The structure of this Thesis is the following. In Chapter1we give a list of notations and acronyms we use in the rest of the Thesis. In Chapter2we give some basic background, known results and notions on the topics we will use in later chapters in our investigation.
Chapter1. Introduction 3 In Chapter3 we study a general class of nonlinear VIEs of the form
x(t) = Z t
0
f(t, s, x(·))ds+h(t), t≥0,
and present a sufficient condition which guarantees the boundedness of the solution (see Theorem 3.2.1below). Our method is based on a special inequality and a comparison technique. We show in Example 3.4.1 below that our method can be applied in a case when the known method of Burton [15] cannot be applied. We formulate sufficient conditions for the cases when the nonlinearity of the VIE has sub-linear, linear and super-linear estimates (see Theorems 3.3.1, 3.3.3and 3.3.4 below, respectively ). In a special class of VIEs we show that our conditions are not only sufficient, but also necessary (see Theorem3.3.2).
In Chapter4 we consider a continuous time nonlinear system with delay ( x(t) =˙ g(t, x(t−σ(t))) +u(t), t >0,
y(t) =Cx(t).
Here we propose a state feedback control law of the form u(t) =−Dx(t−τ) +r(t), where Dis a diagonal matrix, τ is a delay in the control mechanism, r is the reference input, to guarantee that the solution of the closed loop system is BIBO stable. We formulate sufficient conditions for the cases when the nonlinearity has sub-linear, linear and super-linear estimates, respectively (see Theorems4.2.1,4.2.3and4.2.6). In the super-linear case we prove a weak form of BIBO stability, which we call local BIBO stability. This new notion is introduced in Definition4.1.1. In practice it is useful if the delay in the control mechanism is relatively large. In our results we give a relation between D,τ and the bound of the nonlinear term which guarantees the BIBO stability of the nonlinear feedback control system. Our method is based on the variation of constant formula and the exponential estimate of the solution of the associated linear homogeneous delay system, and uses our general results on the boundedness of VIEs given in Chapter3. In some proofs we use the technique involving the integral of the absolute value of the fundamental solution of a homogeneous linear delay equation which was introduced in [47,50,51] in the study of stability of perturbed delay equations. The results of Chapter 3 and Chapter 4 on BIBO stability was published in [9]. In Section4.3we also give sufficient conditions in terms of the Lipschitz constant of the nonlinearity and the integral of the absolute value of the fundamental solution in the case of an autonomous nonlinear feedback control system (see Theorem4.3.1). The theoretical results are illustrated by several numerical examples (see Section4.4). Our numerical investigations show that our sufficient conditions give a good upper bound on the absolute value of the solutions. We also observe that in the case when the associated linear homogeneous equation has oscillatory solutions (the case when the delay is ”large” in the control mechanism) the solutions of the nonlinear feedback control system are also oscillatory.
Chapter1. Introduction 4 In Chapter 5 we give a sufficient condition to guarantee the boundedness of solutions of a general class of VDEs of the form
x(n+ 1) =
n
X
j=0
f(n, j, x(j−σ(j))) +h(n), n≥0.
Our main result of this chapter is formulated in Theorem 5.3.1. The results of this chapter are the discrete versions of those of Chapter3for VIEs. We also present special sufficient conditions for the cases of sub-linear, linear and super-linear nonlinearities (see Theorems 5.4.1, 5.4.5 and 5.4.7). We also discuss several special scalar VDEs in this chapter, and at some cases we obtain necessary and sufficient conditions (see Theorems5.4.3and 5.6.1and Corollaries 5.5.3-5.5.8). In Section 5.7we give several illustrative examples for the applicability of our results. The results given in this chapter are analogous with those of Chapter3, and they are extension of our earlier work [49] where an ordinary VDEs (without time delay) was investigated.
In Chapter6 we study a discrete time nonlinear system with time delay x(n+ 1)−x(n) =g(n, x(n−σ(n))) +u(n), n≥0
y(n) =Cx(n), n≥0.
We give sufficient conditions for the BIBO stability of the closed loop system for the sub-linear and linear cases (see Theorems6.2.1and 6.2.2). The results are the discrete counterparts of the results of Chapter4 given for continuous time controlled systems.
In Chapter 7 we present some applications of our results. First we discuss a model equation describing the El Ni˜no Southern Oscillation phenomenon observed in the equatorial Pacific. It is observed that there is a regular discrepancy of the see-level temperature of the ocean from the average temperature, and this anomaly shows a regular oscillatory behavior with a relatively large (about 1.5oC) amplitude. The mathematical model describing this temperature anomaly is investigated e.g. in [12,109]. In Section7.1we show that our method of Chapter3can be applied for this model, and we can give a sufficient condition which guarantees the boundedness of the solutions, moreover, for some cases the estimates of the upper bound of the solution obtained by our method gives a good approximation of the amplitudes of the oscillation in the solution of the nonlinear delay equation. Moreover, the oscillatory behavior of the associated linear homogeneous equation explains the appearance of the oscillatory solutions of the nonlinear equation for ”large”
delays. Numerical examples illustrate the theoretical results.
In Section 7.3 a class of population model equations is considered. This general model was introduced by Cooke and Kaplan [25], and later studied by many authors [17, 18, 46, 100].
In this section we show that we can find a control term of the form u(t) = ax(t) +r such that the corresponding closed-loop system has a unique, predefined equilibrium c, moreover, the solutions are bounded in a neighborhood of this equilibrium. This control term can be interpreted as a harvesting in the population proportional to the size of the population and a continuous
Chapter1. Introduction 5 immigration to the population with a constant rate. We show that the main method of Chapter 3 can be used to give a condition which implies the boundedness of the solution with initial population size close to the equilibrium for large classes of nonlinearities in the model equation.
In Chapter 8 we summarize the new results. Also the list of publications and conference lectures of E. Awwad related to the topic of this Thesis is given. Some technical or long proofs are moved to Appendix A.