Algebra of sets of dynamic processes with differential realizations in a Hilbert space

Solvability of differential realization of minimum dynamic order for a family of nonlinear input-output processes in Hilbert space

“…. , A 0 ∈ L(X, X) and B ∈ L(Y, X) in the autonomous differential realization (2) are unique if and only if the family N of dynamic processes satisfies the condition D n = U n . Remark 2.…”

“…Lemma 3. The processes (1 ) are solutions of some equation (2) if and only if there exist operators A n−1 , . .…”

“…Remark 2. Each of assumptions (i) and (ii) in Theorem 2 is equivalent to the existence of a (not necessarily time-independent) realization (2), and each of them can reasonably be used as an analytical basis depending on the context of the problem in hand. In this connection, the "pure" autonomy (without the boundedness of the operators A n−1 , .…”

“…λ∈ L 2 (T + , ν, R), 2 Considerations are based on the following two assertions: • L2(T+, μ, R), we obtain the inequality T + χS(τ )ϕ 2 (τ ) μ(dτ ) < ∞; then…”

“…The study carried out in the present paper can be viewed as a continuation of the analysis of solvability of the problem on a nonlinear differential realization of complex behavioristic [1, p. 16] systems with state equations in an infinite-dimensional state space, which was initiated in the papers [2][3][4] dealing with the qualitative analysis of problems concerning the existence of realizations for time-varying models. In the present paper, we consider similar problems for autonomous equations with a nonlinear positional control.…”