Analysis Under Nested Criteria

Abstract

The small-gain theorem provides a sufficient condition for the stability of two feedback interconnected dynamical systems for which input-to-state (or input–output) gains can be defined. Roughly speaking, to apply this theorem, the resulting gains’ composition is required to be continuous, increasing, and strictly smaller than the identity function.

Appendix of Chap. 3

3.1.1 Technical Lemma

Lemma 3.19

Let $k\ge 0$ and $p>0$ be constant integers. Given a function $h\in \mathscr {C}^k(\mathbb R^n,\mathbb R^p)$, and compact set $\mathbf {K}\subset \mathbb R^n$ such that, for every $y\in \mathbf {K}$, $h(y)\ne 0$. Then, there exists $\tilde{h}\in \mathscr {C}^k(\mathbb R^n,\mathbb R^p)$ such that $\mathtt {supp}(\tilde{h})\supset \mathbf {K}$ and, for every $y\in \mathbf {K}$, $\tilde{h}(y)=h(y)$.

The proof of Lemma 3.19 is based on [24, p. 370] and can be found in [32].

3.1.2 The Divergence Theorem for Level Sets of a Lyapunov Function

The following definition is recalled from [23]:

Definition 3.20

(Gamma function) The function

$$\begin{aligned} \begin{array}{rrcl} \varGamma :&{}\mathbb {R}_{\ge 0}&{}\rightarrow &{}\mathbb R\\ &{}s&{}\mapsto &{}\displaystyle \int \limits _0^\infty t^{s-1}e^{-t}\,dt \end{array} \end{aligned}$$

is called gamma function.

The next definition is recalled from [21].

Definition 3.21

(Hausdorff measure) Let $\mathbf {E}\subset \mathbb R^n$, the diameter of the set $\mathbf {E}$ is the function

$$\begin{aligned} \begin{array}{rrcl} {{\mathrm{\mathtt {diam}}}}:&{}\mathbf {E}\times \mathbf {E}&{}\rightarrow &{}\mathbb {R}_{\ge 0}\\ &{}(x,y)&{}\mapsto &{}\sup \{|x-y|\}. \end{array} \end{aligned}$$

Let $0\le n<\infty$ and define, for $0<\delta \le \infty$, the value

$$\begin{aligned} \mathscr {H}^n_\delta (\mathbf {E})=\inf \left\{ \sum _{j\in \mathbb {N}}{{\mathrm{\mathtt {diam}}}}(\mathbf {E}_j)^n:\mathbf {E}\subset \bigcup _{j\in \mathbb {N}}\mathbf {E}_j,{{\mathrm{\mathtt {diam}}}}(\mathbf {E}_j)<\delta ,\mathbf {E}_j\subset \mathbb R^n\right\} . \end{aligned}$$

The n-dimensional unnormalized Hausdorff measure of $\mathbf {E}$ is the limit

$$\begin{aligned} \widetilde{\mathscr {H}}^n(\mathbf {E})=\lim _{\delta \rightarrow 0}\mathscr {H}^n_\delta (\mathbf {E})=\sup _{\delta >0}\mathscr {H}^n_\delta (\mathbf {E}). \end{aligned}$$

The n-dimensional Hausdorff measure of $\mathbf {E}$ is given by

$$\begin{aligned} \mathscr {H}^n(\mathbf {E})=\dfrac{\alpha (s)}{2^n}\widetilde{\mathscr {H}}^n(\mathbf {E}), \end{aligned}$$

where

$$\begin{aligned} \alpha (n)=\dfrac{\varGamma \left( \tfrac{n}{2}\right) }{\varGamma \left( \tfrac{n}{2}+1\right) }. \end{aligned}$$

The relation between Hausdorff and Lebesgue measures is explained in the following remark which is based on [10, Sect. 2.2] and [20]:

Remark 3.22

Note that the n-dimensional Lebesgue measure of a set $\mathbf {E}\subset \mathbb R^n$ is the n-fold product of unidimensional Lebesgue measures (cf. Definitions A.5 and A.7) while the Hausdorff measure is computed in terms of arbitrarily coverings of $\mathbf {E}$ with small diameters. Moreover, the Lebesgue measure in $\mathbb R^n$ is equivalent to the n-dimensional Hausdorff measure, i.e., $\mu =\mathscr {H}^n$. Also, if $\mathscr {H}^n(\mathbf {E})<\infty$, then $\mathscr {H}^{n-1}(\mathbf {E})=\infty$ and $\mathscr {H}^{n+1}(\mathbf {E})=0$.

The next concept, recalled from [21, p. 50], is a measure-theoretical notion of boundaries of a set.

Definition 3.23

(Essential boundaries) For a set $\mathbf {E}\subset \mathbb R^n$,

• The essential exterior is the set

  $$\begin{aligned} {{\mathrm{\mathtt {ext}}}}_*(\mathbf {E})= \left\{ x\in \mathbb R^n:\lim _{r\rightarrow 0}\dfrac{\mu \left( \mathbf {E}\cap \mathbf {B}_{\le r}(x)\right) }{\mu \left( \mathbf {B}_{\le r}(x)\right) }=0\right\} ; \end{aligned}$$

• The essential interior is the set ${{\mathrm{\mathtt {int}}}}_*(\mathbf {E})={{\mathrm{\mathtt {ext}}}}_*(\mathbb R^n\setminus \mathbf {E})$;

• The essential closure is the set $\mathtt {cl}_*\{\mathbf {E}\}=\mathbb R^n\setminus {{\mathrm{\mathtt {ext}}}}_*(\mathbf {E})$;

• The essential boundary is the set $\partial _*\mathbf {E}=\mathtt {cl}_*\{\mathbf {E}\}\setminus {{\mathrm{\mathtt {int}}}}_*(\mathbf {E})$.

The following properties hold⁸:

$$\begin{aligned} \begin{array}{l} {{\mathrm{\mathtt {int}}}}_*(\mathbf {E})\subset \mathtt {cl}_*\{\mathbf {E}\},{{\mathrm{\mathtt {int}}}}_*(\mathbb R^n\setminus \mathbf {E})={{\mathrm{\mathtt {ext}}}}_*(\mathbf {E}),\\ {\partial _*\mathbf {E}=\mathtt {cl}_*(\mathbf {E})\cap \mathtt {cl}_*\{\mathbb R^n\setminus \mathbf {E}\}=\partial _*(\mathbb R^n\setminus \mathbf {E})=\mathbb R^n\setminus ({{\mathrm{\mathtt {int}}}}_*(\mathbf {E})\cup {{\mathrm{\mathtt {ext}}}}_*(\mathbf {E})).} \end{array} \end{aligned}$$

Definition 3.23 is related to the usual topological concepts as follows:

$$\begin{aligned} {{\mathrm{\mathtt {int}}}}(\mathbf {E})\subset {{\mathrm{\mathtt {int}}}}_*(\mathbf {E}),\quad \mathtt {cl}_*\{\mathbf {E}\}\subset \mathtt {cl}\{\mathbf {E}\},\quad \partial _*\mathbf {E}\subset \partial \mathbf {E}. \end{aligned}$$

Moreover,

$$\begin{aligned} \partial _*\mathbf {E}=\partial \mathbf {E}\Leftrightarrow {{\mathrm{\mathtt {int}}}}(\mathbf {E})={{\mathrm{\mathtt {int}}}}_*(\mathbf {E})\quad \text {and}\quad \mathtt {cl}_*\{\mathbf {E}\}=\mathtt {cl}\{\mathbf {E}\}, \end{aligned}$$

and the inclusion

$$\begin{aligned} {{\mathrm{\mathtt {int}}}}_*(\mathbf {E})\subset \left\{ x\in \mathbb R^n:\lim _{r\rightarrow 0}\dfrac{\mu \left( \mathbf {E}\cap \mathbf {B}_{\le r}(x)\right) }{\mu \left( \mathbf {B}_{\le r}(x)\right) }=1\right\} \end{aligned}$$

becomes an inequality, when $\mathbf {E}$ is measurable.

The following measure-theoretical notion of perimeter of a set is recalled from [21, Definition 4.5.1]:

Definition 3.24

(Perimeter of a set) The perimeter of a set $\mathbf {E}\subset \mathbb R^n$ is the measure

$$\begin{aligned} P(\mathbf {E})=\mathscr {H}^{n-1}(\partial _*\mathbf {E}). \end{aligned}$$

The perimeter is finite if $\mu (\mathbf {E})+P(\mathbf {E})<\infty$.

The notion of a perimeter of a set is an important concept for the next theorem, adapted from [1, Theorem 2.5].

Theorem 3.25

Let $k\ge 0$ be a constant integer, and consider the Lipschitz map $V\in \mathscr {C}^k(\mathbb R^2,\mathbb {R}_{\ge 0})$ with $\mathtt {supp}(V)$ compact. The following statements hold, for almost every $c\in \mathbb {R}_{\ge 0}$:

• 1. $\varOmega _{=c}(V)$ is 1-rectifiable⁹ and $\mathscr {H}^1\left( \varOmega _{=c}(V)\right) <\infty$;

• 2. For $\mathscr {H}^1$-almost every $x\in \varOmega _{=c}(V)$, the map V is differentiable at x;

• 3. Every connected component C of $\varOmega _{=c}(V)$ is either a point or a closed simple curve with a Lipschitz parametrization $p:[a,b]\rightarrow C$ which is injective and satisfies, for almost every $t\in [a,b]$,

  $$\dfrac{dp}{dt}(t)=\tau (p(t)),$$

  where, for every $x\in C$, $\tau (x)$ is the vector tangent to C.

From item 2 and since the level set $\varOmega _{=c}(V)$ is either a point or a simple closed curve of $\mathbb R^2$, $\partial \varOmega _{=c}(V)=\mathtt {cl}\{\varOmega _{=c}(V)\}=\varOmega _{=c}(V)$. Moreover, $\partial _*\varOmega _{=c}(V)\subset \mathtt {cl}\{\varOmega _{=c}(V)\}$. From item 1, the sublevel set $\varOmega _{\le c}(V)$ has finite perimeter. Thus the inequality

$$\begin{aligned} \int \limits _{\varOmega _{= c}(V)}\,d\mathscr {H}^1<\infty \end{aligned}$$

holds. Note that, from Remark 3.22, this integral is defined in the Lebesgue sense in $\mathbb R^1$.

The next definition, based on [20, Definition 1.6] and [21, pp. 127–128], recalls the concept of vector being an outward normal to a set.

Definition 3.26

(Outward normal) For every $x\in \partial _*\mathbf {E}$, denote by $n_\mathbf {E}(x)$ the unit vector of $\mathbb R^n$ such that

$$\begin{aligned} \mathbf {H}_\pm (\mathbf {E},x)=\{y\in \mathbb R^n:\pm n_\mathbf {E}(x)\cdot (y-x)\ge 0\}. \end{aligned}$$

The function $n_\mathbf {E}$ is called outward unit normal of $\mathbf {E}\subset \mathbb R^n$ if, for every $x\in \partial _*\mathbf {E}$,

$$\begin{aligned} \begin{array}{rcl} \displaystyle \lim _{r\rightarrow 0}\dfrac{\mu \left( \mathbf {B}_{\le r}(x)\cap \mathbf {H}_+(\mathbf {E},x)\cap \mathbf {E}\right) }{\mu \left( \mathbf {B}_{\le r}(x)\right) }&{}=&{}0,\\ \\ \displaystyle \lim _{r\rightarrow 0}\dfrac{\mu \left( \mathbf {B}_{\le r}(x)\cap \left( \mathbf {H}_-(\mathbf {E},x)\setminus \mathbf {E}\right) \right) }{\mu \left( \mathbf {B}_{\le r}(x)\right) }&{}=&{}0 \end{array} \end{aligned}$$

(3.36)

hold.

From item 2 of Theorem 3.25, for $\mathscr {H}^1$-almost every $x\in \varOmega _{=c}(V)$, ${{\mathrm{\mathtt {grad}}}}\,V(x)$ exists. Thus, the vector field

$$\begin{aligned} \begin{array}{rrcl} n:&{}\varOmega _{= c}(V)&{}\rightarrow &{}\mathbb R^2\\ &{}x&{}\mapsto &{}\left\{ \begin{array}{rcl} \dfrac{{{\mathrm{\mathtt {grad}}}}\,V(x)}{|{{\mathrm{\mathtt {grad}}}}\,V(x)|},&{}\text {if}&{}{{\mathrm{\mathtt {grad}}}}\,V(x)\ \text {exists,}\\ 0,&{}\text {if}&{}\text {otherwise} \end{array} \right. \end{array} \end{aligned}$$

(3.37)

is $\mathscr {H}^1$-almost everywhere an outward normal to the set $\varOmega _{\le c}(V)$. Since the outward normal to sets of finite perimeter is unique (cf. [11, Theorem 3.4]), the vector n satisfies the limits (3.36).

For a further reading on the sets of finite perimeters and on the construction of outward normals for them, the interested reader is invited to see [21, Chaps. 5 and 6].

The next result shows the relationship between the line integral along a closed curve and the integral on the domain bounded by this curve.

Theorem 3.27

(Generalized divergence theorem)¹⁰ Under the assumptions of Theorem 3.25. Let $k\ge 0$ be a constant integer, and consider the map $f\in \mathscr {C}^k(\mathbb R^2,\mathbb R^2)$. Then, the formula

$$\begin{aligned} \iint \limits _{\varOmega _{\le c}(V)}{{\mathrm{\mathtt {div}}}}\,f(x)\,dx=\oint \limits _{[a,b]}f(p(s))\cdot n(p(s))\,ds \end{aligned}$$

holds, where the integral of the left-hand side (resp. right-hand side) is taken in the Lebesgue (resp. 1-dimensional Hasudorff) measure on $\mathbb R^2$ (resp. $\mathbb R$), and $p:[a,b]\rightarrow \varOmega _{=c}(V)$ is a parametrization of $\varOmega _{=c}(V)$.

Before showing a sketch of the proof of Theorem 3.27, the following lemma, based on [31, p. 106], is needed. For a detailed proof in $\mathbb R^n$, the interested reader may consult [21, Chaps. 1–6].

Lemma 3.28

(Green’s Theorem) Let $C\subset \mathbb R^2$ be a positively oriented, piecewise smooth, simple closed curve with finite length, let $\mathbf {D}_C$ be the region bounded by C, and let $f=(f_1,f_2):\mathbb R^2\rightarrow \mathbb R^2$. If $f_1:\mathbb R^2\rightarrow \mathbb R$ and $f_2:\mathbb R^2\rightarrow \mathbb R$ are defined on an open region containing $\mathbf {D}_C$, and f is differentiable in such a region, then

$$\begin{aligned} \oint \limits _C(f_1(x_1,x_2)\,dx_1 + f_2(x_1,x_2)\,dx_2)=\iint \limits _{\mathbf {D}_C}\left( \dfrac{\partial f_2}{\partial x_1}-\dfrac{\partial f_1}{\partial x_2} \right) \,dx_1dx_2, \end{aligned}$$

(3.38)

where the path of integration along C is counterclockwise.

Sketch of the (of Lemma 3.28 )

This proof is based on [31, p. 108]. Since C is a simple closed curve in the plane, the region $\mathbf {D}_C$ is bounded. The projection of the curve in the x-axis (resp. y-axis) yields an interval [a, b] (resp. [e, f]). Consider the points of $A,B\in C$ (resp. $E,F\in C$) corresponding to the points a and b (resp. e and f) on the x-axis, the curve C can be seen as the union of the curves AEB and AFB. Figure 3.3 illustrates the curve C, and the intervals [a, b], and [e, f].

Let the equation of the curve containing the points AEB (resp. AFB) be given by a piecewise continuous function $\eta _1:[a,b]\rightarrow \mathbb R^2$ (resp. $\eta _2:[a,b]\rightarrow \mathbb R^2$).

Integrating the partial derivative of $f_1$ with respect to $x_2$ in $\mathbf {D}_C$ yields

$$\begin{aligned} \displaystyle \iint \limits _{\mathbf {D}_C}\dfrac{\partial f_1}{\partial x_2}(x_1,x_2)\,dx_1dx_2=\displaystyle \int \limits _a^b\int \limits _{\eta _1(x_1)}^{\eta _2(x_1)}\dfrac{\partial f_1}{\partial x_2}(x_1,x_2)\,dx_2dx_1, \end{aligned}$$

where the equality is due to Fubini’s theorem (cf. [9, Theorem 14.1]). Moreover, since C has finite length, the equality

$$\begin{aligned} \begin{array}{rcl} \displaystyle \iint \limits _{\mathbf {D}_C}\dfrac{\partial f_1}{\partial x_2}(x_1,x_2)\,dx_1dx_2&{}=&{}\displaystyle \int \limits _a^b (f_1(x_1,\eta _2(x_1))-f_1(x_1,\eta _1(x_1)))\,dx_1\\ &{}=&{}-\displaystyle \int \limits _a^b f_1(x_1,\eta _1(x_1))\,dx_1-\displaystyle \int \limits _b^a f_1(x_1,\eta _2(x_1))\,dx_1\\ &{}=&{}-\displaystyle \oint \limits _C f_1(x_1,x_2)\,dx_1. \end{array} \end{aligned}$$

holds.

Analogously, integrating the partial derivative of $f_2$ with respect to $x_1$ in $\mathbf {D}_C$ yields

$$\begin{aligned} \displaystyle \iint \limits _{\mathbf {D}_C}\dfrac{\partial f_2}{\partial x_1}(x_1,x_2)\,dx_1dx_2=\displaystyle \oint \limits _C f_2(x_1,x_2)\,dx_2. \end{aligned}$$

From where the conclusion follows.

Fig. 3.3

Illustration of the curve C

Now, it is possible to present an idea of the proof of Theorem 3.27.

From Theorem 3.25,

• The curve $\varOmega _{=c}(V)$ is piecewise $\mathscr {C}^1$, because it is rectifiable. Moreover, it is also simple and closed;

• Since V is $\mathscr {H}^1$-a.e. differentiable in $\varOmega _{=c}(V)$, the outward normal vector defined by (3.37) is $\mathscr {H}^1$-a.e. non-nil;

• The curve $\varOmega _{=c}(V)$ has finite length, because $\mathscr {H}^1(\varOmega _{=c}(V))<\infty$;

• There exists a injective and Lipschitz continuous parametrization $p:[a,b]\rightarrow \varOmega _{=c}(V)$ that is a.e. differentiable.

Consider the vector field $\tilde{f}=(-f_2,f_1)$ that is perpendicular to $f=(f_1,f_2)$. Since $f=(f_1,f_2)\in \mathscr {C}^1(\mathbb R^2,\mathbb R^2)$, $\tilde{f}\in \mathscr {C}^1(\mathbb R^2,\mathbb R^2)$. Together with the above, from¹¹ Lemma 3.28,

$$\begin{aligned} \displaystyle \oint \limits _{\varOmega _{=c}(V)}(-f_2(x_1,x_2)\,dx_1+f_1(x_1,x_2)\,dx_2)=\displaystyle \oint \limits _{\varOmega _{=c}(V)}(-f_2(x_1,x_2),f_1(x_1,x_2))\cdot (dx_1,dx_2) \end{aligned}$$

Consider a point $\bar{x}=(\bar{x}_1,\bar{x}_2)\in C$ for which there exists $\bar{s}\in [a,b]$ such that $p(\bar{s})=(\bar{x}_1,\bar{x}_2)$ and $p'(\bar{s})$ is defined. The unit tangent vector to C at $\bar{x}$ is given by . The unit normal vector at $\bar{x}$ is given by $N(\bar{s})=n(p(s))=(\sigma (\bar{s}),-\tau (\bar{s}))$. For almost every $s\in [a,b]$,

$$\begin{aligned} \begin{pmatrix}dx_1\\ dx_2 \end{pmatrix}=T(s)\,ds=\begin{pmatrix}\tau (s)\\ \sigma (s)\end{pmatrix}\,ds. \end{aligned}$$

Thus,

$$\begin{aligned} \begin{array}{rcl} \displaystyle \oint \limits _{\varOmega _{=c}(V)}(-f_2(x_1,x_2)\,dx_1+f_1(x_1,x_2)\,dx_2)&{}=&{}\displaystyle \oint \limits _{[a,b]}(-f_2(p(s)),f_1(p(s)))\cdot (\tau (s),\sigma (s))\,ds\\ &{}=&{}\displaystyle \oint \limits _{[a,b]}(f_1(p(s)),f_2(p(s)))\cdot (\sigma (s),-\tau (s))\,ds\\ &{}=&{}\displaystyle \oint \limits _{[a,b]}(f_1(p(s)),f_2(p(s)))\cdot n(p(s))\,ds \end{array} \end{aligned}$$

From (3.38),

$$\begin{aligned} \displaystyle \iint \limits _{\varOmega _{\le c}(V)}\left( \dfrac{\partial f_1}{\partial x_1}(x_1,x_2)+\dfrac{\partial f_2}{\partial x_2}(x_1,x_2)\right) \,dx_1dx_2=\displaystyle \oint \limits _{[a,b]}(f_1(p(s)),f_2(p(s)))\cdot n(p(s))\,ds. \end{aligned}$$

This concludes the sketch of the proof of Theorem 3.27.

3.1.3 Integration Along Solutions of an ODE

Before recalling the main result, the following lemma which is recalled from [22, Lemma A.1] is needed.

Lemma 3.29

(Liouville’s Theorem) Let $k\ge 1$ and $p\ge 1$ be constant integers, the function $\rho \in (\mathscr {C}^k\cap \mathscr {L}^p)(\mathbb R^{n+m},\mathbb {R}_{\ge 0})$. Let also Y(t, y) be a solution of (3.8) starting in $y\in \mathbb R^{n+m}$ and computed at time $t\in \mathbb {R}_{\ge 0}$. For a measurable set $\mathbf {Z}$, let $Y(\cdot ,\mathbf {Z})=\{Y(\cdot ,z):z\in \mathbf {Z}\}$. Then,

$$\begin{aligned} \int \limits _{Y(t,\mathbf {Z})}\rho (y)\,dy-\int \limits _\mathbf {Z}\rho (y)\,dy=\int \limits _0^t\int \limits _{Y(\tau ,\mathbf {Z})}{{\mathrm{\mathtt {div}}}}\,(\rho h)(y)\,dyd\tau . \end{aligned}$$

The main result in this section is recalled from [22, Theorem 1].

Theorem 3.30

(Almost attractivity) Let $k\ge 1$ and $p\ge 1$ be constant integers. Suppose that there exists $\rho \in (\mathscr {C}^k\cap \mathscr {L}^p)(\mathbb R^n,\mathbb {R}_{\ge 0})$ such that,

$$\begin{aligned} \int _{\mathbf {B}_{\ge 1}(0)}\dfrac{(h\rho )(y)}{|y|}\,dy<\infty \end{aligned}$$

and

$$\begin{aligned} y\in \mathbb R^{n+m},\quad {{\mathrm{\mathtt {div}}}}\,(h\rho )(y)>0. \end{aligned}$$

Then, for almost every initial condition $y\in \mathbb R^{n+m}$,

$$\begin{aligned} \limsup _{t\rightarrow \infty }|Y(t,y)|=0. \end{aligned}$$

Moreover, if the origin is stable, then the conclusion remains valid when $\rho$ takes negative values.