Controllability and Observability in Time-Variable Linear Systems

Controllability and Controller-Observer Design for a Class of Linear Time-Varying Systems

“…As pointed out in [11], the above theorem does not require for the coefficient a(t) to be n − 2 times differentiable, which is an advantage over the controllability criterion of [3], [21], [19], [20]. However, the opposite is also true: the systems with n − 2 times differentiable coefficient a(t), that do not satisfy condition (2), can be considered by the criterion of [3], [21], [19], [20].…”

“…However, the opposite is also true: the systems with n − 2 times differentiable coefficient a(t), that do not satisfy condition (2), can be considered by the criterion of [3], [21], [19], [20]. Clearly, it is desirable to remove the constraint (2).…”

“…Attempts have been made to obtain simpler criteria for assessing this fundamental system property. When the system coefficients have a certain number of derivatives or are analytic, then algebraic criteria that do not require solving the Lyapunov equation are available [3], [21], [19], [20]. In [11] an algebraic criterion for controllability of (1) is introduced that does not require the differentiability of a(t).…”

Optimal LQ-feedback control for a class of first-order hyperbolic distributed parameter systems

“…In order to prove the exponential stability of this C 0 -semigroup, the complete observability (see [25]) of some related matrix pair is needed.…”

“…This relation is observed thanks to the fact that the solution of a related operator Lyapunov equation can be built from the solution of the corresponding matrix Lyapunov equation: see Section 3.2. First let us define the controllability matrix of a matrix pair (M, N ) (see [25], p. 66):…”