Lemma 1 For any constant matrix W ∈ R m × m, W > 0, scalar 0 < h(t) < h, vector function ω : [0, h] → R m such that the integrations concerned are well defined, then.
For any constant symmetric matrix (Min R^{mtimes m}), (M=M^{T}>0), (gamma>0), the vector function (omega: [0,gamma ]rightarrowmathbb{R}^{m}) such that the integrations concerned are well defined biggl( int_{0}^{gamma}omega^{T}(s),ds biggr)^{T}M biggl( int_{0}^{gamma }omega(s),ds biggr) leq gamma int_{0}^{gamma}omega ^{T}(s)Momega(s),ds. [45].
For any symmetric positive definite matrix M ∈ M n × n, scalar μ > 0 and vector function ω : [ 0, μ ] → R n such that the integrations concerned are well defined, the following inequality holds: ( ∫ 0 μ ω ( s ) d s ) T M ( ∫ 0 μ ω ( s ) d s ) ≤ μ ( ∫ 0 μ ω T ( s ) M ω ( s ) d s ).
For any symmetric positive definite matrix M > 0, scalar σ > 0 and vector function ω : [ 0, σ ] → R n such that the integrations concerned are well defined, the following inequality holds: ( ∫ 0 σ ω ( s ) d s ) T M ( ∫ 0 σ ω ( s ) d s ) ≤ σ ( ∫ 0 σ ω T ( s ) M ω ( s ) d s ).
Given any real matrix (M>0) of appropriate dimension and a vector function (omega(cdot):[a,b]rightarrow R^{n}), such that the integrations concerned are well defined, then biggl[ int_{b}^{a}omega(s),dsbiggr]^{T}M biggl[ int_{b}^{a}omega(s),dsbiggr]leq b-a) int_{b}^{a}omega ^{T}(s)Momega(s),dsbiggr]leq b-a
For any constant matrix (Omegainmathbb{R}^{n times n}), (Omega = Omega^{T} > 0), scalar (gamma> 0), vector function (omega:[0, gamma] rightarrowmathbb{R}^{n}), such that the integrations concerned are well defined, then frac{1}{gamma} biggl( int_{0}^{gamma} omega(s),ds biggr)^{T} Omega biggl( int_{0}^{gamma} omega(s),ds biggr) leq int _{0}^{gamma} omega^{T}(s) Omega omega(s),ds.
For any positive-definite matrix M > 0, scalar γ > 0, and vector function ω : [ 0, γ ] → R n such that the integrations concerned are well defined, then the following inequality holds: ( ∫ 0 γ ω ( s ) d s ) T M ( ∫ 0 γ ω ( s ) d s ) ≤ γ ( ∫ 0 γ ω T ( s ) M ω ( s ) d s ).
Let (M >0in R^{ntimes n}), a positive scalar (vartheta>0), vector function (x: [ 0,vartheta ] rightarrow R^{n}) such that the integrations concerned are well defined, and they exist: biggl( int_{0}^{vartheta} x ( s),mathrm{d} s biggr)^{mathrm{T}} M biggl( int_{0}^{vartheta} x ( s),mathrm{d} s biggr) boldsymbol{leq} vartheta biggl( int_{0}^{vartheta} x ( s) Mx ( s),mathrm{d} s biggr).
For any positive definite matrix (Phiin R^{n times n}), scalar (gamma> 0), vector function (w:[0,gamma] to R^{n}) such that the integrations concerned are well defined, then biggl( int_{0}^{gamma} w(s),dsbiggr)^{T}Phi biggl( int_{0}^{gamma} w(s),dsbiggr) le gamma int_{0}^{gamma} w^{T}(s)Phi w(s),ds.
For any constant Hermitian matrix (M inmathbf{C}^{ntimes n}) and (M>0), a vector function (Phi (s):[p,q]rightarrowmathbf{C}^{n}) with scalars (p< q) such that the integrations concerned are well defined, then biggl( int_{p}^{q}Phi(s),ds biggr)^{ast}M biggl( int_{p}^{q}Phi(s),ds biggr) leq q-p) int_{p}^{q}Phi^{ast}(s)MPhi(s),ds.
