PDF - Universiteit Twente

2.2 Discrete-time case
Next, we define what we mean by stability in discrete time.
2.2. Discrete-time case
Definition 2.11 The operator sequence (A n d )n≥0 is bounded if there exists
a constant M ≥ 1 such that
�A n d � ≤ M, for all n ≥ 0.
The operator sequence (A n d )n≥0 is power stable if there exist constants M ≥
1 and r ∈ (0, 1) such that
�A n d � ≤ Mr n , for all n ≥ 0. (2.21)
The operator sequence (A n d )n≥0 is strongly stable if for all x0 ∈ X,
A n d x0 → 0, as n → ∞.
So the difference between strongly stable and exponentially stable is that
in the second case the solutions of equation (2.3) converge to zero exponentially.
We have defined stability as a property of the operator sequence (An d )n≥0.
However, since the sequence is directly linked to the abstract difference
equation (2.3), we will sometime say that the equation (2.3) is bounded,
power stable or strongly stable.
For the power sequence of the adjoint difference operator (A∗n d )n≥0, the
following holds.
Remark 2.12 The following properties hold for the adjoint difference operator:
• (A n d )n≥0 is bounded if and only if (A ∗n
d )n≥0 is bounded,
• (A n d )n≥0 is power stable if and only if (A ∗n
d )n≥0 is power stable.
For strong stability such an equality does not hold. For example, the operator
which applies a left translation of size ∆ on X = L 2 (0, ∞), given by
(Adf)(s) = f(s + ∆), f ∈ L 2 (0, ∞), s ≥ 0 (2.22)
is strongly stable. However its adjoint, given by
(A ∗ �
f(s − ∆) s ≥ ∆,
df)(s) =
0 s < ∆,
is not strongly stable, since �A∗ df� = �f�.
(2.23)
23