Differential Equations with Deviating Argument

equations involving a variable (argument) as well as an unknown function and its derivatives, which are taken, generally speaking, for different values of the variable (as distinguished from ordinary differential equations). Examples are

(1) x’(t) = ax(t - τ)

and

(2) x’(t) = ax(kt)

where the constants a, τ, and k are given; τ = t - (t - τ) in equation (1) and τ = t - kt in equation (2) are deviations of the argument. Such equations appeared at the end of the 18th century. They were analyzed repeatedly both for their own sake and in connection with solving geometric problems. Later they were studied in conjunction with various applications, the first of which was control theory. The construction of a systematic theory of differential equations with deviating argument was begun in the 1950’s. Since the 1960’s this theory has represented a significant branch of mathematical analysis.

Linear, homogenous, and autonomous (that is, with constant coefficients and constant deviations of the argument) differential equations with deviating argument have been studied the most. These include such equations, for example, as (1). Here we have a sufficiently complete system of solutions of the type x = e^p , whereby p is a root of a transcendental, characteristic equation P(p) = 0. Specifically, P(p) is a sum of terms of the type Ap^mc^αp where m ≥ 0 is an integer (for example, for (1) we have P(p) ≡ p – ae^–tp). Generally speaking, this equation has an infinite number of complex roots. Other solutions of our differential equation with deviating argument can be represented as series in the indicated, simplest solutions. Therefore, the basic properties of the set of solutions, particularly their stability, are a consequence of the distribution of the zeros of the function P(p).

The most important and most frequently studied class of differential equations with deviating argument are differential equations with retarded argument, in which the highest derivative of the unknown function for a certain value of the variable is defined by this very function and its lower derivatives evaluated at lesser or equal values of the variable. For example, equation (1) with τ ≥ 0 (≥ is the retardation or delay) and equation (2) with k ≤ 1 and t ≥ 0. If time is the variable, such equations and systems of such equations describe processes with time lag, the speed of which at any moment is defined by their state not only at that very moment (as for ordinary differential equations) but at preceding moments as well. Such a situation occurs particularly in systems with automatic control where there is delay in the control unit. Equations with retarded argument in many respects resemble ordinary differential equations; however, they also differ from them in a number of ways. For example, if the solution of equation (1) is constructed for t ≥ t₀, then as an initial condition, x(t) must be given for t₀– t ≤ t ≤ t₀; the solution may be successively constructed for intervals t₀ ≤ t ≤ t₀ + τ, t₀ + τ ≤ t₀ + 2τ, using at each step the result of computations from the preceding step. Methods of operational calculus may be applied to such equations in linear autonomous cases.