝑠.

(𝑖𝑖𝑖)     𝑇(𝑡, 𝑠)is completely continuous in t.

The family { (𝑡, 𝑠) 𝑓𝑜𝑟 𝑡 > 𝑠}is a semi − group of linear transformations,

𝐬𝐞𝐞 𝐂𝐡𝐮𝐤𝐰𝐮(𝟏𝟗𝟖𝟑) for these properties.  Now writing (1,4.8)in terms of 𝑇(𝑡, 𝑠),             𝑤𝑒 ℎ𝑎𝑣𝑒 , 𝑥(𝑡, 𝑡 , ф, 𝑓) = [𝑇(𝑡, 𝑡 )] ф(0) + ∫ 𝑋(𝑠, 𝑥 )𝑑𝑠.                                                (1.5.0)

We now define the following:

Definition: 1.11.1:  (stability)

The trivial solution x = 0,   of system (1.4.1) is 𝐬𝐭𝐚𝐛𝐥𝐞 if for any given 𝑡 є E    and a positive number ε > 0, 𝑡ℎ𝑒𝑟𝑒 𝑒𝑥𝑖𝑠𝑡𝑠 𝛿 = 𝛿 ( 𝑡 , ε) such that ф є B(0, ε),  implies that

𝑥 (𝑡 , ф) є β(0, δ)  for all  t ≥ t0 , фєC and 𝐁(𝟎, 𝐫) is a ball centered at 0, with radius r.

Definition 1.11.2:  (uniform stability)

The trivial solution x = 0   of the system (1.4.1) is said to be uniformly stable if for any ε >0,  there exists δ = δ(ε) (independent of 𝑡 ) such that фє B(0, ε)  implies

𝑥 (𝑡 , ф) є B(0, δ) ,   for all 𝑡 > 𝑡

Definition 1.11.3: (asymptotic stability)

The trivial solution x = 0 , of  the system (1.4.1)  is asymptotically stable if it is stable   such that ф є B(0 , δ)   implies that   𝑥 (𝑡 , ф) →0  ,    as  t →∞

Definition 1.11.4:  (uniform asymptotic stability)

The trivial solution of the system (1.4.1) is uniformly asymptotically stable if the system is uniformly stable and for фє B(0, δ) ,  implies 𝑥 (𝑡 , ф) → 0      as t→∞       .

The solution 𝑥 (𝑡 , ф) of system (1.4.1) is exponentially asymptotical stable if there exist constants  𝑘 > 0 ,and 𝑐 > 0, such that the solution satisfies

𝑥 (𝑡 , ф) = ф and ║ 𝑥 (𝑡 , ф)║   ≤  𝑘𝑒 ^{(    )}

Remark: 1.11.1

If the choice of the initial function ф є C ([-h,0] , Eⁿ)  is arbitrary, then

Definition (1.11.4) is said to be exponentially asymptotically stable in the large and so we have global results that generally defines global conditions.

A homogenous linear neutral equation is given by

[𝐷(𝑡, 𝑥 )  ]    = 𝐿(𝑡, 𝑥 )                                                                (1.5.1.)

𝐷(𝑡, 𝑥 )  is called the functional difference operator. We now give the condition for the uniform stability of the functional difference operator.

 

1.12.   Existence of Mild Solution of Nonlinear Neutral Differential

Equations in Banach Spaces

The primary motivation of the study of neutral functional differential equations is that

it has wide range of applications (see Balachandran and Anandhi (2003), Balachandran and Dauer (1996)). Balachandran and Dauer (1996) have pointed out their application in transmission line theory. They explained that the mixed initial boundary hyperbolic differential equation which arises in the study of lossless transmission lines can be replaced by an associated neutral differential equation. Asuquo and Usah (2008), Chukwu (1992), and Iheagwam and Nse (2007), have provided complex economic models governed by neutral differential equations and have provided broad policy guidelines for the control of regional economy to equilibrium state.

Neutral systems have also been cited to have applications in population studies and Engineering, in nuclear reactor dynamics, (see Balachandran and Dauer (1996), Fu and

Ezzinbi (2003)).

Many studies have investigated the conditions for the existence of solutions of linear and non-linear neutral systems. Notably, among them are in Balachandran and Dauer (2002), Balachandran and Leelamani (2006), Fu and Ezzinbi (2003). The presentation in; Balachandran and Anandhi (2003), Balachandran and Dauer (1996 ), Balachandran

and Sakthivel (1999 ) have introduced a twist in the study of neutral systems by investigating linear and non-linear neutral volterra integro differential equations. These efforts provided

more advanced method of integration yielding the variation of parameters (see Balachandran and Anandhi (2003)). The theory of neutral differential equation is currently

being carried out in Banach spaces, (see Balachandran and Anandhi (2003),

Balachandran and Leelamani (2006), Umana (2008)).

𝑑

[𝑥(𝑡) + 𝑔(𝑡, 𝑥(𝑡), 𝑥 𝑢 (𝑡) , … , 𝑥(𝑢 (𝑡)))]

𝑑𝑡

= 𝐿(𝑡, 𝑥 ) + ℎ𝑡, 𝑥(𝑡), 𝑥 𝑣 (𝑡) , … , 𝑥 𝑣 (𝑡)

𝑥(𝛳) = 0; 𝛳𝜖[−ℎ, 0] , 𝑡𝜖[0. 𝑡 ] , 𝑡 > 0                                                                       (1.5.2),

With the purpose of obtaining mild solutions of the system(1.5.2) in the Banach spaces ,using the Schaefer’s Fixed Point Theorem.

In the system (1.5.2), g, L, h are the systems parameters. L is the infinitesimal generator of a compact analytic

𝐿(𝑡, 𝑥 )  = ∫ 𝑑𝜂(. , 𝑠, ф) 𝑥(𝑡 + 𝑠) = ∑ 𝐴 (𝑡 − 𝑤 ) + ∫ A(t, θ)x(t + θ) 𝑑  is a bounded linear operator, where the n x n matrix functions 𝐴 , A(t, θ) are measurable in

((t, s) є E x E,  θє [- ∞, 0).   η is normalized such that η (t, s, ф) = 0, s ≥ 0 for all ф η (t, s, ф) = η (t, -h, θ) for all s ≤ -h

η (t, s, ф)  is continuous from the left in s on (-∞, 0] and has bounded variations on  (-∞, 0] for each t, ф and there is an integrable function M such that

║ 𝐿(𝑡, 𝑥 )   ║ ≤   M(t) ║𝑥 ║       ,for all t є ( -∞,∞ ), ф є ( -∞,0].

We assume     L(t, ф) is continuous.     Let   0єD(L), then the fractional power L^a  for   ,

0 < a < 1   as closed linear operator on its domain   D ( L^a ) is dense in X. Furthermore,

D( L^a ) is  a Banach space under the norm

║x ║_a   =  ║ L^a x║       for  all  x є D ( L^a )    and it is denoted by   𝑋 .   h is a

function defined on the product space  Jx𝑋            into  X g : [0,𝑡 ] x 𝑋       → X,   is a continuous function.

The delays 𝑢 (𝑡), 𝑣 (𝑡), are continuous scalar valued functions defined on J such that    𝑢 (𝑡) ≤ t     and 𝑣 (𝑡) ≤ t.

That is, these are values preceding t. We define the supremum norm on X  by

║x║ = max _є     |x (t)|.

The imbedding   𝑋 → 𝑋      for 0< b < a < 1 is compact whenever the resolvent operator L is compact. For semi-group {T(t) }, the following properties will be used:

• There is a number 𝑁 > 1 such that ║T (t)║ ≤ 𝑁  for all t є [0, 𝑡 ] .
• For any a > 0, there exists a positive constant 𝑁 such that

║L^a T (t)║ ≤     ,           0 < t < τ

To study system (1.5.2), we assume the hereditary property of the function.

Let          x: (-∞,τ ] → X ,𝑥    is a function defined on the delay interval  ( -∞ ,0 ]   such that       𝑥 (θ)   = x (t + θ)    , belongs to some abstract phase space (-∞, 0]   .

In this work, the state space will be the abstract phase space   C ([-∞, 0] )

Definition 1.12.1:   (mild solution)

A function    𝑥(. )   is called a mild solution of the system (1.5.2) if

𝑥(𝑡)   = 0      ,    for t є ( -∞ ,0 ] ,

the restriction of  𝑥(𝑡)   to the interval  [ 0, τ ]    is continuous and for each  [ 0, τ ] , the function  𝑥(𝑡)    satisfies  system( 4.4.3).  That is, the function  𝑥(𝑡)  satisfies the following integral equation:

𝑥(𝑡)   =  T(t){0 + g[0 ,x (𝑢 (0)),…,x(𝑢 (0))] – g{t ,x(t),x(𝑢 (t)),…,x(𝑢 (t))}

− ∫ LT (t − s) g(s, x(s), x (u (s)) … x (u     (s))) ds

• ∫ T(t − s)h(  s , x(s), x(𝑣 (s)), … , x(𝑣 (s)))ds                                (4.4.3) where   LT (t − s) g(s, x(s), x (u (s)) … x (u (s)))  is integrable for    s є ( -∞, t ] .

CONTROLLABILITY RESULTS FOR NON-LINEAR NEUTRAL FUNCTIONAL DIFFERENTIAL EQUATIONS

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