2 Notation and Terminology

$\bf R^{n}_{}$

is the space of n-dimensional vectors x = (x₁,..., x_n)';

C[-$\tau$, 0]

is the space of n-dimensional continuous on [- $\tau$, 0] functions q( ^. ) with the norm | q( ^. )|_C = $$\max_{-\tau\le s\le 0}^{}$$| q(s)|;

Q[- $\tau$, 0]

is the space of n-dimensional functions q( ^. ) continuous everywhere on [- $\tau$, 0] except, perhaps, a finite set of points of discontinuity of the first kind (at which q( ^. ) is continuous on the right), with the norm | q( ^. )|_Q = $$\sup_{-\tau\le s\le 0}^{}$$| q(s)|;

Q[- $\tau$, 0)

is the restriction of the space Q[- $\tau$, 0] on the half-interval [- $\tau$, 0), i.e. Q[- $\tau$, 0) consists of n-dimensional functions y(s), - $\tau$$\le$s < 0, with the properties:

1)

y( ^. ) is continuous on the half-interval [- $\tau$, 0) except, perhaps, a finite set of points of discontinuity of the first kind (at which q( ^. ) is continuous on the right);

2)

y( ^. ) is bounded on [- $\tau$, 0);

3)

there exists finite left-side limit at zero $$\lim_{s \to 0-}^{}$$y(s);

(the norm in Q[- $\tau$, 0) is defined as | y( ^. )|_$\tau$ = $$\sup_{-\tau\le s < 0}^{}$$| y(s)|);

H = $\bf R^{n}_{}$×Q[- $\tau$, 0)

is the space of pairs h = {x, y( ^. )} with the norm | h|_H = max { | x|,| y( ^. )|_$\tau$ }.

We denote by prime ' the transposition of vectors or matrices. The scalar product of vectors x, y $\in$ $\bf R^{n}_{}$ is x'y, and the corresponding norm is defined by the formula | x| = $\sqrt{x' x }$.

Remark 2.1. It is necessary to note, the spaces H and Q[- $\tau$, 0] are isometric and the corresponding isometric mapping $\pi$ : Q[- $\tau$, 0]$\to$H transfers functions q( ^. ) $\in$ Q[- $\tau$, 0] into pairs {q(0);q(s),$\tau$$\le$s < 0} $\in$ H. So sometimes we will use the term "function" and for pairs {x, y( ^. )} $\in$ H. We use the presentation of the space Q[- $\tau$, 0] in the form H = $\bf R^{n}_{}$×Q[- $\tau$, 0) because in case of delay differential equations the point q(0) and the points {q(s),$\tau$$\le$s < 0} (for q( ^. ) $\in$ Q[- $\tau$, 0]) play, generally speaking, different roles. So it is convenient to use different notation for the point q(0) and points {q(s),$\tau$$\le$s < 0} of the function q( ^. ) $\in$ Q[- $\tau$, 0]; for example: x = q(0) and y( ^. ) = {q(s),$\tau$$\le$s < 0}, and moreover to consider x and y( ^. ) as independent. $\Box$

The term basic functions is used for the subset of C[- $\tau$, 0] which consists of elementary functions, i.e. polynomials, trigonometrical functions, logarithms, exponential.

ARE

- Algebraic Riccati Equation;

DDE

- Delay Differential Equations, also called systems with delays, hereditary systems, functional differential equations;

ES-LQR

- Explicit Solution of Linear Quadratic Regulator (problem);

GRE

- Generalized Riccati Equations;

LTI

- Linear Time Invariant (system);

MIMO

- Multi Input - Multi Output;

ODE

- Ordinary Differential Equations.