ABSTRACT
The stability investigations in this book are based on applications on Bohl-Perron theorem. To apply this theorem for exponential stability of a given equation one must prove that for any bounded on a semi-axes right-hand side all solutions of the equation are also bounded. To show this property of this equation one uses in the chapter the socalled W-method. The method is in the following. An integral substitution into equation x ( t )=∫0 t w ( t , s ) z ( s ) ds , where W(t; s) is a fundamental function of a known exponentially stable equation, transforms the given differential equation to an operator equation z = Tz + f with a bounded linear operator on some functional space on the semi-axes. If this equation has a unique solution on this space then by Bohl-Perron theorem the given functional-differential equation is exponentially stable.
To use successfully this method to obtain explicit exponential stability conditions one has to choose a “model” equation for which its fundamental function W(t; s) is simple and has an explicit form. In this chapter, a model equation is the following ordinary differential equation with constant coefficients x ′′ + ax ′ + bx = 0.
Delay-independent and delay-dependent stability conditions obtained in the chapter are independent of conditions obtained in the previous chapters.
