Models of Neutral Differential Systems

Abstract

Consider a linear neutral integrodelay differential equation of the form $$\dot{x}(t)+\sum_{j=1}^{m}b_{j}\dot{x}(t-\sigma_{1})+\beta \int_{0}^{\infty}K_{2}(s)\dot{x}(t-s)ds+a_{0}x(t)+\sum_{j=1}^{n}a_{j}x(t-\tau_{j})+\alpha \int_{0}^{\infty}K_{1}(s)x(t-s)ds=0$$ (5.1.1) in which ẋ(t) denotes the right derivative of x at t. (Throughout this chapter we use an upper dot to denote right derivative and this is convenient in writing neutral differential equations systematically). Asymptotic stability of the trivial solution of (5.1.1) and several of its variants have been considered by many authors. There exists a well developed fundamental theory for neutral delay differential equations (e.g. existence, uniqueness, continuous dependence of solutions on various data; see, for instance, the survey article by Akhmerov et al. [1984]); however, there exist no “easily verifiable” sufficient conditions for the asymptotic stability of the trivial solution of (5.1.1). By the phrase “easily verifiable” we mean a verification which is as easy as in the case of Routh-Hurwitz criteria, the diagonal dominance condition or the positivity of principal minors of a matrix etc. Certain results which are valid for linear autonomous ordinary and delay-differential equations cannot be generalized (or extended) to neutral equations. It has been shown by Gromova and Zverkin [1986] that a linear neutral differential equation can have unbounded solutions even though the associated characteristic equation has only purely imaginary roots (see also Snow [1965], Gromova [1967], Zverkin [1968], Brumley [1970], and Datko [1983]); such a behavior is not possible in the case of ordinary or (non-neutral) delay differential equations. It is known (Theorem 6.1 of Henry [1974]) that if the characteristic equation associated with a linear neutral equation has roots only with negative real parts and if the roots are uniformly bounded away from the imaginary axis, then the asymptotic stability of the trivial solution of the corresponding linear autonomous equation can be asserted. However, verification of the uniform boundedness away from the imaginary axis of all the roots of the characteristic equation is usually difficult. An alternative method for stability investigations is to resort to the technique of Lyapunov-type functional and functions; this will be amply illustrated in this chapter.