By A.V. Kim

Beginning with the works of N.N.Krasovskii [81, eighty two, 83], which clari fied the sensible nature of platforms with delays, the sensible method presents a starting place for a whole conception of differential equations with delays. in response to the practical process, assorted elements of time-delay method conception were constructed with virtually an identical completeness because the corresponding box of ODE (ordinary differential equations) the ory. The time period useful differential equations (FDE) is used as a syn onym for platforms with delays 1. The systematic presentation of those re sults and extra references are available in a few first-class books [2, 15, 22, 32, 34, 38, forty-one, forty five, 50, fifty two, seventy seven, seventy eight, eighty one, ninety three, 102, 128]. during this monograph we current easy proof of i-smooth calculus ~ a brand new differential calculus of nonlinear functionals, in keeping with the idea of the invariant spinoff, and a few of its purposes to the qualitative idea of useful differential equations. usage of the hot calculus is the most contrast of this booklet from different books dedicated to FDE thought. different distinguishing beneficial properties of the amount are the subsequent: - the relevant idea that we use is the separation of finite dimensional and limitless dimensional parts within the constructions of FDE and functionals; - we use the conditional illustration of practical differential equa tions, that is handy for program of tools and buildings of i~smooth calculus to FDE theory.

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**Additional info for Functional Differential Equations: Application of i-smooth calculus**

**Example text**

Similar to the classic finite dimensional calculus, this operation is based on a notion of i-primitive of a functional. These constructions are not used in other chapters of the book so we consider only some basic ideas. )] = R n x Q[-r,O) --+ R. 1. 1) : 1) at point h* E RTL x Q[-r, 0) if ayW[h*] = V[h*] ; 2) on H = R1L x Q[-r,O) if ayW[h] = V[h] for every h E H. 2) o If an i-primitive W of the functional V is known then, obviously, for any C = const a functional W + C also is the i-primitive of V, because the invariant derivative of the constant C is equal to zero.

4) where g(t,X,f-L): (a, b) x R n x (-c,c) -+ Rn; a, band c are some positive constants. 1. S1},ppose that: . ) ag(t,X,f-L) 1) mappmgs 9 (t, x, f-L, ax are continuous on A == (a, b) x RT! x (-c, c), . ag(t,X,f-L) ag(t,X,f-L) b d d d . fJerentiable on (a, b) x (0,(3) x U x (-c,c). Proof. 1) Continuous differentiability of x(t; to, x O, f-L) with respect to t and xO follows from the classic theorems on differentiability of ODE solution. 2) Let us fix arbitrary initial point (to, xU) E (0, (3) x U and show the differentiability of x(t; to, x O, f-L) == ¢( t, f-L) with respect to f-L.

1. 1. 1) considered in phase space H. 1. ) disappears, then the results turn into the corresponding results of ODE theory (with exactness in notation). It allows us to carry out a methodological analysis of results and methods of FDE theory. 4. ), because they play different roles in the 44 CHAPTER 4 dynamic of time-delay systems. One can use a fruitful analogy between a train and the presentation {x(t);x(t + S), -T ~ S < O} = {x,Y(')h of the element of trajectory of the system. ) presents the cars that follow the locomotive.