Stability of Switched Feedback Time-Varying Dynamic Systems Based on the Properties of the Gap Metric for Operators

and Applied Analysis 3 2. Preliminary Results on Closed and Bounded Sequences of Linear Operators The subsequent result relies on the convergence of sequences of operators to limits so that we give the following simple result. Theorem 2.1. P∗1 / P∗2 ⇒ S P∗1 / S P∗2 ; for all P∗1, P∗2 ∈ L X ∩ clFP . Proof. Pk ∈ S P∗1 ⇔ Pk → P∗1 as k → ∞ in the strong operator topology. Thus, for any ε1 > 0, ∃n1 n1 ε1 ∈ N such that, in the strong operator topology, ‖Pk − P∗1‖ : ‖Pkx − P∗1x‖X sup x∈X, ‖x‖ 1 ‖Pkx − P∗1x‖ < ε1, ∀k > n1 2.1 as x ranges over the unit ball in X. Now, assume that Pk → P∗1 in the uniform operator topology as k → ∞. Thus, for any ε2 > 0, ∃n2 n2 ε2 ∈ N such that ‖Pkx − P∗1x‖X < ε1; for all k > n2. Since P∗1 / P∗2, δ ‖P∗1 − P∗2‖ > 0, then if Pk → P∗2 as k → ∞ in the uniform operator topology, 0 < δ − ε1 ≤ |‖Pk − P∗1‖ − ‖P∗1 − P∗2‖| ‖Pk − P∗2‖ ≤ ‖Pk − P∗2‖ ‖ Pk − P∗1 P∗1 − P∗2 ‖ < ε2 2.2 for ε2 ε2 δ < δ, ε1 ε1 δ, ε2 > δ − ε2, and k > n : max n1, n2 n ε1, ε2, δ . But the choice of ε1 > 0 is arbitrary and then the above constraint fails if 0 < ε2 ≤ δ − ε1 for any given ε1 < δ. Hence, P∗1 P∗2 is what contradicts the assumption Pk → P∗2 / P∗1 as k → ∞ in the uniform operator topology so that there is some infinite subsequence {Pnk} of {Pk} such that {Pnk} ⊂ SP∗1 ⇒ {Pnk} / ∈ S P∗2 . Thus, either S P∗1 ⊃ S P∗2 with improper set inclusion or S P∗1 ∩ S P∗2 ∅. Here, we are considering the sequences as sets what is trivially consistent . By reversing the roles of S P∗1 and S P∗2 one concludes that S P∗1 ⊂ S P∗2 with improper set inclusion or S P∗1 ∩S P∗2 ∅. Then, P∗1 / P∗2 ⇒ S P∗1 / S P∗2 . Note that proof of the above result might be easily readdressed with the uniform operator topology as well with the replacements δ → δ x : ‖P∗1x − P∗2x‖x∈X ≤ δ‖x‖ and ‖Pkx − P∗ix‖x∈X < εi x ≤ εi‖x‖ for any x ∈ X. Note that P∗1 / P∗2 ⇒ S P∗1 / S P∗2 is compatible with the existence of distinct convergent sequences S P∗1 , S P∗2 to either P∗1 or to P∗2 / P∗1 which can contain common operators Pk ∈ L X . The above result is linked with the so-called gap metric 1, 2 , as follows. IfMi i 1, 2 are closed subspaces of X then the directed gap δ M1,M2 fromM1 toM2 is defined by δ M1,M2 : sup { inf ∥x − y∥ : y ∈ M2 ] : x ∈ M1, ‖x‖ 1 } ‖ I − P2 P1‖, 2.3 where Pi i 1, 2 are the corresponding projection operators. Note that the directed gap is not symmetric, in general. The gap metric is defined by δ M1,M2 : max ( δ M1,M2 , δ M2,M1 ) ‖P1 − P2‖ sup x∈X, ‖x‖ 1 ‖P1x − P2x‖, 2.4 4 Abstract and Applied Analysis where the last identity holds see, e.g., 38 . Note that the gap metric has the symmetry property so that it is a well-posed metric. Then, the metric space X, δ , with the gap metric δ, is a complete metric space which is also the Banach space X, ‖ · ‖ , defined for the above norm, which induces the strong topology on X. We can also use the above concept to define distances between closed operators P via the gap metric. Since closed operators P on X of domain D P X are bounded, the gap metric is also useful to quantify the “separation” between linear bounded then continuous operators whose domain is thewhole vector space X of the Banach space X, ‖ · ‖ and which are not necessarily projection operators. If P1 and P2 are closed linear operators on X, then the gap distance between them is δ G P1 , G P2 , where G Pi ⊂ X × X is the graph of Pi; i 1, 2 and we will denote such a gap distance by δ P1, P2 for the sake of simplicity. Note that the linear operator P : D P ⊂ X → X is closed if its graph G P is closed in the direct sum X ⊕X. The following result involves the proofs of some properties of convergent sequences on a Hilbert space X under contractive conditions for the gap metric. Theorem 2.2. Consider a sequence of linear closed operators {Pk}k∈N0 on a Hilbert space X, where N0 is the set of nonnegative integers, such that P0 is also bounded. Then, the following properties hold: i The operators in {Pk}k∈N0 are also bounded if δ Pk 1, Pk 2 < 1/ √ 1 ‖Pk 1‖. Also, the sequence {Pk}k∈N0 converges to a unique bounded operator P on H, which is bounded and unique and δ Pk 1, Pk → 0 as k → ∞, if the following constraints hold for some real sequence {Kk}k∈N0 : ‖Pk 1 − Pk‖ ≤ 1 Kk for Kk ∈ 0, K ⊂ 0, 1 ; ∀k ∈ N0, 2.5 δ Pk 2, Pk 1 < Kk‖Pk 1 − Pk‖ 1 ‖Pk 1‖ Kk‖Pk 1 − Pk‖ ( 1 ‖Pk 1‖ )1/2 , ∀k ∈ N0. 2.6 ii Assume that P0 is invertible with bounded inverse P−1 0 and δ Pk 1, Pk 2 < 1/ √ 1 ‖Pk 1‖. Then, the sequence {P−1 k }k∈N0 exists consisting of bounded operators on X, In addition, such a sequence of inverse operators converges to a unique bounded operator P−1 onH, which is the bounded inverse operator of P onX in Theorem 2.2(i), if 2.5 2.6 hold. iii If the operators in {Pk}k∈N0 are linear and closed and δ Pk 1, Pk 2 < 1/ √ 1 ‖Pk 1‖ then the operators in {Pk}k∈N0 are also densely defined. If, in addition, 2.5 2.6 hold then the densely defined operators of the sequence {Pk}k∈N0 converge to a limit operator P which is also densely defined, that is, their domains D Pk ; k ∈ N0 and D P are dense subsets of X and their images R Pk ; k ∈ N0 and R P are contained in X. Proof. Let δ Pk, Pk 1 be an abbreviated notation for δ G Pk , G Pk 1 where the graph of Pk, is the range of the operator [ I Pk ] defined on D Pk , that is, G Pk R [ I Pk ] { x Pkx ] : x ∈ D Pk } ⊂ X ⊕ X of the operator [ I Pk ] defined on D Pk . Proceed by complete induction Abstract and Applied Analysis 5and Applied Analysis 5 by taking any k ∈ N0 and assuming that {Pj}j ≤k 1 ∈N0 is bounded, x ∈ D Pk 1 such that ‖ x Pk 2x ‖ 1 and take