Modal controllability for systems of neutral type in classes of differential-difference controllers

Finite spectrum assignment problem for bilinear systems with several delays

“…. , C * A n−1 B are linearly independent, is equivalent to the condition (16), where the matrices A 0j , j = 0, r 0 , in (14) are defined by equalities (6), (7). It follows from Lemma 2 below.…”

“…Taking into account that det G = 1 = 0, we see that the system (28) is resolvable with respect to u 0 (over K) for any pregiven γ i ∈ K, i = 1, n, iff condition (16) Remark 1. Let us show that Theorem 1 is a generalization of [29,Theorem 2].…”

“…Then the system (31) is arbitrary n-point spectrum assignable by constant control iff conditions (16), (17) hold.…”

“…These are works on stabilization of an object with delay [1][2][3][4][5], stabilization of a group of objects by a single controller [6], assignment of a given finite spectrum [7][8][9], spectral reducibility [10][11][12], i.e., reduction of systems to a finite (but not given) spectrum, modal controllability [13][14][15][16][17]. At present, for retarded and neutral type systems, as well as for completely regular differential-algebraic systems with several delays, spectral criteria of modal controllability have been obtained [13,15,16] that coincide in form with the criterion of complete controllability (see, for example, [18]). They are also solvability conditions for the problem of assigning a finite spectrum [7,9].…”

“…In the case of practical realization, integrals containing distributed delay are replaced by finite sums, which, even when using quadrature formulas of high accuracy, can lead to undesirable effects [4,19,20]. As an alternative to the situation described, some works [13,14,16] (see also the introduction in [13]) offer sufficient conditions for modal controllability in the class of differential-difference controllers. Such works are based on the transformation of the object under study to a certain special form, which makes it easy to obtain the required controller, and formulate solvability conditions in terms of a controllability matrix or its analogue.…”