1.3    Notations and Preliminaries

In this section, we introduce notations and some important definitions that will be used frequently in the book.

1.3.1    Notation

The notation will be standard most of the times except where stated otherwise. Moreover, N will denote the set of natural numbers, while Z will denote the set of integers. Similarly,ℜ, ℜⁿ will denote respectively, the real line and the n-dimensional real vector space, t ∈ ℜ will denote the time parameter.

$\mathcal{X}$, M, N,… will denote differentiable-manifolds of dimension n which are locally Euclidean and $TM={\cup }_{x\in M}{T}_{x}M,T^{\star }M={\cup }_{x\in M}{T}_x^{\star }M$ will denote respectively the tangent and cotangent bundles of M with dimensions 2n. Moreover, π and π^⋆ will denote the natural projections T M → M and T ^⋆ M → M respectively.

A C^r(M) vector-field is a mapping f : M → T M such that π ◦ f = I_M (the identity on M), and f has continuously differentiable partial-derivatives of arbitrary order r. The vectorfield is often denoted as $f={\text{Σ}}_{i=1}^n{f}_i\frac{\partial }{\partial x_i}$ or simply as (f₁,…, f_n)^T where f_i, i = 1,…, n are the components of f in the coordinate system x₁,…, x_n. The vector space of all C^∞ vector-fields over M will be denoted by V ^∞(M).

A vector-field f also defines a differential equation (or a dynamic system) $\dot{x}$(t) = f(x), x ∈ M, x(t₀) = x₀. The flow (or integral-curve) of the differential equation φ(t, t₀, x₀), t ∈ ℜ, is the unique solution of the differential equation for any arbitrary initial condition x₀ over an open interval I ⊂ ℜ. The flow of a differential-equation will also be referred as the trajectory of the system and will be denoted by x(t, t₀, x₀) or x(t) when the initial condition is immaterial. We shall also assume throughout this book that the vector-fields are complete, and hence the domain of the flow extends over (−∞, ∞).

The Lie-bracket of two vector-fields $f={\text{Σ}}_{i=1}^n{f}_i\frac{\partial }{\partial x_i},g={\text{Σ}}_{i=1}^n{g}_i\frac{\partial }{\partial x_i}$ is the vector-field [f, g] : V ^∞(M) × V ^∞(M) → V ^∞(M) defined by

$a{d}_{fg}\triangleq \left[f,g\right]={\displaystyle \sum _{j=1}^n\left({\displaystyle \sum _{i=1}^n\frac{\partial g_j}{\partial x_i}{f}_i-\frac{\partial f_j}{\partial x_i}{g}_i}\right)\frac{\partial }{\partial x_j}}.$

Furthermore, an equilibrium-point of the vector-field f or the differential equation defined by it, is a point $\overline{x}$ such that $f\left(\overline{x}\right)=0\text{or}\varphi \left(t,t_0,\overline{x}\right)=\overline{x}\forall t\in ℛ$.

An invariant-set for the system $\dot{x}$(t) = f(x), is any set $\mathcal{A}$ such that, for any $x_0\in \mathcal{A},\Rightarrow \varphi \left(t,t_0,x_0\right)\in \mathcal{A}$for all t ∈ ℜ. A set S ⊂ ℜⁿ is the ω-limit set of a trajectory φ(., t₀, x₀) if for every y ∈ S, ∃ a sequence t_n → ∞ such that φ(t_n, t₀, x₀) → y.

A differential k-form ${\omega }_x^k$, k = 1, 2,…, at a point x ∈ M, is an exterior product from T_xM to ℜ, i.e., ${\omega }_x^k$ : T_xM ×. .. × T_xM (κ copies) →, which is a κc>-linear skew-symmetric function of κ-vectors on T_xM. The space of all smooth κ-forms on M is denoted by Ω^κ(M).

The set M^p×q(M) will denote the ring of p × q matrices over M.

The ℱ-derivative (Fréchet-derivative) of a real-valued function V : ℜⁿ → >ℜ is defined as the function DV ∈ $\mathcal{L}$(ℜⁿ) (the space of linear operators from ℜⁿ to ℜⁿ) such that ${\lim }_{\upsilon \to 0}\frac{1}{\Vert \upsilon \Vert }\left[V\left(x+\upsilon \right)-V\left(x\right)-\langle DV,\upsilon \rangle \right]=0,$, for any v ∈ ℜⁿ.

For a smooth function V : ℜⁿ → ℜ, $V_x=\frac{\partial V}{\partial x}$ is the row-vector of first partial-derivatives of V (.) with respect to (wrt) x. Moreover, the Lie-derivative (or directional-derivative) of the function V with respect to a vector-field X is defined as

$L_{X}V\left(x\right)=V_x\left(x\right)X\left(x\right)=X\left(V\right)\left(x\right)={\displaystyle \sum _{i=1}^n\frac{\partial V}{\partial x_i}\left(x_1,\dots ,x_n\right)X_i\left(x_1,\dots ,x_n\right).}$

$\Vert .\Vert$ : W ⊆ℜⁿ → ℜ will denote the Euclidean-norm on W, while ${\mathcal{L}}_2$([t₀, T],ℜⁿ), ${\mathcal{L}}_2$([t₀, ∞),ℜⁿ), ${\mathcal{L}}_{\infty }$(([t₀,∞), ℜⁿ),${\mathcal{L}}_{\infty }$(([t₀, T],ℜⁿ) will denote the standard Lebesguespaces of vector-valued square-integrable and essentially bounded functions over [t₀, T] and [t₀, ∞) respectively, and where for any v : [t₀, T] → ℜⁿ, ν : [t₀, ∞) → ℜⁿ

$\begin{array}{l}{\Vert \upsilon \left(\left[t_0,T\right]\right)\Vert }_{{ℒ}_2}^2\triangleq {\displaystyle \underset{t_0}{\overset{T}{\int }}{\displaystyle \sum _{i=1}^n\left|v_i{\left(t\right)}^{2}dt,\right|}}\\ {\Vert v\left(\left[t_0,\infty \right]\right)\Vert }_{{ℒ}_2}^2\triangleq \underset{T\to \infty }{\lim }{\displaystyle \underset{t_0}{\overset{T}{\int }}{\displaystyle \sum _{i=1}^n{\left|v_i\left(t\right)\right|}^{2}dt,}}\\ {\Vert v\left(\left[t_0,T\right]\right)\Vert }_{{ℒ}_{\infty }}\triangleq ess\underset{\left[t_0,T\right]}{\sup }\left\{\left|v_i\left(t\right)\right|,i=1,\dots ,n\right\}\\ {\Vert v\left([t_0,\infty )\right)\Vert }_{{ℒ}_{\infty }}\triangleq ess\underset{[t_0,\infty )}{\sup }\left\{\left|v_i\left(t\right)\right|,i=1,\dots ,n\right\}.\end{array}$

Similarly, the spaces ℓ₂([k₀, K], ℜⁿ), ℓ₂([k₀, ∞),ℜ>ⁿ), ℓ_∞([k₀, K],ℜⁿ), ℓ_∞([k₀, ∞),ℜⁿ) will denote the corresponding discrete-time spaces, which are the spaces of square-summable and essentially-bounded sequences with the corresponding norms defined for sequences {v_k} : [k₀K] → ℜⁿ, {ν_k} : [k₀, ∞) → ℜⁿ:

$\begin{array}{l}{\Vert v\left(\left[k_0,K\right]\right)\Vert }_{l_2}^2\triangleq {\displaystyle \sum _{k=K_0}^K{\displaystyle \sum _{i=1}^n{\left|v_{ik}\right|}^2},}\\ {\Vert v\left(\left[k_0,\infty \right]\right)\Vert }_{l_2}^2\triangleq \underset{K\to \infty }{\lim }{\displaystyle \sum _{k=K_0}^K{\displaystyle \sum _{i=1}^n{\left|v_{ik}\right|}^2},}\\ {\Vert v\left([k_0,K)\right)\Vert }_{l_{\infty }}\triangleq ess\underset{\left[k_0,K\right]}{\sup }\left\{\left|v_{ik}\right|,i=1,\dots ,n\right\},\\ {\Vert v\left([k_0,\infty )\right)\Vert }_{l_{\infty }}\triangleq ess\underset{[k_0,\infty )}{\sup }\left\{\left|v_{ik}\right|,i=1,\dots ,n\right\}.\end{array}$

Most of the times we shall denote the above norms by ${\Vert (.)\Vert }_2$, respectively ${\Vert (.)\Vert }_{\infty }$ when there is no chance of confusion.

In addition, the spaces ${\mathcal{L}}_2$(jℜ) and ${\mathcal{L}}_{\infty }$(jℜ) (equivalently (ℓ₂(jω) and ℓ_∞(jω) are the frequency-domain counterparts of ${\mathcal{L}}_2$([t₀, ∞) and ${\mathcal{L}}_{\infty }$(([t₀, ∞),ℜⁿ), (respectively ℓ₂([k₀,∞), ℜⁿ), ℓ_∞([k₀,∞), ℜⁿ)) will be rarely used in the text. However, the subspaces of these spaces which we denote with $\mathcal{H}$, e.g. ${\mathcal{H}}_2$(jℜ) and ${\mathcal{H}}_{\infty }$(jℜ) represent those elements of ${\mathcal{L}}_2$(jℜ) and ${\mathcal{L}}_{\infty }$(jℜ) respectively that are analytic on the open right-half complex plane, i.e., on Re(s) > 0. These will be used to represent symbolically asymptotically-stable input-output maps. Indeed, these spaces, also called “Hardy-spaces” (after the name of the mathematician who discovered them), are the origin of the name ${\mathcal{H}}_{\infty }$-control. Moreover, the discrete-time spaces are also equivalently represented by ${\mathcal{H}}_2$(jω) and ${\mathcal{H}}_{\infty }$(jω).

For a matrix $A\in {\Re }^{n\times n}$, λ(A) will denote a spectral (or eigen)-value and $\sigma \left(A\right)={\text{λ}}^{\frac{1}{2}}\left(A^{T}A\right)$ the singular-value of A. A ≥ B (A > B) for an n × n matrix B, implies that A − B is positive-semidefinite (positive-definite respectively).

Lastly, C₊, C₋ and C^r, r = 0, 1,…, ∞ will denote respectively the open right-half, left-half complex planes and the set of r-times continuously differentiable functions.

1.3.2    Stability Concepts

In this subsection, we define some of the various notions of stability that we shall be using in the book. The proofs of all the results can be found in [157, 234, 268] from which the material in this subsection is based on. For this purpose, we consider a time-invariant (or autonomous) nonlinear state-space system defined on a manifold $\mathcal{X}\subseteq {\Re }^n$ in local coordinates (x₁,…, x_n):

$\dot{x}=f\left(x\right);x\left(t_0\right)=x_0,$

(1.39)

or

$x\left(k+1\right)=f\left(x\left(k\right)\right);x\left(k_0\right)=x^0,$

(1.40)

where $x\in \mathcal{X}\subset {\Re }^n$ is the state vector and $f:\mathcal{X}\to V^{\infty }\mathcal{X}$ is a smooth vector-field (equivalently $f:\mathcal{X}\to \mathcal{X}$ is a smooth map) such that the system is well defined, i.e., satisfies the existence and uniqueness theorem for ordinary differential-equations (ODE) (equivalently difference-equations (DE)) [157]. Further, we assume without any loss of generality (wlog) that the system has a unique equilibrium-point at x = 0. Then we have the following definitions.

Definition 1.3.1 The equilibrium-point x = 0 of (1.39) is said to be

•  stable, if for each ϵ > 0, there exists δ = δ(ϵ) > 0 such that

$\Vert x\left(t_0\right)\Vert <k\Rightarrow \Vert x\left(t\right)\Vert <ϵ\forall t\ge t_0;$

•  asymptotically-stable, if it is stable and

$\Vert x\left(t_0\right)\Vert <\delta \Rightarrow \underset{t\to \infty }{\lim }x\left(t\right)=0;$

•  exponentially-stable if there exist constants κ > 0, γ > 0, such that

$\Vert x\left(t\right)\Vert \le k{e}^{\text{-γ}\left(t-t_0\right)}\Vert x\left(t_0\right)\Vert ;$

•  unstable, if it is not stable.

Equivalently, the equilibrium-point x = 0 of (1.40) is said to be

•  stable, if for each ϵ^′ > 0, there exists δ^′ = δ^′ (^ϵ′ ) > 0 such that

$\Vert x\left(k_0\right)\Vert <\delta \text{'}\Rightarrow \Vert x\left(k\right)\Vert <\in \text{'}\forall k\ge k_0;$

•  asymptotically-stable, if it is stable and

$\Vert x\left(k_0\right)\Vert <\delta \text{'}\Rightarrow \underset{t\to \infty }{\lim }x\left(k\right)=0;$

•  exponentially-stable if there exist constants κ^′ > 0, 0 < γ^′ < 1, such that

$\Vert x\left(k\right)\Vert \le k\text{'}{\text{γ'}}^{\left(k-k_0\right)}\Vert x\left(k_0\right)\Vert ;$

•  unstable, if it is not stable.

Definition 1.3.2 A continuous function α : [0, a) ⊂ ℜ₊ → ℜ is said to be of class $\mathcal{K}$ if it is strictly increasing and α(0) = 0. It is said to be of class $\mathcal{K}$_∞ if a = ∞ and α(r) → ∞ as r → ∞.

Definition 1.3.3 A function V : [0, a) × D ⊂ $\mathcal{X}\to ℛ$ is locally positive-definite if (i) it is continuous, (ii) $V\left(t,0\right)=0\mathrm{\forall }t\ge 0,$ and (iii) there exists a constant μ > 0 and a function ψ of class κ such that

$\psi \left[\Vert x\Vert \right]\le V\left(t,x\right),\mathrm{\forall }t\le 0,\mathrm{\forall }x\in B_{\mu }$

where ${ℬ}_u=\left\{x\in {ℛ}^n:\Vert x\Vert \le \mu \right\}.$.V (., .) is positive definite if the above inequality holds for all x ∈ℜⁿ. The function V is negative-definite if −V is positive-definite. Further, if V is independent of t, then V is positive-definite (semidefinite) if V > 0(≥ 0) ∀x ≠ 0 and V (0) = 0.

Theorem 1.3.1 (Lyapunov-stability I). Let x = 0 be an equilibrium-point for (1.39). Suppose there exists a C¹-function $V:D\subset \mathcal{X}\to ℛ,0\in D,V\left(0\right)=0$ such that

$V\left(x\right)>0\mathrm{\forall }x\ne 0,$

(1.41)

$\dot{V}\left(x\right)>0\mathrm{\forall }x\in D.$

(1.42)

Then, the equilibrium-point x = 0 is locally stable. Furthermore, if

$\dot{V}\left(x\right)<0\mathrm{\forall }x\in D\left\{0\right\},$

then x = 0 is locally asymptotically-stable.

Equivalently, if V is such that

$V\left(x_k\right)>0\mathrm{\forall }{x}_k\ne 0,$

(1.43)

$V\left(x_{k+1}\right)-V\left(x_k\right)\le 0\mathrm{\forall }{x}_k\in D.$

(1.44)

Then, the equilibrium-point x = 0 of (1.40) is locally stable. Furthermore, if

$V\left(x_{k+1}\right)-V\left(x_k\right)\le 0\mathrm{\forall }{x}_k\in D\left\{0\right\},$

then x = 0 is locally asymptotically-stable.

Remark 1.3.1 The function V in Theorem 1.3.1 above is called a Lyapunov-function.

Theorem 1.3.2 (Barbashin-Krasovskii). Let $\mathcal{X}={\Re }^n$, x = 0 be an equilibrium-point for (1.39). Suppose there exists a C¹ function V : ℜⁿ → ℜ, V (0) = 0, such that

$V\left(x\right)>0\mathrm{\forall }x\ne 0,$

(1.45)

$\Vert x\Vert \to \infty \Rightarrow V\left(x\right)\to \infty ,$

(1.46)

$\dot{V}\left(x\right)<0\mathrm{\forall }x\ne 0.$

(1.47)

Then, x = 0 is globally asymptotically-stable.

Equivalently, if V is such that

$V\left(x_k\right)>0\mathrm{\forall }{x}_k\ne 0,$

(1.48)

$\Vert x_k\Vert \to \infty \Rightarrow V\left(x_k\right)\to \infty ,$

(1.49)

$V\left(x_{k+1}\right)-V\left(x_k\right)<0\mathrm{\forall }{x}_k\ne 0.$

(1.50)

Then, the equilibrium-point x = 0 of (1.40) is globally asymptotically-stable.

Remark 1.3.2 The function V in Theorem 1.3.2 is called a radially unbounded Lyapunov-function.

Theorem 1.3.3 (LaSalle’s Invariance-Principle). Let Ω ⊂ $\mathcal{X}$ be compact and invariant with respect to the solutions of (1.39). Suppose there exists a C¹-function V : Ω → ℜ, such that $\dot{V}\left(x\right)\le 0\mathrm{\forall }x\in \text{Ω}$ and let $\mathcal{O}=\left\{x\in \text{Ω}|\dot{V}\left(x\right)=0\right\}$. Suppose Γ is the largest invariant set in $\mathcal{O}$, then every solution of (1.39) starting in Ω approaches Γ as t → ∞.

Equivalently, suppose Ω (as defined above) is invariant with respect to the solutions of (1.40) and V (as defined above) is such that $V\left(x_{k+1}\right)-V\left(x_k\right)\le 0\mathrm{\forall }{x}_k\in \text{Ω}$. Let $\mathcal{O}\text{'}=\left\{x_k\in \text{Ω}|V\left(x_{k+1}\right)-V\left(x_k\right)=0\right\}$ and suppose Γ^′ is the largest invariant set in O^′ , then every solution of (1.40) starting in Ω approaches Γ^′ as k → ∞.

The following corollaries are consequences of the above theorem, and are often quoted as the invariance-principle.

Corollary 1.3.1 Let x = 0 be an equilibrium-point of (1.39). Suppose there exists a C¹ function $V:D\subseteq \mathcal{X}\to \Re ,0\in D$, such that $\dot{V}\le 0\mathrm{\forall }x\in D$. Let $\mathcal{O}=\left\{x\in D|\dot{V}\left(x\right)=0\right\}$, and suppose that $\mathcal{O}$ contains no nontrivial trajectories of (1.39). Then, x = 0 is asymptotically-stable.

Equivalently, let x = 0 be an equilibrium-point of (1.40) and suppose V (as defined above) is such that $V\left(x_{k+1}\right)-V\left(x_k\right)\le 0\mathrm{\forall }{x}_k\in D$. Let $\mathcal{O}\text{'}=\left\{x_k\in D|V\left(x_{k+1}\right)-V\left(x_k\right)=0\right\}$, and suppose that $\mathcal{O}\text{'}$ contains no nontrivial trajectories of (1.40). Then, x = 0 is asymptotically-stable.

Corollary 1.3.2 Let $\mathcal{X}={\Re }^n$ x = 0 be an equilibrium-point of (1.39). Suppose there exists a C¹ radially-unbounded positive-definite function V : ℜⁿ → ℜ, such that $\dot{V}$ ≤ 0 ∀x ∈ ℜ ⁿ. Let $\mathcal{O}=\left\{x\in D|\dot{V}\left(x\right)=0\right\}$, and suppose that O contains no nontrivial trajectories of (1.39). Then x = 0 is globally asymptotically-stable.

Equivalently, let $\mathcal{X}={ℛ}^n$, x = 0 be an equilibrium-point of (1.40), and suppose V as defined above is such that $V\left(x_{k+1}\right)-V\left(x_k\right)\le 0\mathrm{\forall }{x}_k\in \mathcal{X}$.Let$\mathcal{O}\text{'}=\left\{x\in \mathcal{X}|V\left(x_{k+1}\right)-V\left(x\right)=0\right\}$, and suppose that $\mathcal{O}\text{'}$ contains no nontrivial trajectories of (1.40). Then x = 0 is globally asymptotically-stable.

We now consider time-varying (or nonautonomous) systems and summarize the equivalent stability notions that we have discussed above for this class of systems. It is not suprising in fact to note that the above concepts for nonautonomous systems are more involved, intricate and diverse. For instance, the δ in Definition 1.3.1 will in general be dependent on t₀ too in this case, and there is in general no invariance-principle for this class of systems.

However, there is something close to it which we shall state in the proceeding.

We consider a nonautonomous system defined on the state-space manifold $\tilde{\mathcal{X}}\subseteq ℛ\times \mathcal{X}:$

$\dot{x}=f\left(x,t\right),x\left(t_0\right)=x_0$

(1.51)

or

$x\left(k+1\right)=f\left(x\left(k\right),k\right),x\left(k_0\right)=x^0$

(1.52)

where $x\in \mathcal{X},\tilde{\mathcal{X}}=\mathcal{X}\times \Re ,f:\tilde{\mathcal{X}}\to V^{\infty }\left(\tilde{\mathcal{X}}\right)$ is C¹ with respect to t (equivalently $f:\mathbf{\text{Z}}\times \mathcal{X}\to \mathcal{X}$ is C^r with respect to x). Moreover, we shall assume with no loss of generality that x = 0 is the unique equilibrium-point of the system such that f(0, t) = 0 ∀t ≥ t₀ (equivalently f(0, k) = 0 ∀k ≥ k₀).

Definition 1.3.4 A continuous function β : J ⊂ ℜ₊ × ℜ₊ → ℜ₊ is said to be of class $\mathcal{K}\mathcal{L}$ if β(r, .) ∈ class $\mathcal{K}$, β(r, s) is decreasing with respect to s, and β(r, s) → 0 as s → ∞.

Definition 1.3.5 The origin x = 0 of (1.51) is

•  stable, if for each ϵ > 0 and any t₀ ≥ 0 there exists a δ = δ(ϵ, t₀) > 0 such that

$\Vert x\left(t_0\right)\Vert <\delta \Rightarrow \Vert x\left(t\right)\Vert <\in \mathrm{\forall }t\ge t_0;$

•  uniformly-stable, if there exists a class $\mathcal{K}$ function α(.) and 0 < c ∈ ℜ₊, such that

$\Vert x\left(t_0\right)\Vert <c\Rightarrow \Vert x\left(t\right)\Vert \le \alpha \left(\Vert x\left(t_0\right)\Vert \right)t\ge t_0;$

•  uniformly asymptotically-stable, if there exists a class $\mathcal{K}\mathcal{L}$ function β(., .) and c > 0 such that

$\Vert x\left(t_0\right)\Vert <c\Rightarrow \Vert x\left(t\right)\Vert \le \beta \left(\Vert x\left(t_0\right)\Vert ,t-t_0\right)\mathrm{\forall }t\ge t_0;$

•  globally uniformly asymptotically-stable, if it is uniformly asymptotically stable for all x(t₀);

•  exponentially stable if there exists a $\mathcal{K}\mathcal{L}$ function β(r, s) = κre^−γs, κ > 0, γ > 0, such that

$\Vert x\left(t_0\right)\Vert <c\Rightarrow \Vert x\left(t\right)\Vert \le k\Vert x\left(t_0\right)\Vert e^{-\text{γ}\left(t-t_0\right)}\mathrm{\forall }t\ge t_0;$

Equivalently, the origin x = 0 of (1.52) is

•  stable, if for each ϵ^′ > 0 and any k₀ ≥ 0 there exists a δ^′ = δ^′ (ϵ′, k₀) > 0 such that

$\Vert x\left(k_0\right)\Vert <\delta \Rightarrow \Vert x\left(k\right)\Vert <{\in }^{\prime }\mathrm{\forall }k\ge k_0;$

•  uniformly-stable, if there exists a class $\mathcal{K}$ function α^′ (.) and c^′ > 0, such that

$\Vert x\left(k_0\right)\Vert <c^{\prime }\Rightarrow \Vert x\left(k\right)\Vert \le {\alpha }^{\prime }\left(\Vert x\left(k_0\right)\Vert \right)\mathrm{\forall }k\ge k_0;$

•  uniformly asymptotically-stable, if there exists a class $\mathcal{K}\mathcal{L}$ function β^′ (., .) and c^′ > 0 such that

$\Vert x\left(k_0\right)\Vert <c^{\prime }\Rightarrow \Vert x\left(k\right)\Vert \le {\beta }^{\prime }\left(\Vert x\left(k_0\right)\Vert ,k-k_0\right)\mathrm{\forall }k\ge k_0;$

•  globally-uniformly asymptotically-stable, if it is uniformly asymptotically stable for all x(k₀);

•  exponentially stable if there exists a class $\mathcal{K}\mathcal{L}$ function β^′ (r, s) = κ^′ rγ^′s, κ^′ > 0, 0 < γ^′ < 1, and c^′ > 0 such that

$\Vert x\left(k_0\right)\Vert <c^{\prime }\Rightarrow \Vert x\left(k\right)\Vert \le {\kappa }^{\prime }\left(\Vert x\left(k_0\right)\Vert {\text{γ}}^{\text{'}\left(k-k_0\right)}\right)\mathrm{\forall }k\ge k_0;$

Theorem 1.3.4 (Lyapunov-stability: II). Let x = 0 be an equilibrium-point of (1.51), and let $\mathcal{B}$(0, r) be the open ball with radius r (centered at x = 0) on $\mathcal{X}$. Suppose there exists a C¹ function (with respect to both its argument) V : $\mathcal{B}$(0, r) × ℜ₊ → ℜ such that:

$\begin{array}{l}{\alpha }_1\left(\Vert x\Vert \right)\le V\left(x,t\right)\le {\alpha }_2\left(\Vert x\Vert \right)\\ \frac{\partial V}{\partial t}+\frac{\partial V}{\partial x}f\left(x,t\right)\le -{\alpha }_3\left(\Vert x\Vert \right)\mathrm{\forall }t\ge t_0,\mathrm{\forall }x\in ℬ\left(0,r\right),\end{array}$

where α₁, α₂, α₃ ∈ class κ defined on [0, r). Then, the equilibrium-point x = 0 is uniformly asymptotically-stable for the system.

If however the above conditions are satisfied for all x ∈ $\mathcal{X}$ (i.e., as r) → ∞ and α₁, α₂, α₃ ∈ class $\mathcal{K}$_∞, then x = 0 is globally-uniformly asymptotically-stable.

The above theorem can also be stated in terms of exponential-stability.

Theorem 1.3.5 Let x = 0 be an equilibrium-point of (1.51), and let $\mathcal{B}$(0, r) be the open ball with radius r on $\mathcal{X}$. Suppose there exists a C¹ function (with respect to both its arguments) V : $\mathcal{B}$(0, r) × ℜ₊→ ℜ and constants c₁, c₂, c₃, c₄ > 0 such that:

$\begin{array}{l}{c}_1{\Vert x\Vert }^2\le V\left(x,t\right)\le c_2\Vert x\Vert \\ \frac{\partial V}{\partial t}+\frac{\partial V}{\partial x}f\left(x,t\right)\le -c_3{\Vert x\Vert }^2\mathrm{\forall }t\ge t_0,\mathrm{\forall }x\in ℬ\left(0,r\right),\\ \Vert \frac{\partial V}{\partial x}\Vert \le c_4\Vert x\Vert .\end{array}$

Then, the equilibrium-point x = 0 is locally exponentially-stable for the system.

If however the above conditions are satisfied for all x ∈ $\mathcal{X}$ (i.e., as r) → ∞, then x = 0 is globally exponentially-stable.

Remark 1.3.3 The above theorem is usually stated as a converse theorem. This converse result can be stated as follows: if x = 0 is a locally exponentially-stable equilibrium-point for the system (1.51), then the function V with the above properties exists.

Finally, the following theorem is the time-varying equivalent of the invariance-principle.

Theorem 1.3.6 Let $\mathcal{B}$(0, r) be the open ball with radius r on $\mathcal{X}$. Suppose there exists a C¹ function (with respect to both its argument) V : $\mathcal{B}$(0, r) × ℜ → ℜ such that:

$\begin{array}{l}{\alpha }_1\left(\Vert x\Vert \right)\le V\left(x,t\right)\le {\alpha }_2\left(\Vert x\Vert \right)\\ \frac{\partial V}{\partial t}+\frac{\partial V}{\partial x}f\left(x,t\right)\le -W\left(x\right)\le 0\mathrm{\forall }t\ge t_0,\mathrm{\forall }x\in ℬ\left(0,r\right),\end{array}$

where α₁, α₂ ∈ class $\mathcal{K}$ defined on [0, r), and W (.) ∈ C¹($\mathcal{B}$(0, r)). Then all solutions of (1.51) with $\Vert x\left(t_{\mathrm{0}}\right)\Vert <{\alpha }_{\mathrm{2}}^{-\mathrm{1}}\left({\alpha }_{\mathrm{1}}\left(r\right)\right)\in$ class $\mathcal{K}$² are bounded and are such that

$W\left(x\left(t\right)\right)\to 0as\text{t}\to \infty \text{.}$

Furthermore, if all of the above assumptions hold for all x ∈ $\mathcal{X}$ and α₁(.) ∈ $\mathcal{K}$_∞, then the above conclusion holds for all x(t₀) ∈ $\mathcal{X}$, or globally.

The discrete-time equivalents of Theorems 1.3.4-1.3.6 can be stated as follows.

Theorem 1.3.7 (Lyapunov-stability II). Let x = 0 be an equilibrium-point of (1.52), and let $\mathcal{B}$(0, r) be the open ball on $\mathcal{X}$. Suppose there exists a C¹ function (with respect to both its argument) V : $\mathcal{B}$(0, r) × Z₊ → ℜ such that:

$\begin{array}{l}{{\alpha }^{\prime }}_1\left(\Vert x\Vert \right)\le V\left(x,k\right)\le {{\alpha }^{\prime }}_2\left(\Vert x\Vert \right)\mathrm{\forall }k\in {\text{Z}}_{+}\\ V\left(x_{k+1},k+1\right)-V\left(x,k\right)\le {{\alpha }^{\prime }}_3\left(\Vert x\Vert \right)\mathrm{\forall }k\in k_0,\mathrm{\forall }x\in ℬ\left(0,r\right),\end{array}$

where ${\alpha }_1^{\text{'}},{\alpha }_2^{\text{'}},{\alpha }_3^{\text{'}}\in$ class $\mathcal{K}$ defined on [0, r), then the equilibrium point x = 0 is uniformly asymptotically-stable for the system.

If however the above conditions are satisfied for all x ∈ $\mathcal{X}$ (i.e., as r → ∞) and${\alpha }_1^{\text{'}},{\alpha }_2^{\text{'}}\in$ class $\mathcal{K}$_∞, then x = 0 is globally-uniformly asymptotically-stable.

Theorem 1.3.8 Let x = 0 be an equilibrium-point of (1.52), and let $\mathcal{B}$(0, r) be the open ball on $\mathcal{X}$. Suppose there exists a C¹ function (with respect to both its arguments) V : $\mathcal{B}$(0, r) × Z₊ → ℜ and constants $c_1^{\text{'}},c_2^{\text{'}},c_3^{\text{'}},c_4^{\text{'}}>\mathrm{0}$ such that:

$\begin{array}{l}{c^{\prime }}_1{\Vert x\Vert }^2\le V\left(x,k\right)\le {c^{\prime }}_2{\Vert x\Vert }^2\\ V\left(x_{k+1},k+1\right)-V\left(x,k\right)\le {c^{\prime }}_3{\Vert x\Vert }^2\mathrm{\forall }k\ge k_0,\mathrm{\forall }x\in ℬ\left(0,r\right),\\ V\left(x_{k+1},k\right)-V\left(x,k\right)\le {c^{\prime }}_4\left(\Vert x\Vert \right)\mathrm{\forall }k\ge k_0\end{array}$

Then the equilibrium-point x = 0 is locally exponentially-stable for the system.

If however the above conditions are satisfied for all x ∈ $\mathcal{X}$ (i.e., as r → ∞), then x = 0 is globally exponentially-stable.

Theorem 1.3.9 Let $\mathcal{B}$(0, r) be the open ball on $\mathcal{X}$. Suppose there exists a C¹ function (with respect to both its argument) V : $\mathcal{B}$(0, r) × Z₊ → ℜ such that:

$\begin{array}{l}{{\alpha }^{\prime }}_1\left(\Vert x\Vert \right)\le V\left(x,k\right)\le {{\alpha }^{\prime }}_2\Vert x\Vert \mathrm{\forall }k\in {\mathbf{\text{Z}}}_{+}\\ V\left(x_{k+1},k\right)-V\left(x,k\right)\le -W\left(x\right)\le 0\mathrm{\forall }k\ge k_0,\mathrm{\forall }x\in ℬ\left(0,r\right),\end{array}$

where ${\alpha }_1^{\text{'}},{\alpha }_2^{\text{'}}\in$ class $\mathcal{K}$ defined on [0, r), and W (.) ∈ C¹($\mathcal{B}$(0, r)). Then all solutions of (1.52) with $\Vert x\left(k_0\right)\Vert <{\alpha }_1^{\text{'}-1}\left({\alpha }_1^{\text{'}}\left(r\right)\right)$ are bounded and are such that

$w\left(x_k\right)\to 0ask\to \infty \text{.}$

Furthermore, if all the above assumptions hold for all x ∈ $\mathcal{X}$ and α₁(.) ∈$\mathcal{K}$_∞, then the above conclusion holds for all x(k₀) ∈ $\mathcal{X}$, or globally.

1.4    Notes and Bibliography

More background material on differentiable manifolds and differential-forms can be found in Abraham and Marsden [1] and Arnold [38]. The introductory material on stability is based on the well-edited texts by Khalil [157], Sastry [234] and Vidyasagar [268].

¹C¹ with respect to both arguments.

²If α_i ∈ class $\mathcal{K}$ defined on [0, r), then ${\alpha }_1^{-1}$ is defined on [0, α_i(r)) and belongs to class $\mathcal{K}$.