# Multifractal dimensions for projections of measures

Bilel SELMI<EMAIL_ADDRESS>Faculty of Sciences of Monastir
Department of Mathematics
5000-Monastir
Tunisia

(May 20, 2011)

###### Abstract

In this paper, we study the multifractal Hausdorff and packing dimensions of
Borel probability measures and study their behaviors under orthogonal
projections. In particular, we try through these results to improve the main
result of M. Dai in [13] about the multifractal analysis of a measure of
multifractal exact dimension.

###### keywords:

Multifractal analysis, Dimensions of measures, Projection.

###### MSC:

[2010] 28A78, 28A80.

††journal: Boletim da Sociedade Paranaense de Matemática

##  1 Introduction

The notion of dimensions is an important tool in the classification of subsets
in $\mathbb{R}^{n}$. The Hausdorff and packing dimensions appear as some of
the most common examples in the literature. The determination of set’s
dimensions is naturally connected to the auxilliary Borel measures supported
by these sets. Moreover, the estimation of a set’s dimension is naturally
related to the dimension of a probability measure $\nu$ in $\mathbb{R}^{n}$.
In this way, thinking particularly to sets of measure zero or one, leads to
the respective definitions of the lower and upper Hausdorff dimensions of
$\nu$ as follows

$\underline{\dim}(\nu)=\inf\Big{\\{}\dim(E);\;E\subseteq\mathbb{R}^{n}\;\text{and}\;\nu(E)>0\Big{\\}}$

and

$\overline{\dim}(\nu)=\inf\Big{\\{}\dim(E);\;E\subseteq\mathbb{R}^{n}\;\text{and}\;\nu(E)=1\Big{\\}},$

where $\dim(E)$ denotes the Hausdorff dimension of $E$ (see[16]). If
$\underline{\dim}(\nu)=\overline{\dim}(\nu)$, this common value is denoted by
${\dim}(\nu)$. In this case, we say that $\nu$ is unidimensional. Similarly,
we define respectively the lower and upper packing dimensions of $\nu$ by

$\underline{\operatorname{Dim}}(\nu)=\inf\Big{\\{}\operatorname{Dim}(E);\;E\subseteq\mathbb{R}^{n}\;\text{and}\;\nu(E)>0\Big{\\}}$

and

$\overline{\operatorname{Dim}}(\nu)=\inf\Big{\\{}\operatorname{Dim}(E);\;E\subseteq\mathbb{R}^{n}\;\text{and}\;\nu(E)=1\Big{\\}},$

where $\operatorname{Dim}(E)$ is the packing dimension of $E$ (see [16]).
Also, if the equality
$\underline{\operatorname{Dim}}(\nu)=\overline{\operatorname{Dim}}(\nu)$ is
satisfied, we denote by ${\operatorname{Dim}}(\nu)$ their common value.

The lower and upper Hausdorff dimensions of $\nu$ were studied by A.H. Fan in
[19, 20]. They are related to the Hausdorff dimension of the support of $\nu$.
A similar approach, concerning the packing dimensions, was developed by
Tamashiro in [45]. There are numerous works in which estimates of the
dimension of a given measure are obtained [2, 7, 13, 16, 21, 24, 25, 26, 27,
41]. When $\overline{\dim}(\nu)$ is small (resp. $\underline{\dim}(\nu)$ is
large), it means that $\nu$ is singular (resp. regular) with respect to the
Hausdorff measure. Similar definitions are used when concerned with the upper
and lower packing dimensions.
Note that, in many works (see for example [16, 24, 25, 26]), the quantities
$\underline{\dim}(\nu)$, $\overline{\dim}(\nu)$,
$\underline{\operatorname{Dim}}(\nu)$ and $\overline{\operatorname{Dim}}(\nu)$
are related to the asymptotic behavior of the function
$\alpha_{\nu}(x,r)=\frac{\log\nu(B(x,r))}{\log r}$.

One of the main problems in multifractal analysis is to understand the
multifractal spectrum, the Rényi dimensions and their relationship with each
other. During the past 20 years, there has been enormous interest in computing
the multifractal spectra of measures in the mathematical literature and within
the last 15 years the multifractal spectra of various classes of measures in
Euclidean space $\mathbb{R}^{n}$ exhibiting some degree of self-similarity
have been computed rigorously (see [16, 33, 38] and the references therein).
In an attempt to develop a general theoretical framework for studying the
multifractal structure of arbitrary measures, Olsen [33] and Pesin [37]
suggested various ways of defining an auxiliary measure in very general
settings. For more details and backgrounds on multifractal analysis and its
applications, the readers may be referred also to the following essential
references [1, 2, 8, 9, 10, 11, 12, 14, 15, 22, 30, 31, 33, 34, 35, 36, 39,
42, 43, 46, 47, 48, 49].

In this paper, we give a multifractal generalization of the results about
Hausdorff and packing dimension of measures. We first estimate the
multifractal Hausdorff and packing dimensions of a Borel probability measure.
We try through these results to improve the main result of M. Dai in [13,
Theorem A] about the multifractal analysis of a measure of exact multifractal
dimension. We are especially based on the multifractal formalism developed by
Olsen in [33]. Then, we investigate a relationship between the multifractal
dimensions of a measure $\nu$ and its projections onto a lower dimensional
linear subspace.

##  2 Preliminaries

We start by recalling the multifractal formalism introduced by Olsen in [33].
This formalism was motivated by Olsen’s wish to provide a general mathematical
setting for the ideas present in the physics literature on multifractals.

Let $E\subset\mathbb{R}^{n}$ and $\delta>0$, we say that a collection of balls
$\big{(}B(x_{i},r_{i})\big{)}_{i}$ is a centered $\delta$-packing of $E$ if

$\forall i,\;0<r_{i}<\delta,\quad x_{i}\in E,\;\text{and}\quad
B(x_{i},r_{i})\cap B(x_{j},r_{j})=\emptyset,\quad{\forall}\;i\neq j.$

Similarly, we say that $\big{(}B(x_{i},r_{i})\big{)}_{i}$ is a centered
$\delta$-covering of $E$ if

$\forall i,\;0<r_{i}<\delta,\quad x_{i}\in E,\quad\text{and}\qquad
E\subset\bigcup_{i}\;B(x_{i},r_{i}).$

Let $\mu$ be a Borel probability measure on $\mathbb{R}^{n}$. For
$q,t\in\mathbb{R}$, $E\subseteq{\mathbb{R}}^{n}$ and $\delta>0$, we define

$\overline{{\mathcal{P}}}^{q,t}_{\mu,\delta}(E)=\displaystyle\sup\left\\{\sum_{i}\mu(B(x_{i},r_{i}))^{q}(2r_{i})^{t}\right\\},$

where the supremum is taken over all centered $\delta$-packings of $E$. The
generalized packing pre-measure is given by

$\overline{{\mathcal{P}}}^{q,t}_{\mu}(E)=\displaystyle\inf_{\delta>0}\overline{{\mathcal{P}}}^{q,t}_{\mu,\delta}(E).$

In a similar way, we define

$\overline{{\mathcal{H}}}^{q,t}_{\mu,\delta}(E)=\displaystyle\inf\left\\{\sum_{i}\mu(B(x_{i},r_{i}))^{q}(2r_{i})^{t}\right\\},$

where the infinimum is taken over all centered $\delta$-coverings of $E$. The
generalized Hausdorff pre-measure is defined by

$\overline{{\mathcal{H}}}^{q,t}_{\mu}(E)=\displaystyle\sup_{\delta>0}\overline{{\mathcal{H}}}^{q,t}_{\mu,\delta}(E).$

Especially, we have the conventions $0^{q}=\infty$ for $q\leq 0$ and $0^{q}=0$
for $q>0$.

Olsen [33] introduced the following modifications on the generalized Hausdorff
and packing measures,

${\mathcal{H}}^{q,t}_{\mu}(E)=\displaystyle\sup_{F\subseteq
E}\overline{{\mathcal{H}}}^{q,t}_{\mu}(F)\quad\text{and}\quad{\mathcal{P}}^{q,t}_{\mu}(E)=\inf_{E\subseteq\bigcup_{i}E_{i}}\sum_{i}\overline{\mathcal{P}}^{q,t}_{\mu}(E_{i}).$

The functions ${\mathcal{H}}^{q,t}_{\mu}$ and ${\mathcal{P}}^{q,t}_{\mu}$ are
metric outer measures and thus measures on the family of Borel subsets of
$\mathbb{R}^{n}$. An important feature of the Hausdorff and packing measures
is that ${\mathcal{P}}^{q,t}_{\mu}\leq{\overline{\mathcal{P}}}^{q,t}_{\mu}$.
Moreover, there exists an integer $\xi\in\mathbb{N}$, such that
${\mathcal{H}}^{q,t}_{\mu}\leq\xi{\mathcal{P}}^{q,t}_{\mu}.$ The measure
${\mathcal{H}}^{q,t}_{\mu}$ is of course a multifractal generalization of the
centered Hausdorff measure, whereas ${\mathcal{P}}^{q,t}_{\mu}$ is a
multifractal generalization of the packing measure. In fact, it is easily seen
that, for $t\geq 0$, one has

$2^{-t}{\mathcal{H}}^{0,t}_{\mu}\leq{\mathcal{H}}^{t}\leq{\mathcal{H}}^{0,t}_{\mu}\quad\text{
and}\quad{\mathcal{P}}^{0,t}_{\mu}={\mathcal{P}}^{t},$

where ${\mathcal{H}}^{t}$ and ${\mathcal{P}}^{t}$ denote respectively the
$t$-dimensional Hausdorff and $t$-dimensional packing measures.

The measures ${\mathcal{H}}^{q,t}_{\mu}$ and ${\mathcal{P}}^{q,t}_{\mu}$ and
the pre-measure ${\overline{\mathcal{P}}}^{q,t}_{\mu}$ assign in a usual way a
multifractal dimension to each subset $E$ of $\mathbb{R}^{n}$. They are
respectively denoted by $\dim_{\mu}^{q}(E)$, $\operatorname{Dim}_{\mu}^{q}(E)$
and $\Delta_{\mu}^{q}(E)$ (see [33]) and satisfy

$\begin{array}[]{lllcr}\dim_{\mu}^{q}(E)&=&\inf\Big{\\{}t\in\operatorname{\mathbb{R}};\quad{\mathcal{H}}^{{q},t}_{\mu}(E)=0\Big{\\}}=\sup\Big{\\{}t\in\operatorname{\mathbb{R}};\quad{\mathcal{H}}^{{q},t}_{\mu}(E)=+\infty\Big{\\}},\\\
\\\
\operatorname{Dim}_{\mu}^{q}(E)&=&\inf\Big{\\{}t\in\operatorname{\mathbb{R}};\quad{\mathcal{P}}^{{q},t}_{\mu}(E)=0\Big{\\}}=\sup\Big{\\{}t\in\operatorname{\mathbb{R}};\quad{\mathcal{P}}^{{q},t}_{\mu}(E)=+\infty\Big{\\}},\\\
\\\
\Delta_{\mu}^{q}(E)&=&\inf\Big{\\{}t\in\operatorname{\mathbb{R}};\quad\overline{\mathcal{P}}^{{q},t}_{\mu}(E)=0\Big{\\}}=\sup\Big{\\{}t\in\operatorname{\mathbb{R}};\quad\overline{\mathcal{P}}^{{q},t}_{\mu}(E)=+\infty\Big{\\}}.\end{array}$

The number $\dim_{\mu}^{q}(E)$ is an obvious multifractal analogue of the
Hausdorff dimension $\dim(E)$ of $E$ whereas $\operatorname{Dim}_{\mu}^{q}(E)$
and $\Delta_{\mu}^{q}(E)$ are obvious multifractal analogues of the packing
dimension $\operatorname{Dim}(E)$ and the pre-packing dimension $\Delta(E)$ of
$E$ respectively. In fact, it follows immediately from the definitions that

$\dim(E)=\dim_{\mu}^{0}(E),\;\;\;\operatorname{Dim}(E)=\operatorname{Dim}_{\mu}^{0}(E)\quad\text{and}\quad\Delta(E)=\Delta_{\mu}^{0}(E).$

##  3 Multifractal Hausdorff and packing dimensions of measures

Now, we introduce the multifractal analogous of the Hausdorff and packing
dimensions of a Borel probability measure.

###### Definition 3.1

The lower and upper multifractal Hausdorff dimensions of a measure $\nu$ with
respect to a measure $\mu$ are defined by

$\underline{\dim}_{\mu}^{q}(\nu)=\inf\Big{\\{}\dim_{\mu}^{q}(E);\;E\subseteq\mathbb{R}^{n}\;\text{and}\;\nu(E)>0\Big{\\}}$

and

$\overline{\dim}_{\mu}^{q}(\nu)=\inf\Big{\\{}\dim_{\mu}^{q}(E);\;E\subseteq\mathbb{R}^{n}\;\text{and}\;\nu(E)=1\Big{\\}}.$

We denote by ${\dim}_{\mu}^{q}(\nu)$ their common value, if the equality
$\underline{\dim}_{\mu}^{q}(\nu)=\overline{\dim}_{\mu}^{q}(\nu)$ is satisfied.

###### Definition 3.2

The lower and upper multifractal packing dimensions of a measure $\nu$ with
respect to a measure $\mu$ are defined by

$\underline{\operatorname{Dim}}_{\mu}^{q}(\nu)=\inf\Big{\\{}\operatorname{Dim}_{\mu}^{q}(E);\;E\subseteq\mathbb{R}^{n}\;\text{and}\;\nu(E)>0\Big{\\}}$

and

$\overline{\operatorname{Dim}}_{\mu}^{q}(\nu)=\inf\Big{\\{}\operatorname{Dim}_{\mu}^{q}(E);\;E\subseteq\mathbb{R}^{n}\;\text{and}\;\nu(E)=1\Big{\\}}.$

When
$\underline{\operatorname{Dim}}_{\mu}^{q}(\nu)=\overline{\operatorname{Dim}}_{\mu}^{q}(\nu)$,
we denote by ${\operatorname{Dim}}_{\mu}^{q}(\nu)$ their common value.

###### Definition 3.3

Let $\mu,\nu$ be two Borel probability measures on $\mathbb{R}^{n}$.

  1. 1.

We say that $\mu$ is absolutely continuous with respect to $\nu$ and write
$\mu\ll\nu$ if, for any set $A\subset\operatorname{\mathbb{R}}^{n}$,
$\nu(A)=0\Rightarrow\mu(A)=0$.

  2. 2.

$\mu$ and $\nu$ are said to be mutually singular and we write $\mu\bot\;\nu$
if there exists a set $A\subset\operatorname{\mathbb{R}}^{n}$, such that
$\mu(A)=0=\nu(\operatorname{\mathbb{R}}^{n}\setminus A).$

The quantities $\underline{\dim}_{\mu}^{q}(\nu)$ and
$\overline{\dim}_{\mu}^{q}(\nu)$ (resp.
$\underline{\operatorname{Dim}}_{\mu}^{q}(\nu)$ and
$\overline{\operatorname{Dim}}_{\mu}^{q}(\nu)$) allow to compare the measure
$\nu$ with the generalized Hausdorff (resp. packing) measure. More precisely,
we have the following result.

###### Theorem 3.1

Let $\mu,\nu$ be two Borel probability measures on $\mathbb{R}^{n}$ and
$q\in\mathbb{R}$. We have,

  1. 1.

$\underline{\dim}_{\mu}^{q}(\nu)=\sup\Big{\\{}t\in\operatorname{\mathbb{R}};\;\nu\ll{\mathcal{H}}^{q,t}_{\mu}\Big{\\}}$
and
$\overline{\dim}_{\mu}^{q}(\nu)=\inf\Big{\\{}t\in\operatorname{\mathbb{R}};\;\nu\bot{\mathcal{H}}^{q,t}_{\mu}\Big{\\}}.$

  2. 2.

$\underline{\operatorname{Dim}}_{\mu}^{q}(\nu)=\sup\Big{\\{}t\in\operatorname{\mathbb{R}};\;\nu\ll{\mathcal{P}}^{q,t}_{\mu}\Big{\\}}$
and
$\overline{\operatorname{Dim}}_{\mu}^{q}(\nu)=\inf\Big{\\{}t\in\operatorname{\mathbb{R}};\;\nu\bot{\mathcal{P}}^{q,t}_{\mu}\Big{\\}}.$

Proof.

  1. 1.

Let’s prove that
$\underline{\dim}_{\mu}^{q}(\nu)=\sup\Big{\\{}t\in\operatorname{\mathbb{R}};\;\nu\ll{\mathcal{H}}^{q,t}_{\mu}\Big{\\}}$.
Define

$s=\sup\Big{\\{}t\in\operatorname{\mathbb{R}};\;\nu\ll{\mathcal{H}}^{q,t}_{\mu}\Big{\\}}.$

For any $t<s$ and $E\subseteq\operatorname{\mathbb{R}}^{n}$, such that
$\nu(E)>0$, we have ${\mathcal{H}}^{q,t}_{\mu}(E)>0$. It follows that
$\dim_{\mu}^{q}(E)\geq t$ and then, $\underline{\dim}_{\mu}^{q}(\nu)\geq t$.
We deduce that $\underline{\dim}_{\mu}^{q}(\nu)\geq s$.

On the other hand, for any $t>s$, there exists a set
$E\subseteq\operatorname{\mathbb{R}}^{n}$, such that $\nu(E)>0$ and
${\mathcal{H}}^{q,t}_{\mu}(E)=0$. Consequently, $\dim_{\mu}^{q}(E)\leq t$ and
so, $\underline{\dim}_{\mu}^{q}(\nu)\leq t$. This leads to
$\underline{\dim}_{\mu}^{q}(\nu)\leq s$.

Now, we prove that
$\overline{\dim}_{\mu}^{q}(\nu)=\inf\Big{\\{}t\in\operatorname{\mathbb{R}};\;\nu\bot{\mathcal{H}}^{q,t}_{\mu}\Big{\\}}.$
For this, we define

$s^{\prime}=\inf\Big{\\{}t\in\operatorname{\mathbb{R}};\;\nu\bot{\mathcal{H}}^{q,t}_{\mu}\Big{\\}}.$

For $t>s^{\prime}$, there exists a set
$E\subseteq\operatorname{\mathbb{R}}^{n}$, such that
${\mathcal{H}}^{q,t}_{\mu}(E)=0=\nu(\operatorname{\mathbb{R}}^{n}\setminus
E)$. Then, $\dim_{\mu}^{q}(E)\leq t$. Since $\nu(E)=1$, then
$\underline{\dim}_{\mu}^{q}(\nu)\leq t$ and
$\underline{\dim}_{\mu}^{q}(\nu)\leq s^{\prime}$.

Now, for $t<s^{\prime}$, take $E\subseteq\operatorname{\mathbb{R}}^{n}$, such
that ${\mathcal{H}}^{q,t}_{\mu}(E)>0$ and $\nu(E)=1$. It can immediately seen
that $\dim_{\mu}^{q}(E)\geq t$. Then, $\underline{\dim}_{\mu}^{q}(\nu)\geq t$.
It follows that $\underline{\dim}_{\mu}^{q}(\nu)\geq s^{\prime}$. This ends
the proof of assertion (1).

  2. 2.

The proof of assertion (2) is given in [27, Theorem 2].$\hfill\square$

###### Remark 3.1

When the upper multifractal Hausdorff (resp. packing) dimension of the measure
is small, it means that the measure $\nu$ is “very singular” with respect to
the generalized multifractal Hausdorff (resp. packing) measure. In the same
way, when the lower multifractal (resp. packing) dimension of the measure is
large, then the measure $\nu$ is “quite regular” with respect to the
generalized multifractal Hausdorff (resp. packing) measure.

The quantities $\underline{\dim}_{\mu}^{q}(\nu)$,
$\overline{\dim}_{\mu}^{q}(\nu)$,
$\underline{\operatorname{Dim}}_{\mu}^{q}(\nu)$ and
$\overline{\operatorname{Dim}}_{\mu}^{q}(\nu)$ are related to the asymptotic
behavior of the function ${\alpha}_{\mu,\nu}^{q}(x,r)$, where

${\alpha}_{\mu,\nu}^{q}(x,r)=\frac{\log\nu\big{(}B(x,r)\big{)}-q\log\mu\big{(}B(x,r)\big{)}}{\log
r}.$

Notice that the chararcterization of the lower und upper packing dimensions by
the function $\alpha_{\mu,\nu}^{q}$ is proved by J. Li in [27, Theorem 3]. In
the following theorem we prove similar results for the Hausdorff dimensions.

###### Theorem 3.2

Let $\mu,\nu$ be two Borel probability measures on $\mathbb{R}^{n}$ and
$q\in\mathbb{R}$. Let

$\underline{\alpha}_{\mu,\nu}^{q}(x)=\displaystyle\liminf_{r\to
0}\;{\alpha}_{\mu,\nu}^{q}(x,r)\quad\text{and}\quad\overline{\alpha}_{\mu,\nu}^{q}(x)=\displaystyle\limsup_{r\to
0}\;{\alpha}_{\mu,\nu}^{q}(x,r).$

We have,

  1. 1.

$\underline{\dim}_{\mu}^{q}(\nu)=\operatorname{ess\,inf}\underline{\alpha}_{\mu,\nu}^{q}(x)\quad\text{and}\quad\overline{\dim}_{\mu}^{q}(\nu)=\operatorname{ess\,sup}\underline{\alpha}_{\mu,\nu}^{q}(x).$

  2. 2.

$\underline{\operatorname{Dim}}_{\mu}^{q}(\nu)=\operatorname{ess\,inf}\overline{\alpha}_{\mu,\nu}^{q}(x)\quad\text{and}\quad\overline{\operatorname{Dim}}_{\mu}^{q}(\nu)=\operatorname{ess\,sup}\overline{\alpha}_{\mu,\nu}^{q}(x),$

where the essential bounds being related to the measure $\nu$.

Proof.

  1. 1.

We prove that
$\underline{\dim}_{\mu}^{q}(\nu)=\text{ess}\inf\underline{\alpha}_{\mu,\nu}^{q}(x)$.

Let $\alpha<\text{ess}\inf\underline{\alpha}_{\mu,\nu}^{q}(x)$. For
$\nu$-almost every $x$, there exists $r_{0}>0$, such that $0<r<r_{0}$ and

$\nu(B(x,r))<\mu(B(x,r))^{q}~{}~{}r^{\alpha}.$

Denote by

$F_{n}=\left\\{x;\;\nu(B(x,r))<\mu(B(x,r))^{q}~{}~{}r^{\alpha},\;\,\text{for}\;\,0<r<\frac{1}{n}\right\\}.$

Let $F=\cup_{n}F_{n}$. It is clear that $\nu(F)=1$. Take $E$ be a Borel subset
of $\operatorname{\mathbb{R}}^{n}$ satisfying $\nu(E)>0$. We have $\nu(E\cap
F)>0$ and there exists an integer $n$, such that $\nu(E\cap F_{n})>0$.

Let $\delta>0$ and $\big{(}B(x_{i},r_{i})\big{)}_{i}$ be a centered
$\delta$-covering of $E\cap F_{n}$. We have

$\displaystyle\sum_{i}\nu(B(x_{i},r_{i}))\leq
2^{-\alpha}\displaystyle\sum_{i}\mu(B(x_{i},r_{i}))^{q}(2r_{i})^{\alpha},$

so that

$2^{\alpha}\nu(E\cap
F_{n})\leq\overline{{\mathcal{H}}}^{q,\alpha}_{\mu,\delta}(E\cap F_{n}).$

Letting $\delta\to 0$ gives that

$2^{\alpha}\nu(E\cap F_{n})\leq\overline{{\mathcal{H}}}^{q,\alpha}_{\mu}(E\cap
F_{n})\leq{\mathcal{H}}^{q,\alpha}_{\mu}(E\cap F_{n}).$

It follows that

${\mathcal{H}}^{q,\alpha}_{\mu}(E)\geq{\mathcal{H}}^{q,\alpha}_{\mu}(E\cap
F_{n})>0\;\Rightarrow\;\dim_{\mu}^{q}(E)\geq\alpha.$

We have proved that

$\underline{\dim}_{\mu}^{q}(\nu)\geq\text{ess}\inf\underline{\alpha}_{\mu,\nu}^{q}(x).$

On the other hand, if
$\text{ess}\inf\underline{\alpha}_{\mu,\nu}^{q}(x)=\alpha$. For
$\varepsilon>0$, let

$E_{\varepsilon}=\Big{\\{}x\in\operatorname{supp}\nu;\;\underline{\alpha}_{\mu,\nu}^{q}(x)<\alpha+\varepsilon\Big{\\}}.$

It is clear that $\nu(E_{\varepsilon})>0$. This means that
$\underline{\dim}_{\mu}^{q}(\nu)\leq{\dim}_{\mu}^{q}(E_{\varepsilon})$. We
will prove that

${\dim}_{\mu}^{q}(E_{\varepsilon})\leq\alpha+\varepsilon,\;\;\forall\;\varepsilon>0.$

Let $E\subset E_{\varepsilon}$ and $x\in E$. Then, for all $\delta>0$ we can
find $0<r_{x}<\delta$, such that

$\nu(B(x,r_{x}))>\mu(B(x,r_{x}))^{q}~{}~{}r_{x}^{\alpha+\varepsilon}.$

Take $\delta>0$. The family $\Big{(}B(x,r_{x})\Big{)}_{x\in E}$ is a centered
$\delta$-covering of $E.$ Using Besicovitch’s Covering Theorem (see [16, 28]),
we can construct $\xi$ finite or countable sub-families
$\Big{(}B(x_{1j},r_{1j})\Big{)}_{j}$,….,$\Big{(}B(x_{\xi j},r_{\xi
j})\Big{)}_{j}$, such that each $E$ satisfies

$E\subseteq\displaystyle\bigcup_{i=1}^{\xi}\bigcup_{j}B(x_{ij},r_{ij})\quad\text{and}\quad\Big{(}B(x_{ij},r_{ij})\Big{)}_{j}\quad\text{is
a }\delta\text{-packing of }E.$

We get

$\displaystyle\displaystyle\sum_{i,j}\mu(B(x_{ij},r_{ij}))^{q}(2r_{ij})^{\alpha+\varepsilon}$
$\displaystyle\leq$ $\displaystyle\xi
2^{\alpha+\varepsilon}\sum_{j}\nu(B(x_{ij},r_{ij}))\leq\xi
2^{\alpha+\varepsilon}\nu(\mathbb{R}^{n}).$

Consequently,

$\overline{\mathcal{H}}^{q,\alpha+\varepsilon}_{\mu,\delta}(E)\leq\xi
2^{\alpha+\varepsilon}\nu(\mathbb{R}^{n})\;\Rightarrow\;\overline{\mathcal{H}}^{q,\alpha+\varepsilon}_{\mu}(E)\leq\xi
2^{\alpha+\varepsilon}\nu(\mathbb{R}^{n}).$

We obtain thus

${\mathcal{H}}^{q,\alpha+\varepsilon}_{\mu}(E_{\varepsilon})\leq\xi
2^{\alpha+\varepsilon}\nu(\mathbb{R}^{n})<\infty.$

Therefore,

$\dim_{\mu}^{q}(E)\leq\alpha+\varepsilon\quad\text{and}\quad\underline{\dim}_{\mu}^{q}(\nu)\leq\text{ess}\inf\underline{\alpha}_{\mu,\nu}^{q}(x).$

We prove in a similar way that
$\overline{\dim}_{\mu}^{q}(\nu)=\text{ess}\sup\underline{\alpha}_{\mu,\nu}^{q}(x)$.

###### Corollary 3.1

Let $\mu,\nu$ be two Borel probability measures on $\mathbb{R}^{n}$ and take
$q,\alpha\in\mathbb{R}$. We have,

  1. 1.

$\underline{\dim}_{\mu}^{q}(\nu)\geq\alpha$ if and only if
$\underline{\alpha}_{\mu,\nu}^{q}(x)\geq\alpha$ for $\nu$-a.e. $x$.

  2. 2.

$\overline{\dim}_{\mu}^{q}(\nu)\leq\alpha$ if and only if
$\underline{\alpha}_{\mu,\nu}^{q}(x)\leq\alpha$ for $\nu$-a.e. $x$.

  3. 3.

$\underline{\operatorname{Dim}}_{\mu}^{q}(\nu)\geq\alpha$ if and only if
$\overline{\alpha}_{\mu,\nu}^{q}(x)\geq\alpha$ for $\nu$-a.e. $x$.

  4. 4.

$\overline{\operatorname{Dim}}_{\mu}^{q}(\nu)\leq\alpha$ if and only if
$\overline{\alpha}_{\mu,\nu}^{q}(x)\leq\alpha$ for $\nu$-a.e. $x$.

Proof. Follows immediately from Theorem 3.2.

###### Example 3.1

We recall the definition of the deranged Cantor set (see [3, 4, 6, 9]).

Let $I_{\emptyset}=[0,1]$. We obtain respectively the left and right sub-
intervals $I_{\varepsilon,1}$ and $I_{\varepsilon,2}$ of $I_{\varepsilon}$ by
deleting the middle open sub-interval of $I_{\varepsilon}$ inductively for
each $\varepsilon\in\\{1,2\\}^{n}$, where $n\in\operatorname{\mathbb{N}}$.

We consider the sequence

$\mathcal{C}_{n}=\displaystyle\bigcup_{\varepsilon\in\\{1,2\\}^{n}}I_{\varepsilon}.$

$\\{\mathcal{C}_{n}\\}_{n\in\operatorname{\mathbb{N}}}$ is a decreasing
sequence of closed sets.

For each $n\in\operatorname{\mathbb{N}}$ and each
$\varepsilon\in\\{1,2\\}^{n}$, we put

$|I_{\varepsilon,1}|/|I_{\varepsilon}|=c_{\varepsilon,1}\quad\text{and}\quad|I_{\varepsilon,2}|/|I_{\varepsilon}|=c_{\varepsilon,2},$

where $|I|$ is the diameter of $I$. The set
$\mathcal{C}=\displaystyle\bigcap_{n\geq 0}\mathcal{C}_{n}$ is called a
deranged Cantor set.

Let $\nu$ be a probability measure supported by the deranged Cantor set
$\mathcal{C}$ and $\mu$ be the Lebesgue measure on $I_{\emptyset}$. For
$\varepsilon_{1},...,\varepsilon_{n}\in\\{1,2\\}$, we denote by
$I_{\varepsilon_{1},...,\varepsilon_{n}}$ the basic set of level $n$. For
$x\in C$, we denote by $I_{n}(x)$ the $n$-th level set containing $x$. We
introduce the sequence of random variables $X_{n}$ defined by

$X_{n}(x)=-\log_{3}\left(\frac{\nu(I_{n}(x))}{\nu(I_{n-1}(x))}\right).$

We have

$\frac{S_{n}(x)}{n}=\frac{X_{1}(x)+...+X_{n}(x)}{n}=\frac{\log(\nu(I_{n}(x))}{\log\mid
I_{n}(x)\mid}.$

By Lemma 1 in [9], we have for all $x\in\mathcal{C}$,

$\liminf_{n\to\infty}\frac{\log(\nu(I_{n}(x))}{\log\mid
I_{n}(x)\mid}=\liminf_{r\to 0}\frac{\log(\nu(B(x,r))}{\log r}$

and

$\limsup_{n\to\infty}\frac{\log(\nu(I_{n}(x))}{\log\mid
I_{n}(x)\mid}=\limsup_{r\to 0}\frac{\log(\nu(B(x,r))}{\log r}.$

The quantities $\underline{\dim}_{\mu}^{q}(\nu)$ and
$\overline{\dim}_{\mu}^{q}(\nu)$ are related to the asymptotic behavior of the
sequence $\displaystyle\frac{S_{n}}{n}$. More precisely, we have the following
two relations

$\underline{\dim}_{\mu}^{q}(\nu)=\text{ess}\inf\left\\{\liminf_{n\to\infty}\frac{S_{n}(x)}{n}-q\right\\}\quad\text{and}\quad\overline{\dim}_{\mu}^{q}(\nu)=\text{ess}\sup\left\\{\liminf_{n\to\infty}\frac{S_{n}(x)}{n}-q\right\\}.$

In the same way, we can also prove that

$\underline{\operatorname{Dim}}_{\mu}^{q}(\nu)=\text{ess}\inf\left\\{\limsup_{n\to\infty}\frac{S_{n}(x)}{n}-q\right\\}\quad\text{and}\quad\overline{\operatorname{Dim}}_{\mu}^{q}(\nu)=\text{ess}\sup\left\\{\limsup_{n\to\infty}\frac{S_{n}(x)}{n}-q\right\\}.$

We say that the measure $\nu$ is $(q,\mu)$-unidimensional if
$\overline{\dim}_{\mu}^{q}(\nu)=\underline{\dim}_{\mu}^{q}(\nu)$. We also say
that $\nu$ has an exact multifractal packing dimension whenever
$\overline{\operatorname{Dim}}_{\mu}^{q}(\nu)=\underline{\operatorname{Dim}}_{\mu}^{q}(\nu)$.
In general, a Borel probability measure is not $(q,\mu)$-unidimensional and
$\overline{\operatorname{Dim}}_{\mu}^{q}(\nu)\neq\underline{\operatorname{Dim}}_{\mu}^{q}(\nu)$.

In the following, we are interested to the $(q,\mu)$-unidimensionality and
ergodicity of $\nu$ and to the calculus of its multifractal Hausdorff and
packing dimensions. Our purpose in the following theorem is to prove the main
Theorem of M. Dai [13, Theorem A] under less restrictive hypotheses.

###### Theorem 3.3

The measure $\nu$ is $(q,\mu)$-unidimensional with
${\dim}_{\mu}^{q}(\nu)=\alpha$ if and only if the following two conditions are
satisfied.

  1. 1.

There exists a set $E$ of $\operatorname{\mathbb{R}}^{n}$ with
${\dim}_{\mu}^{q}(E)=\alpha$, such that $\nu(E)=1$.

  2. 2.

$\nu(E)=0$, for every Borel set $E$ satisfying ${\dim}_{\mu}^{q}(E)<\alpha$.

Proof. We can deduce from Theorems 3.1 and 3.2 that $\nu$ is
$(q,\mu)$-unidimensional if and only if we have the following assertions.

  1. 1.

$\nu$ is absolutely continuous with respect to
${\mathcal{H}}^{q,\alpha-\varepsilon}_{\mu}$, for all $\varepsilon>0$.

  2. 2.

$\nu$ and ${\mathcal{H}}^{q,\alpha+\varepsilon}_{\mu}$ are mutually singular,
for all $\varepsilon>0$.

Then, the proof of Theorem 3.3 becomes an easy consequence of the following
lemma.

###### Lemma 3.1

[13] The following conditions are equivalent.

  1. 1.

We have,

    1. (a)

there exists a set $E$ of $\operatorname{\mathbb{R}}^{n}$ with
${\dim}_{\mu}^{q}(E)=\alpha$, such that $\nu(E)=1$.

    2. (b)

$\nu(E)=0$, for every Borel set $E$ satisfying ${\dim}_{\mu}^{q}(E)<\alpha$.

  2. 2.

We have,

    1. (a)

$\nu\ll{\mathcal{H}}^{q,\alpha-\varepsilon}_{\mu}$ for all $\varepsilon>0$.

    2. (b)

$\nu\bot{\mathcal{H}}^{q,\alpha+\varepsilon}_{\mu}$ for all $\varepsilon>0$.

###### Remark 3.2

Theorem 3.3 improves Dai’s result [13, Theorem A] (we need not to assume that
$\mu$ is a doubling measure).

The symmetrical results are true as well.

###### Theorem 3.4

Let $\mu,\nu$ be two Borel probability measures on $\mathbb{R}^{n}$ and take
$\alpha,q\in\operatorname{\mathbb{R}}$. The following conditions are
equivalent.

  1. 1.

$\overline{\operatorname{Dim}}_{\mu}^{q}(\nu)=\underline{\operatorname{Dim}}_{\mu}^{q}(\nu)=\alpha$.

  2. 2.

We have,

    1. (a)

there exist a set $E\subset\operatorname{\mathbb{R}}^{n}$ with
$\operatorname{Dim}_{\mu}^{q}(E)=\alpha$, such that $\nu(E)=1$,

    2. (b)

if $\;E\subset\operatorname{\mathbb{R}}^{n}$ satisfies
$\operatorname{Dim}_{\mu}^{q}(E)<\alpha$, then $\nu(E)=0$.

  3. 3.

We have,

    1. (a)

$\nu\ll{\mathcal{P}}^{q,\alpha-\epsilon}_{\mu}$, for all $\epsilon>0$.

    2. (b)

$\nu\bot{\mathcal{P}}^{q,\alpha+\epsilon}_{\mu}$, for all $\epsilon>0$.

Proof. We can deduce from Theorems 3.1 and 3.2 that the assertions (1) and (3)
are equivalent. We only need to prove the equivalence of the assertions (2)
and (3).

Assume that the measure $\nu$ satisfies the hypothesis (a) and (b) of (2). Let
$\;E\subset\operatorname{\mathbb{R}}^{n}$ and suppose that
${\mathcal{P}}^{q,\alpha-\epsilon}_{\mu}(E)=0$, for all $\epsilon>0$. Then, we
have that $\operatorname{Dim}_{\mu}^{q}(E)\leq\alpha-\epsilon<\alpha$. By
condition (b) of (2), we obtain $\nu(E)=0$. Thus,

$\nu\ll{\mathcal{P}}^{q,\alpha-\epsilon}_{\mu},\quad\text{for all
}\epsilon>0.$

Thanks to condition (a) of (2), there exists a set
$E\subset\operatorname{\mathbb{R}}^{n}$ of multifractal packing dimension
$\alpha$, such that $\nu(E)=1$ and
$\operatorname{Dim}_{\mu}^{q}(E)=\alpha<\alpha+\epsilon$, for all
$\epsilon>0$. Then, ${\mathcal{P}}^{q,\alpha+\epsilon}_{\mu}(E)=0$. Thus,

$\nu\bot{\mathcal{P}}^{q,\alpha+\epsilon}_{\mu},\quad\text{for all
}\epsilon>0.$

Now, assume that $\nu$ satisfies conditions (a) and (b) of (3). This means
that $\nu\ll{\mathcal{P}}^{q,\alpha-\epsilon}_{\mu}$, for all $\epsilon>0$.
Taking a Borel set $E$ with $\operatorname{Dim}_{\mu}^{q}(E)=\beta<\alpha$ and
$\epsilon=\frac{\alpha-\beta}{2}$, we get
${\mathcal{P}}^{q,(\alpha+\beta)/2}_{\mu}(E)=0.$ Then, $\nu(E)=0$.

Since $\nu\bot{\mathcal{P}}^{q,\alpha+\epsilon}_{\mu}$, for all $\epsilon>0$,
there exists a set $F_{\epsilon}$ with
${\mathcal{P}}^{q,\alpha+\epsilon}_{\mu}(F_{\epsilon})=0$ and
$\nu(F_{\epsilon})=1$. Hence,
$\operatorname{Dim}_{\mu}^{q}(F_{\epsilon})\leq\alpha+\epsilon$. Choose a
sequence $(\epsilon_{k})_{k}$ such that $\epsilon_{k}\to 0$ as $k\to+\infty$
and consider the set $F=\displaystyle\bigcap_{k\geq 1}F_{\epsilon_{k}}$. It is
clear that $\nu(F)=1$ and

$\operatorname{Dim}_{\mu}^{q}(F)\leq\displaystyle\liminf_{k\to\infty}\operatorname{Dim}_{\mu}^{q}(F_{\epsilon_{k}})\leq\displaystyle\liminf_{k\to\infty}(\alpha+\epsilon_{k})=\alpha.$

If $\operatorname{Dim}_{\mu}^{q}(F)=\alpha$, then the condition (a) of (2) is
satisfied for $E=F$.
If $\operatorname{Dim}_{\mu}^{q}(F)<\alpha$, then putting $E=F\cup G$, for
some Borel set $G$ of multifractal packing dimension $\alpha$, we obtain

$\nu(E)=1\qquad\text{and}\qquad\operatorname{Dim}_{\mu}^{q}(E)=\max\Big{\\{}\operatorname{Dim}_{\mu}^{q}(F),\operatorname{Dim}_{\mu}^{q}(G)\Big{\\}}=\alpha.$

###### Proposition 3.1

Let $\mu$ be the Lebesgue measure on $\mathbb{R}^{n}$, $\nu$ be a compactly
supported Borel probability measure on $\mathbb{R}^{n}$ and
$T:\operatorname{supp}\nu\rightarrow\operatorname{supp}\nu$ a $K$-lipschitz
function. Suppose that $\nu$ is $T$-invariant and ergodic on
$\operatorname{supp}\nu$. Then,

$\overline{\dim}_{\mu}^{q}(\nu)=\underline{\dim}_{\mu}^{q}(\nu)\quad\text{and}\quad\overline{\operatorname{Dim}}_{\mu}^{q}(\nu)=\underline{\operatorname{Dim}}_{\mu}^{q}(\nu).$

Proof. $T$ is a $K$-lipschitz function, then $T(B(x,r))\subseteq B(T(x),Kr)$.
Since $\nu$ is $T$-invariant, then we can deduce that

$\nu\big{(}B(x,r)\big{)}\leq\nu\big{(}T^{-1}\big{(}T(B(x,r))\big{)}\big{)}\leq\nu\big{(}T^{-1}\big{(}B(T(x),Kr)\big{)}\big{)}=\nu\big{(}B(T(x),Kr)\big{)}.$

It follows that,

$\displaystyle\frac{\log\nu\big{(}B(x,r)\big{)}-q\log\mu\big{(}B(x,r)\big{)}}{\log
r}$ $\displaystyle=$ $\displaystyle\frac{\log\nu\big{(}B(x,r)\big{)}}{\log
r}-q$ $\displaystyle\geq$
$\displaystyle\frac{\log\nu\big{(}B(T(x),Kr)\big{)}}{\log(Kr)}\times\frac{\log(Kr)}{\log
r}-q,$

which proves that

$\underline{\alpha}_{\mu,\nu}^{q}(x)\geq\underline{\alpha}_{\mu,\nu}^{q}(T(x))\quad\text{and}\quad\overline{\alpha}_{\mu,\nu}^{q}(x)\geq\overline{\alpha}_{\mu,\nu}^{q}(T(x)).$

Since $\nu$ is ergodic, then the function
$\underline{\alpha}_{\mu,\nu}^{q}(x)-\underline{\alpha}_{\mu,\nu}^{q}(T(x))$
(resp.
$\overline{\alpha}_{\mu,\nu}^{q}(x)-\overline{\alpha}_{\mu,\nu}^{q}(T(x))$) is
positive and satisfies

$\int\Big{(}\underline{\alpha}_{\mu,\nu}^{q}(x)-\underline{\alpha}_{\mu,\nu}^{q}(T(x))\Big{)}d\nu(x)=0\quad\left(\text{resp.}\;\;\int\Big{(}\overline{\alpha}_{\mu,\nu}^{q}(x)-\overline{\alpha}_{\mu,\nu}^{q}(T(x))\Big{)}d\nu(x)=0\right).$

We can conclude that,

$\underline{\alpha}_{\mu,\nu}^{q}(x)=\underline{\alpha}_{\mu,\nu}^{q}(T(x))\quad\text{and}\quad\overline{\alpha}_{\mu,\nu}^{q}(x)=\overline{\alpha}_{\mu,\nu}^{q}(T(x))\quad\text{for
}\;\nu\;\text{-a.e. }x$

and that the functions $\underline{\alpha}_{\mu,\nu}^{q}$,
$\overline{\alpha}_{\mu,\nu}^{q}$ are $T$-invariant. On the other hand, the
measure $\nu$ is ergodic and

$-q\leq\underline{\alpha}_{\mu,\nu}^{q}\leq\overline{\alpha}_{\mu,\nu}^{q}\leq
n-q.$

It follows that $\underline{\alpha}_{\mu,\nu}^{q}$
($\overline{\alpha}_{\mu,\nu}^{q}$) is $\nu$-almost every where constant,
which says that

$\overline{\dim}_{\mu}^{q}(\nu)=\underline{\dim}_{\mu}^{q}(\nu)\quad\text{and}\quad\overline{\operatorname{Dim}}_{\mu}^{q}(\nu)=\underline{\operatorname{Dim}}_{\mu}^{q}(\nu).$

###### Remark 3.3

In the case where
$\overline{\alpha}_{\mu,\nu}^{q}(x)=\underline{\alpha}_{\mu,\nu}^{q}(x)=\alpha$
for $\nu$-almost all $x$, we have
${\dim}_{\mu}^{q}(\nu)={\operatorname{Dim}}_{\mu}^{q}(\nu)=\alpha.$ The
results developed by Heurteaux in [24, 25, 26] and Fan et al. in [19, 20, 21]
are obtained as a special case of the multifractal Theorems when $q$ equals
$0$.

###### Example 3.2

We say that the probability measure $\mu$ is a quasi-Bernoulli measure on the
Cantor set $\mathcal{C}=\\{0,1\\}^{\operatorname{\mathbb{N}}^{*}}$ if we can
find $C\geq 1$ such that

$\forall x,y\in\mathcal{F}\qquad C^{-1}\mu(x)\mu(y)\leq\mu(xy)\leq
C\mu(x)\mu(y),$

where $\mathcal{F}$ is the set of words written with the alphabet $\\{0,1\\}$.
Let $\mathcal{F}_{n}$ be the set of words of length $n$, and take
$x=x_{1}x_{2}...\in\mathcal{C}$, let $I_{n}(x)$ be the unique cylinder
$\mathcal{F}_{n}$ that contains $x$. Let us introduce the function
$\tau_{\mu}$ defined for $p\in\operatorname{\mathbb{R}}$, by

$\tau_{\mu}(p)=\displaystyle\lim_{n\to\infty}\tau_{\mu}(p,n)\quad\text{with}\quad\tau_{\mu}(p,n)=\displaystyle\frac{1}{n\log
2}\log\left(\sum_{x\in\mathcal{F}_{n}}\mu(x)^{p}\right).$

Let $\mu$ and $\nu$ be two probability measures on $\mathcal{C}$ such that,
$\nu\ll\mu$ and $\mu$ is a quasi-Bernoulli measure. Then,
$\tau^{\prime}_{\mu}(1)$ exists and we have

$\displaystyle\displaystyle\lim_{n\to\infty}\frac{\log\mu\big{(}B(x,2^{-n})\big{)}}{\log(2^{-n})}=\displaystyle\lim_{n\to\infty}\frac{\log_{2}\mu\big{(}I_{n}(x)\big{)}}{-n}=-\tau^{\prime}_{\mu}(1)\text{
for}\;\mu\text{-a.e. }x\in\mathcal{C},$ (3.1)

and

$\displaystyle\displaystyle\lim_{n\to\infty}\;\;\frac{\log_{2}\nu\big{(}I_{n}(x)\big{)}}{-n}=-\tau^{\prime}_{\nu}(1)=-\tau^{\prime}_{\mu}(1)\quad\text{for}\;\nu\text{-a.e.
}x\in\mathcal{C}.$ (3.2)

For more details about (3.1) and (3.2), the reader can see [23, 25]. We have
immediately from (3.1) and (3.2) that the measure $\nu$ is
$(q,\mu)$-unidimensional and

${\dim}_{\mu}^{q}(\nu)={\operatorname{Dim}}_{\mu}^{q}(\nu)=(q-1)\tau^{\prime}_{\mu}(1)=(q-1)\tau^{\prime}_{\nu}(1).$

##  4 Projections results

In this section, we show that the multifractal Hausdorff and packing
dimensions of a measure $\nu$ are preserved under almost every orthogonal
projection. Casually, we briefly recall some basic definitions and facts which
will be repeatedly used in subsequent developments. Let $m$ be an integer with
$0<m<n$ and $G_{n,m}$ the Grassmannian manifold of all $m$-dimensional linear
subspaces of $\mathbb{R}^{n}$. Denote by $\gamma_{n,m}$ the invariant Haar
measure on $G_{n,m}$, such that $\gamma_{n,m}(G_{n,m})=1$. For $V\in G_{n,m}$,
define the projection map $\pi_{V}:\mathbb{R}^{n}\longrightarrow V$ as the
usual orthogonal projection onto $V$. Then, the set $\\{\pi_{V},\;V\in
G_{n,m}\\}$ is compact in the space of all linear maps from $\mathbb{R}^{n}$
to $\mathbb{R}^{m}$ and the identification of $V$ with $\pi_{V}$ induces a
compact topology for $G_{n,m}$. Also, for a Borel probability measure $\mu$
with compact support on $\mathbb{R}^{n}$, denoted by $\operatorname{supp}\mu$,
and for $V\in G_{n,m}$, define the projection $\mu_{V}$ of $\mu$ onto $V$, by

$\mu_{V}(A)=\mu(\pi_{V}^{-1}(A))\quad\forall A\subseteq V.$

Since $\mu$ is compactly supported and
$\operatorname{supp}\mu_{V}=\pi_{V}(\operatorname{supp}\mu)$ for all $V\in
G_{n,m}$, then for any continuous function $f:V\longrightarrow\mathbb{R}$, we
have

$\displaystyle\int_{V}fd\mu_{V}=\int f(\pi_{V}(x))d\mu(x)$

whenever these integrals exist. For more details, see for example [17, 18, 28,
29, 40, 44]. The convolution is defined, for $1\leq m<n$ and $r>0$, by

$\begin{array}[]{llll}\overline{\phi}_{r}^{m}:&\mathbb{R}^{n}&\longrightarrow&\mathbb{R}\\\
&x&\longmapsto&\gamma_{n,m}\Big{\\{}V\in G_{n,m};\;|\pi_{V}(x)|\leq
r\Big{\\}}.\end{array}$

Moreover, define

$\begin{array}[]{llll}\phi_{r}^{m}:&\mathbb{R}^{n}&\longrightarrow&\mathbb{R}\\\
&x&\longmapsto&\min\Big{\\{}1\>,\>r^{m}|x|^{-m}\Big{\\}}.\end{array}$

We have that $\phi_{r}^{m}(x)$ is equivalent to $\overline{\phi}_{r}^{m}(x)$
and write $\phi_{r}^{m}(x)\asymp\overline{\phi}_{r}^{m}(x)$.

For a probability measure $\mu$ and for $V\in G_{n,m}$, we have

$\mu\ast\phi_{r}^{m}(x)\asymp\mu\ast\overline{\phi}_{r}^{m}(x)=\int\mu_{V}(B(x_{V},r))dV$

and

$\mu\ast\phi_{r}^{m}(x)=\int\min\Big{\\{}1\>,\>r^{m}|x-y|^{-m}\Big{\\}}d\mu(y).$

So, integrating by parts and converting into spherical coordinates (see [18]),
we obtain

$\mu\ast\phi_{r}^{m}(x)=mr^{m}\int_{r}^{+\infty}u^{-m-1}\mu(B(x,u))du.$

We present the tools, as well as the intermediate results, which will be used
in the proofs of our main results. The following straightforward estimates
concern the behaviour of the convolution $\mu\ast\phi_{r}^{m}(x)$ as $r\to 0$.

###### Lemma 4.1

[18] Let $1\leq m\leq n$ and $\mu$ be a compactly supported Borel probability
measure on $\operatorname{\mathbb{R}}^{n}$. For all $x\in\mathbb{R}^{n}$, we
have

$cr^{m}\leq\mu\ast\phi_{r}^{m}(x)$

for all sufficiently small $r$, where $c>0$ independent of $r$.

###### Definition 4.1

Let $E\subseteq\mathbb{R}^{n}$ and $0<s<+\infty$. We say that $E$ is
$s$-Ahlfors regular if it is closed and if there exists a Borel measure $\mu$
on $\mathbb{R}^{n}$ and a constant $1\leq C_{E}<+\infty$, such that $\mu(E)>0$
and

$C_{E}^{-1}r^{s}\leq\mu(B(x,r))\leq C_{E}r^{s},\quad\text{for all}\;\;x\in
E\;\;\;\text{and}\;\;\;0<r\leq 1.$

###### Lemma 4.2

Let $0<m\leq n$

  1. 1.

Let $\mu$ be a compactly supported Borel probability measure on
$\operatorname{\mathbb{R}}^{n}$. Then, for all $x\in\mathbb{R}^{n}$ and $r>0$,

$\mu(B(x,r))\leq\mu\ast\phi_{r}^{m}(x).$

  2. 2.

Suppose that $\mu$ is a compactly supported Borel probability measure on
$\mathbb{R}^{n}$ with support contained in an $s$-Ahlfors regular set for some
$0<s\leq m$. Then, for all $\varepsilon>0$ and $\mu$-almost all $x$ there is
$c>0$ such that

$c~{}r^{-\varepsilon}\mu(B(x,r))\geq\mu\ast\phi_{r}^{m}(x).$

for sufficiently small $r$.

Proof. The proof of assertion (1) is exactly the same as that given in [18].
The assertion (2) is nothing but Lemma 5.8 of [32].

We use the properties of $\mu\ast\phi_{r}^{n}(x)$ to have a relationship
between the kernels and the projected measures.

###### Lemma 4.3

[18] Let $1\leq m\leq n$ and $\mu$ be a compactly supported Borel probability
measure on $\operatorname{\mathbb{R}}^{n}$. We have,

  1. 1.

Let $\varepsilon>0$. For all $V\in G_{n,m}$, for $\mu$-almost all $x$ and for
sufficiently small $r$,

$r^{\varepsilon}\mu\ast\phi_{r}^{m}(x)\leq\mu_{V}(B(x_{V},r)).$

  2. 2.

Let $\varepsilon>0$. For $\gamma_{n,m}$-almost all $V\in G_{n,m}$, for all
$x\in\mathbb{R}^{n}$ and for sufficiently small $r$,

$r^{-\varepsilon}\mu\ast\phi_{r}^{m}(x)\geq\mu_{V}(B(x_{V},r)).$

Throughout this section, we consider a compactly supported Borel probability
measure $\mu$ on $\mathbb{R}^{n}$ with support contained in an $s$-Ahlfors
regular set for some $0\leq s\leq m<n$ and $\nu$ be a compactly supported
Borel probability measure on $\mathbb{R}^{n}$ such that
$\operatorname{supp}\nu\subseteq\operatorname{supp}\mu$ and $\nu\ll\mu$.

We introduce the function $\underline{\alpha}_{\mu,\nu}^{q,m}$ and
$\overline{\alpha}_{\mu,\nu}^{q,m}$, by

$\underline{\alpha}_{\mu,\nu}^{q,m}(x)=\displaystyle\liminf_{r\to
0}\;\;\frac{\log\nu\ast\phi_{r}^{m}(x)-q\log\mu\ast\phi_{r}^{m}(x)}{\log r},$

and

$\overline{\alpha}_{\mu,\nu}^{q,m}(x)=\displaystyle\limsup_{r\to
0}\;\;\frac{\log\nu\ast\phi_{r}^{m}(x)-q\log\mu\ast\phi_{r}^{m}(x)}{\log r}.$

###### Proposition 4.1

Let $q\in\mathbb{R}$. We have that for $\nu$-almost all $x$

  1. 1.

If $q>0$, then

$\underline{\alpha}_{\mu,\nu}^{q,m}(x)=\underline{\alpha}_{\mu,\nu}^{q}(x).$

  2. 2.

If $q\leq 0$ and $\underline{\alpha}_{\mu,\nu}^{q}(x)\leq m(1-q)$, then

$\underline{\alpha}_{\mu,\nu}^{q,m}(x)=\underline{\alpha}_{\mu,\nu}^{q}(x).$

Proof.

  1. 1.

We will prove that for $\nu$-almost all $x$, we have
$\underline{\alpha}_{\mu,\nu}^{q,m}(x)\leq\underline{\alpha}_{\mu,\nu}^{q}(x).$
The proof of the other inequality is similar.

By using Lemma 4.2, we have

$\log\nu(B(x,r))\leq\log\nu\ast\phi_{r}^{m}(x).$

Since $\nu$ is absolutely continuous with respect to $\mu$ and $q>0$, we have
that for $\nu$-almost all $x$

$-q\big{(}\log c-\varepsilon\log
r+\log\mu(B(x,r))\big{)}\leq-q\log\mu\ast\phi_{r}^{m}(x).$

So, for $\nu$-almost all $x$,

$\frac{\log\nu(B(x,r))-q\big{(}\log c-\varepsilon\log
r+\log\mu(B(x,r))\big{)}}{\log
r}\geq\frac{\log\nu\ast\phi_{r}^{m}(x)-q\log\mu\ast\phi_{r}^{m}(x)}{\log r}.$

Finally, Letting $\varepsilon\to 0$, we get
$\underline{\alpha}_{\mu,\nu}^{q}(x)\geq\underline{\alpha}_{\mu,\nu}^{q,m}(x).$

  2. 2.

The inequality $\underline{\alpha}_{\mu,\nu}^{q,m}(x)\leq m(1-q)$ follows
immediately from Lemma 4.1.

By using Lemma 4.2, we have

$\log\nu(B(x,r))\leq\log\nu\ast\phi_{r}^{m}(x).$

Since $q\leq 0$, then

$-q\log\mu(B(x,r))\leq-q\log\mu\ast\phi_{r}^{m}(x).$

It follows that
$\underline{\alpha}_{\mu,\nu}^{q,m}(x)\leq\underline{\alpha}_{\mu,\nu}^{q}(x)$.
The proof for other inequality is similar to that given for assertion (1).

The following proposition is a consequence of Lemma 4.3.

###### Proposition 4.2

Let $q\in\mathbb{R}$. For $\gamma_{n,m}$-almost all $V\in G_{n,m}$ and
$\nu$-almost all $x$, we have

$\underline{\alpha}_{\mu_{V},\nu_{V}}^{q}(x_{V})=\underline{\alpha}_{\mu,\nu}^{q,m}(x)$

and

$\overline{\alpha}_{\mu_{V},\nu_{V}}^{q}(x_{V})=\overline{\alpha}_{\mu,\nu}^{q,m}(x).$

The following theorem presents general relations between the multifractal
Hausdorff and the multifractal packing dimension of a measure $\nu$ and that
of its orthogonal projections.

###### Theorem 4.1

Let $q\in\mathbb{R}$.

  1. 1.

For $\gamma_{n,m}$-almost all $V\in G_{n,m}$, we have

$\underline{\operatorname{Dim}}_{\mu_{V}}^{q}(\nu_{V})=\operatorname{ess\,inf}\overline{\alpha}_{\mu,\nu}^{q,m}(x)\quad\text{
and}\quad\overline{\operatorname{Dim}}_{\mu_{V}}^{q}(\nu_{V})=\operatorname{ess\,sup}\overline{\alpha}_{\mu,\nu}^{q,m}(x)$

where the essential bounds being related to the measure $\nu$.

  2. 2.

For $\gamma_{n,m}$-almost all $V\in G_{n,m}$, we have

    1. (a)

for $q>0$,

$\begin{array}[]{ccll}\underline{\dim}_{\mu_{V}}^{q}(\nu_{V})=\underline{\dim}_{\mu}^{q}(\nu)\quad\text{and}\quad\overline{\dim}_{\mu_{V}}^{q}(\nu_{V})=\overline{\dim}_{\mu}^{q}(\nu).\end{array}$

    2. (b)

for $q\leq 0$ and $\overline{\dim}_{\mu}^{q}(\nu)\leq m(1-q),$

$\underline{\dim}_{\mu_{V}}^{q}(\nu_{V})=\underline{\dim}_{\mu}^{q}(\nu)\quad\text{and}\quad\overline{\dim}_{\mu_{V}}^{q}(\nu_{V})=\overline{\dim}_{\mu}^{q}(\nu).$

Proof. Follows immediately from Propositions 4.1 and 4.2 and Corollary 3.1.

###### Remark 4.1

Due to Proposition 5.10 in [32], the result is optimal. If in addition, $q=0$,
then the results of Falconer and O’Neil hold (see [18]).

## Acknowledgments

The author would like to thank the referee for carefully reading of the
manuscript and for various suggestions and improvements.

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# Structured Deep Neural Networks-Based Backstepping Trajectory Tracking
Control for Lagrangian Systems

Jiajun Qian, Liang Xu, Xiaoqiang Ren, Xiaofan Wang The work was supported in
part by the National Natural Science Foundation of China under Grant 62273223,
62373239, 62333011, and the Project of Science and Technology Commission of
Shanghai Municipality under Grant 22JC1401401.Jiajun Qian, Xiaoqiang Ren,
Xiaofan Wang are with the School of Mechatronic Engineering and Automation,
Shanghai University, Shanghai, China. Emails: {qianjiajun, xqren,
<EMAIL_ADDRESS>Xu is with the School of Future Technology, Shanghai
University, Shanghai, China. Email<EMAIL_ADDRESS>(corresponding author)

###### Abstract

Deep neural networks (DNN) are increasingly being used to learn controllers
due to their excellent approximation capabilities. However, their black-box
nature poses significant challenges to closed-loop stability guarantees and
performance analysis. In this paper, we introduce a structured DNN-based
controller for the trajectory tracking control of Lagrangian systems using
backing techniques. By properly designing neural network structures, the
proposed controller can ensure closed-loop stability for any compatible neural
network parameters. In addition, improved control performance can be achieved
by further optimizing neural network parameters. Besides, we provide explicit
upper bounds on tracking errors in terms of controller parameters, which
allows us to achieve the desired tracking performance by properly selecting
the controller parameters. Furthermore, when system models are unknown, we
propose an improved Lagrangian neural network (LNN) structure to learn the
system dynamics and design the controller. We show that in the presence of
model approximation errors and external disturbances, the closed-loop
stability and tracking control performance can still be guaranteed. The
effectiveness of the proposed approach is demonstrated through simulations.

###### Index Terms:

deep neural networks, backstepping control, trajectory tracking, stability
guarantees, Lagrangian systems.

##  I Introduction

Learning-based control methods have gained significant attention in the
control community due to the development of machine learning techniques.
Moreover, neural network-based control has now become prevalent due to the
excellent function approximation capability of neural networks. Traditional
neural network-based control uses shallow neural networks [1]. In contrast,
deep neural networks (DNNs) are superior to shallow NNs in representing
function compositions [2, 3] and avoid the curse of dimensionality [4, 5],
therefore motivating the applications of DNNs in control systems [6, 7, 8, 9,
10, 11, 12, 13, 14, 15, 16, 17]

DNNs in control systems are mainly used to learn dynamics or learn
controllers. In the first category, researchers use DNN to learn complex
models [6, 7, 8, 9, 10] or combine them with the first principles to capture
the uncertainty and residual terms of the system [11, 12]. The second
application involves using neural networks to learn controllers or control
certificates. However, since DNNs are black-box models, providing formal
stability guarantees for neural network-based controllers is difficult. Many
approaches have been proposed to address this challenge.

Neural Lyapunov control methods [13, 14, 15, 16, 17] use neural networks to
learn both a Lyapunov function and a control law. This framework was first
proposed in [14], which comprises a learner and a falsifier. The learner
generates a Lyapunov candidate, while the falsifier aims to identify points
where the Lyapunov candidate fails to satisfy the Lyapunov condition.
Subsequently, the identified points are added to the training dataset and the
neural Lypaunov candidate and the neural controller are re-trained. This
process is repeated until the falsifier cannot find any violation points. The
method has been applied in various fields, such as robot control [18] and
learning safe control strategies [19]. However, identifying points where the
Lyapunov candidate does not satisfy the Lyapunov condition is computationally
complex. Moreover, the training and falsification process may undergo several
rounds before a Lyapunov function and a neural network controller can be
found. As a result, the neural Lypaunov control method is computationally
demanding.

Therefore, effectively reducing the computational complexity while providing
formal stability guarantees for DNN based controllers is challenging.
Structured DNN controllers are proposed for Hamiltonian or Lagrangian systems
as an alternative [20, 21, 22, 23, 24]. Closed-loop stability under structured
DNN controllers can be guaranteed as long as the neural networks satisfy
certain structures. These methods require only the solution of a simple
training problem, significantly reducing computational complexity. However,
there are few structured DNN-based controllers for the trajectory tracking
control problem.

In this paper, we present a structured DNN-based controller for the trajectory
tracking control of Lagrangian systems. Our proposed DNN-based controller is
constructed using the backstepping technique [25, 26]. We demonstrate that
closed-loop stability is ensured for any compatible value of the neural
network parameters. Moreover, we explicitly provide an upper bound for
tracking errors in terms of the control parameters. In scenarios where
obtaining model information is difficult, we propose a modified Lagrangian
Neural Network (LNN) [8] to learn the system model. Moreover, we show that in
the presence of model learning uncertainties and disturbances, we can still
guarantee a bounded tracking error. Finally, we substantiate the effectiveness
of our proposed tracking controller through a series of simulations.

This paper is organized as follows. In Section II, some preliminaries are
provided. Section III gives the tracking controller design and performance
analysis. Section IV shows how to use improved LNNs to learn dynamics and
control designs that can guarantee closed-loop stability. Several simulation
results are given in Section V. Some conclusions are provided in Section VI.

Notations: $\mathbb{R}$ ($\mathbb{R}^{n}$) denote the set of real numbers
($n$-dimensional real vectors). $\lambda_{\min}(\cdot)$ denotes the minimum
eigenvalue of a symmetric matrix. $I$ denote the identity matrix. $W_{1:k}$
denote the sequence $\\{W_{1},\ldots,W_{k}\\}$. $\|\cdot\|$ denotes the
standard Euclidean norm.

##  II Preliminaries

In this section, we briefly introduce the Euler-Lagrange equation, Fully
Connected Neural Network (FCNN), Lagrangian Neural Networks (LNNs) [8] and the
Input Convex Neural Network (ICNN) [27], which will be used in subsequent
sections.

###  II-A Euler-Lagrange Equation

The Euler-Lagrange equation describes the motion of a mechanical system, which
is

$\displaystyle\frac{d}{dt}\left(\frac{\partial
L}{\partial\bm{\dot{q}}}\right)-\frac{\partial
L}{\partial\bm{q}}=\bm{u}+\bm{\tau^{d}},$ (1) $\displaystyle
L(\bm{q},\bm{\dot{q}})=T(\bm{q},\bm{\dot{q}})-V(\bm{q}),$

where $L$ is the Lagrangian, $T$ is the kinetic energy, $V$ is the potential
energy, $\bm{q}$ is the generalized coordinates, $\bm{u}$ is the generalized
non-conservative force, and $\bm{\tau^{d}}$ is the disturbance acting on the
system. Equation (1) can be equivalently represented as

$\displaystyle\bm{M}(\bm{q})\bm{\ddot{q}}+\bm{C}(\bm{q},\bm{\dot{q}})\bm{\dot{q}}+\bm{G}(\bm{q})=\bm{u}+\bm{\tau^{d}},$
(2)

where
$\bm{M}(\bm{q})=\frac{\partial^{2}(T(\bm{q},\bm{\dot{q}}))}{\partial\bm{\dot{q}}\partial\bm{\dot{q}}}$
is the inertia matrix,
$\bm{C}(\bm{q},\bm{\dot{q}})=\bm{\dot{M}}(\bm{q})-\frac{1}{2}\bm{\dot{M}}(\bm{q})^{\top}$
is the Coriolis matrix, and
$\bm{G}(\bm{q})=-\frac{\partial(V(\bm{q}))}{\partial\bm{q}}$ is the
gravitational force vector. The Euler-Lagrange equation has the following
properties [28].

######  Property 1.

$\bm{M}$ is positive definite and is bounded by

$\displaystyle a_{1}\|\bm{x}\|^{2}\leq\bm{x}^{\top}\bm{M}\bm{x}\leq
a_{2}\|\bm{x}\|^{2},\quad\forall\bm{x}\in\mathbb{R}^{n}$

where $a_{1},a_{2}\in\mathbb{R}^{+}$ are positive constants.

######  Property 2.

$\bm{\dot{M}}-2\bm{C}$ is skew symmetric satisfying
$\bm{x}^{\top}(\bm{\dot{M}}-2\bm{C})\bm{x}=0,\quad\forall\bm{x}\in\mathbb{R}^{n}$.

###  II-B Fully Connected Neural Network

Fully Connected Neural Network (FCNN), also known as Multilayer Perceptron
(MLP), is a DNN structure. It contains an input layer, one or more hidden
layers, and an output layer. Each neuron in each layer is connected to every
neuron in the adjacent layer. FCNN has the following expression

$\displaystyle\bm{y_{i+1}}$
$\displaystyle=\sigma_{i}\left(\bm{W_{i}}\bm{y_{i}}+\bm{b_{i}}\right),\quad
i=0,\ldots,k-1,$ (3) $\displaystyle f(\bm{x};\bm{\gamma})$
$\displaystyle=\bm{y_{k}},$

where $\bm{y}_{i}$ are the layer output with $\bm{y}_{0}=\bm{x}$ being the
neural network input, $\sigma_{i}$ represents the layer activation function,
$\bm{\gamma}=\\{\bm{W_{0:k-1}},\bm{b_{0:k-1}}\\}$ are the trainable parameters
of FCNN.

###  II-C Lagrangian Neural Network

Lagrangian Neural Network (LNN) [8] is used to learn the Lagrangian function
of mechanical systems from data. Unlike conventional supervised learning, LNN
is constructed to respect the Euler-Lagrange equation in an unsupervised
manner. By endowing the laws of physics, LNN has better inductive biases.

In LNN, the Lagrangian function is approximated by an FCNN
$\bm{\mathcal{L}}(\bm{q},\bm{\dot{q}};\bm{\gamma})$ with the parameters
$\bm{\gamma}$. During training, $N+1$ samples
$\\{\bm{q}(i),\bm{\dot{q}}(i),\bm{\ddot{q}}(i),\bm{u}(i)\\}_{i=0}^{N}$ are
collected. Then the LNN is trained by solving the following optimization
problem

$\displaystyle\min_{\bm{\gamma}}$
$\displaystyle\sum_{i=0}^{N}\left(\bm{\hat{\ddot{q}}}(i)-\bm{\ddot{q}}(i)\right)^{\top}\bm{Q}\left(\bm{\hat{\ddot{q}}}(i)-\bm{\ddot{q}}(i)\right)$
(4) $\displaystyle\mathrm{s.t.},$
$\displaystyle\bm{\hat{\ddot{q}}}(i)=\frac{\partial^{2}\mathcal{L}(\bm{q}(i),\bm{\dot{q}}(i);\bm{\gamma})}{\partial\bm{\dot{q}}(i)\partial\bm{\dot{q}}(i)}^{-1}[\bm{u}(i)$
$\displaystyle+\frac{\partial\mathcal{L}(\bm{q}(i),\bm{\dot{q}}(i);\bm{\gamma})}{\partial\bm{q}(i)}-\frac{\partial^{2}\mathcal{L}(\bm{q}(i),\bm{\dot{q}}(i);\bm{\gamma})}{\partial\bm{q}(i)\partial\bm{\dot{q}}(i)}\bm{\dot{q}}(i)]$

where $\bm{Q}=\bm{Q}^{\top}>0$ is the weight matrix, and the Jacobian and
Hessian matrix of $\mathcal{L}$ are calculated using automatic differentiation
[29]. By solving (4), one can obtain the optimal parameter $\bm{\gamma^{*}}$.
Then, using automation differentiation, we can obtain an approximation of
$\bm{M}$ as
$\bm{\hat{M}}=\frac{\partial^{2}\mathcal{L}(\bm{q},\bm{\dot{q}};\bm{\gamma^{*}})}{\partial\bm{\dot{q}}\partial\bm{\dot{q}}}$,
$\bm{C}$ as
$\bm{\hat{C}}=\bm{\dot{\hat{M}}}(\bm{q})-\frac{1}{2}\bm{\dot{\hat{M}}}(\bm{q})^{\top}$,
and $\bm{G}$ as
$\bm{\hat{G}}=-\frac{\partial\mathcal{L}(\bm{q},\bm{\dot{q}};\bm{\gamma^{*}})}{\partial\bm{q}}+\frac{1}{2}\bm{\dot{\hat{M}}}(\bm{q})^{\top}\bm{\dot{q}}$.

To effectively solve the optimization problem (4), a special initialization
strategy is necessary [27], which is defined according to the depth and width
of the LNN

$\displaystyle\left.\nu=\frac{1}{\sqrt{n}}\left\\{\begin{array}[]{cc}2.2&\text{First
layer}\\\ 0.58i&\text{Hidden layer }i\in\\{1,\ldots\\}\\\ n&\text{Output
layer}\end{array}\right.\right.,$ (5)

where $i$ represents the $i$-th layer of the neural network, $n$ is the number
of neurons in this layer. The parameters of every layer are then initialized
according to $\mathcal{N}(0,\nu^{2})$, where $\mathcal{N}$ represents a
Gaussian distribution.

###  II-D Input Convex Neural Network

Input Convex Neural Network (ICNN) is a feedforward neural network
architecture with constraints on the neural network parameters to ensure that
the output of the neural network is convex with respect to all inputs (Fully
Input Convex Neural Networks or FICNN) or with respect to a subset of inputs
(partially input Convex Neural Network or PICNN).

FICNN uses the following architecture for $i=0,\ldots,k-1$

$\displaystyle\bm{y_{i+1}}$
$\displaystyle=\sigma_{i}\left(\bm{W_{i}^{(y)}}\bm{y_{i}}+\bm{W_{i}^{(x)}}\bm{x}+\bm{b_{i}}\right),$
(6) $\displaystyle f(\bm{x};\bm{\theta})$ $\displaystyle=\bm{y_{k}},$

where $\sigma_{i}$ represents layer activation functions, $\bm{x}$ is the
FICNN input, $\bm{y_{0}}=0,\bm{W_{0}^{(y)}}=0$,
$\bm{\theta}=\\{\bm{W_{1:k-1}^{(y)}},\bm{W_{0:k-1}^{(x)}},\bm{b_{0:k-1}}\\}$
are network parameters. One can show that $f(\bm{x};\bm{\theta})$ is convex in
$\bm{x}$ if $\bm{W_{1:k-1}^{(y)}}$ are nonnegative and the activation
functions $\sigma_{i}$ are convex and non-decreasing [27].

PICNN uses the following architecture for $i=0,\ldots,k-1$

$\displaystyle\bm{v_{i+1}}=\widetilde{\sigma}_{i}(\bm{\widetilde{W}_{i}}\bm{v_{i}+\bm{\widetilde{b}_{i}}}),$
(7)
$\displaystyle\bm{y_{i+1}}=\sigma_{i}\left(\bm{W_{i}^{(y)}}\left(\bm{y_{i}}\circ(\bm{W^{(yv)}_{i}}\bm{v_{i}}+\bm{b^{(y)}_{i}})\right)\right.$
$\displaystyle\quad\left.+\bm{W_{i}^{(x)}}\left(\bm{x}\circ(\bm{W^{(xv)}_{i}}\bm{v_{i}}+\bm{b^{(x)}_{i}})\right)+\bm{W_{i}^{(v)}}\bm{v_{i}}+\bm{b_{i}}\right),$
$\displaystyle f(\bm{\widetilde{x},\bm{x}};\bm{\theta})=\bm{y_{k}},$

where $\sigma_{i}$ and $\widetilde{\sigma}_{i}$ represent layer activation
functions, $\bm{y_{0}}=0$, $\bm{v_{0}}=\bm{\widetilde{x}}$,
$\bm{W_{0}^{(y)}}=0,$
$\bm{\theta}=\\{\bm{W_{1:k-1}^{(y)}},\bm{W_{1:k-1}^{(yv)}},\bm{W_{0:k-1}^{(x)}},\bm{W_{0:k-1}^{(xv)}}$$,\bm{\widetilde{W}_{0:k-1}},\bm{b^{(y)}_{1:k-1}},$
$\bm{b_{0:k-1}}$ $,\bm{b_{0:k-1}}^{(x)},\bm{\widetilde{b}_{0:k-1}}\\}$ are the
network parameters and $\circ$ denotes the Hadmard product. One can show that
$f(\bm{\widetilde{x}},\bm{x};\bm{\theta})$ is convex in $\bm{x}$ if
$\bm{W_{1:k-1}^{(y)}}$ are nonnegative and the activation functions are convex
and non-decreasing [27]. It should be noted that ICNN can only guarantee the
convexity and not the strong convexity.

##  III Main Results

In this section, we propose a neural backstepping controller (NBS controller)
for the trajectory tracking control of (2) under the assumption that model
information $\bm{M},\bm{C},\bm{G}$ is available. The design of the tracking
controller is given in Section III-A. Its performance is analyzed in Section
III-B and the final learning optimization problem is formulated in Section
III-C.

###  III-A Neural Backstepping Tracking Control Design

Suppose that the reference trajectory is $\bm{q}^{d}(t)$, which is
continuously differentiable. We aim to design a controller such that the state
$\bm{q}(t)$ follows $\bm{q^{d}}(t)$. We define the following errors

$\displaystyle\bm{z_{1}}=\bm{q}-\bm{q^{d}},\bm{z_{2}}=\bm{\dot{q}}-\bm{\phi},$
(8)

where $\bm{\phi}$ is a virtual signal to be designed. The controller $\bm{u}$
and the virtual signal $\bm{\phi}$ are defined as

$\displaystyle\bm{u}=$
$\displaystyle\bm{G}(\bm{q})+\bm{M}(\bm{q})\bm{\dot{\phi}}+\bm{C}(\bm{q},\bm{\dot{q}})\bm{\phi}$
(9)
$\displaystyle\qquad-\frac{\partial\bm{\Phi}(\bm{z_{1}};\bm{\theta_{1}},\bm{S})}{\partial\bm{z_{1}}}-\bm{D}(\bm{z_{2}};\bm{\theta_{2}},m)\bm{z_{2}},$
$\displaystyle\bm{\phi}=$
$\displaystyle\bm{\dot{q}^{d}}-\frac{\partial\bm{\Phi}(\bm{z_{1}};\bm{\theta_{1}},\bm{S})}{\partial\bm{z_{1}}}.$

In the above control design, $\bm{\Phi}$ is constructed as

$\displaystyle\bm{\Phi}(\bm{z_{1}};\bm{\theta_{1}},\bm{S})=\bm{\psi}(\bm{z_{1}};\bm{\theta_{1}})+\bm{z_{1}}^{\top}\bm{S}\bm{z_{1}},$
(10)

where $\bm{S}$ is a positive definite matrix;
$\bm{\psi}(\bm{z_{1}};\bm{\theta_{1}})$ is a FICNN (6) with input $\bm{z_{1}}$
and parameters $\bm{\theta}_{1}$. We let the parameters $\bm{W_{1:k-1}^{(y)}}$
in $\bm{\psi}(\bm{z_{1}};\bm{\theta_{1}})$ be the output of the ReLu function,
whose input is some free parameters. Moreover, we set $\bm{b_{i}}$ in
$\bm{\psi}(\bm{z_{1}};\bm{\theta_{1}})$ to $0$, which ensures $\psi(0)=0$.
Furthermore, by selecting sReLu [30] as the activation function of
$\bm{\psi}$, we can ensure that $\bm{\psi}(z_{1})\geq 0$, which further
guarantees that $\Phi$ has only one minimum at $\bm{z_{1}}=0$, satisfying
$\bm{\Phi}(0)=0$. This design ensures that $\bm{\Phi}$ is strongly convex with
respect to the input $\bm{z_{1}}$.

In (9), $\bm{D}(\bm{z_{2}};\bm{\theta_{2}},m)$ is a positive definite matrix,
constructed from deep neural networks with parameters $\bm{\theta_{2}}$,
hyperparameter $m$, and input $\bm{z_{2}}$. We first use two independent FCNNs
to generate the diagonal and off-diagonal elements of a lower triangular
matrix $\bm{T}$. The activation function for each FCNN is chosen as
$\tanh(\cdot)$. We then filter the output of the diagonal elements of $T$
through a ReLu function to ensure that the diagonal elements of $T$ are
nonnegative. In addition, a positive number $m$ is added to the diagonal
elements of $T$. Then $\bm{D}(\bm{z_{2}};\bm{\theta_{2}},m)$ is constructed as
$D=T^{\top}T$. Through this design, $D$ is guaranteed to be positive definite.

The structure of the tracking controller is illustrated in Fig. 1, the
Jacobian $\dot{\phi},\frac{\partial\bm{\Phi}(\bm{z_{1}})}{\partial\bm{z_{1}}}$
are obtained through automatic differentiation.

Figure 1: The structure for proposed NBS controller.

###  III-B Stability of the NBS Controller

In the following theorems, we analyze the performance of the NBS tracking
controller. First, we consider the case without disturbances.

######  Theorem 1.

Consider the system (2) without disturbance $\bm{\tau^{d}}$, if the controller
is designed as (9), where $\bm{\Phi}(\bm{z_{1}})$ is strongly convex in
$\bm{z_{1}}$ with a unique minimum at $\bm{z_{1}}=0$ satisfying
$\bm{\Phi}(0)=0$, and $\bm{D}(\bm{z_{2}})$ is positive definite, then the
closed-loop system is globally asymptotically stable at
$\bm{z_{1}}=0,\bm{z_{2}}=0$.

###### Proof.

For simplifing notations, we shall ignore the DNN parameters
$\bm{\theta_{1}},\bm{\theta_{2}}$ and hyperparameters $\bm{S},m$ in (9).
According to (8) and (9), (2) can be reformulated as

$\displaystyle\bm{M}\bm{\dot{z}_{2}}+\bm{C}\bm{z_{2}}+\bm{G}=\bm{M}\bm{\ddot{q}}-\bm{M}\bm{\dot{\phi}}+\bm{C}\bm{\dot{q}}-\bm{C}\bm{\phi}+\bm{G}$
(11)
$\displaystyle=\bm{u}-\bm{M}\bm{\dot{\phi}}-\bm{C}\bm{\phi}=\bm{G}-\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}-\bm{D}(\bm{z_{2}})\bm{z_{2}},$

Consider the following candidate Lyapunov function

$\displaystyle\bm{V}(\bm{z_{1}},\bm{z_{2}})=\bm{\Phi}(\bm{z_{1}})+\frac{1}{2}\bm{z_{2}}^{\top}\bm{M}\bm{z_{2}}.$

Since $\bm{\Phi}(\bm{z_{1}})$ is strongly convex in $\bm{z_{1}}$ and has only
one minimum at $\bm{z_{1}}=0$ with $\bm{\Phi}(0)=0$, in view of Property 1, we
have $\bm{V}(0,0)=0,\bm{V}(\bm{z_{1}},\bm{z_{2}})>0,\forall\bm{z_{1}}\neq
0,\bm{z_{2}}\neq 0$, and
$\|\bm{z_{1}}\|,\|\bm{z_{2}}\|\to\infty\Rightarrow\bm{V}(\bm{z_{1}},\bm{z_{2}})\to\infty$.
The time derivative of $\bm{V}$ is

$\displaystyle\bm{\dot{V}}=\bm{z_{2}}^{\top}\bm{M}\bm{\dot{z}_{2}}+\frac{1}{2}\bm{z_{2}}^{\top}\bm{\dot{M}}\bm{z_{2}}+\bm{\dot{z}}_{1}^{\top}\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}$
$\displaystyle=\bm{z_{2}}^{\top}\left(\bm{u}-\bm{M}\bm{\dot{\phi}}-\bm{C}\bm{\phi}-\bm{C}\bm{z_{2}}-\bm{G}+\frac{1}{2}\bm{\dot{M}}\bm{z_{2}}\right)+\bm{\dot{z}_{1}}^{\top}\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}$
$\displaystyle=\bm{z_{2}}^{\top}\left(-\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}-\bm{D}\bm{z_{2}}\right)+\bm{\dot{z}_{1}}^{\top}\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}+\frac{1}{2}\bm{z_{2}}^{\top}(\bm{\dot{M}}-2\bm{C})\bm{z_{2}}.$

Based on the definitions of (8), we can derive
$\bm{z_{2}}=\bm{\dot{q}}-\bm{\dot{q}^{d}}+\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}=\bm{\dot{z}_{1}}+\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}$.
According to Property 2 and the positive definiteness of $\bm{D}$, we have

$\displaystyle\bm{\dot{V}}=-\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}^{\top}\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}-\bm{z_{2}}^{\top}\bm{D}\bm{z_{2}}<0.$

Therefore, we can conclude that $\bm{z}_{1},\bm{z}_{2}$ will converge to
$\\{\bm{z_{1}},\bm{z_{2}}|\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}^{\top}\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}+\bm{z_{2}}^{\top}\bm{D}\bm{z_{2}}=0\\}$.
Since $\bm{\Phi}(\bm{z_{1}})$ is strongly convex in $\bm{z_{1}}$ and is with a
minimum at $\bm{z_{1}}=0$, $\bm{D}$ is positive, we have that
$\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}^{\top}\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}+\bm{z_{2}}^{\top}\bm{D}\bm{z_{2}}=0$
if and only if $\bm{z_{1}}=0$ and $\bm{z_{2}}=0$. Therefore, we have

$\displaystyle\lim_{t\rightarrow\infty}\bm{z_{1}}=0,\lim_{t\rightarrow\infty}\bm{z_{2}}=0.$

∎

Theorem 1 demonstrates the stability of the closed-loop system under the NBS
tracking controller when there are no disturbances. It is unconditionally
stable for all compatible neural network parameters. That is, the controller
can guarantee closed-loop stability as long as the neural networks $\Phi$ and
$D$ satisfy the conditions in Theorem 1, irrespective of the neural network
parameters. In the following theorem, we will show that the NBS tracking
controller can achieve a bounded tracking error in the presence of bounded
disturbances.

######  Theorem 2.

Consider the system (2) with a bounded disturbance $\|\bm{\tau^{d}}\|^{2}\leq
d$. If the controller is designed as (9) and
$\bm{D}(\bm{z_{2}})\geq\frac{1}{2}\bm{I}$, then the tracking error will
converge to
$\\{\bm{z_{1}}|\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}^{\top}\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}\leq\frac{1}{2}d\\}$.

###### Proof.

Consider the candidate Lyapunov function

$\displaystyle
V(\bm{z_{1}},\bm{z_{2}})=\Phi(\bm{z_{1}})+\frac{1}{2}\bm{z_{2}}^{\top}\bm{M}\bm{z_{2}}.$

The time derivative of $V$ is

$\displaystyle\bm{\dot{V}}$
$\displaystyle=-\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}^{\top}\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}-\bm{z_{2}}^{\top}\bm{D}(\bm{z_{2}})\bm{z_{2}}+\bm{z_{2}}^{\top}\bm{\tau^{d}}$
$\displaystyle\leq-\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}^{\top}\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}-\bm{z_{2}}^{\top}\bm{D}(\bm{z_{2}})\bm{z_{2}}+\frac{1}{2}\bm{z_{2}}^{\top}\bm{z_{2}}+\frac{1}{2}\|\bm{\tau^{d}}\|^{2}.$

Since $\bm{D}(\bm{z_{2}})\geq\frac{1}{2}\bm{I}$, we have

$\displaystyle\bm{\dot{V}}\leq-\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}^{\top}\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}+\frac{1}{2}d.$
(12)

Then $\bm{z_{1}}$ will converge to the set
$\\{\bm{z_{1}}|\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}^{\top}\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}\leq\frac{1}{2}d\\}$.
∎

We can achieve $\bm{D}(\bm{z_{2}})\geq\frac{1}{2}\bm{I}$ by adjusting the
hyperparameter $m$. In Theorem 2, the bound of the tracking error is given as
a function of the Jacobian of $\bm{\Phi}$. In the following, we will provide
an explicit tracking error bound assuming structures of $\bm{\Phi}$.

######  Theorem 3.

Consider the system (2) with a bounded disturbance $\|\bm{\tau^{d}}\|^{2}\leq
d$. If $\bm{D}(\bm{z_{2}})\geq\frac{1}{2}\bm{I}$ and
$\left.\frac{\partial^{2}\bm{\Phi}}{\partial\bm{z_{1}}^{2}}\right|_{\bm{z_{1}}=0}\geq\alpha\bm{I}$,
then the tracking error under the controller (9) will converge to the set
$\\{\bm{z_{1}}|\|\bm{z_{1}}\|^{2}\leq\frac{1}{2}\frac{1}{\alpha^{2}}d\\}$.

###### Proof.

The Taylor expansion of $\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}$ at
$\bm{z_{1}}=0$ is

$\displaystyle\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}=\left.\frac{\partial^{2}\bm{\Phi}}{\partial\bm{z_{1}}^{2}}\right|_{\bm{z_{1}}=0}\bm{z_{1}}+\delta$
(13)

where $\delta$ contains high-order terms. We can approximate
$\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}$ with
$\left.\frac{\partial^{2}\bm{\Phi}}{\partial\bm{z_{1}}^{2}}\right|_{\bm{z_{1}}=0}\bm{z_{1}}$.
Then from
$\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}^{\top}\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}\leq\frac{1}{2}d$,
we have
$\bm{z_{1}}^{\top}\left(\left.\frac{\partial^{2}\bm{\Phi}}{\partial\bm{z_{1}}^{2}}\right|_{\bm{z_{1}}=0}\right)^{2}\bm{z_{1}}\leq\frac{1}{2}d$.
Since
$\left.\frac{\partial^{2}\bm{\Phi}}{\partial\bm{z_{1}}^{2}}\right|_{\bm{z_{1}}=0}\geq\alpha\bm{I}$,
therefore $\alpha^{2}\|\bm{z_{1}}\|^{2}\leq\frac{1}{2}d$, that is,
$\|\bm{z_{1}}\|^{2}\leq\frac{1}{2}\frac{1}{\alpha^{2}}d$. ∎

These analyses imply that we can constrain the Hessian matrix of $\bm{\Phi}$
at $\bm{z_{1}}=\bm{0}$ to improve the tracking error performance under
disturbances. This can be achieved by adding a regularizer
$\text{ReLu}\left(\lambda_{\max}\left(\alpha\bm{I}-\left.\frac{\partial^{2}\bm{\Phi}}{\partial\bm{z_{1}}^{2}}\right|_{\bm{z_{1}}=0}\right)\right)$
during training or by letting the hyperparameter $S$ satisfy
$\bm{S}\geq\alpha\bm{I}$ in $\bm{\Phi}(\bm{z_{1}};\bm{\theta_{1}},\bm{S})$.

###  III-C Learning Optimization Formulation

Since the controller (9) is stable for all compatible DNN parameters, we can
further optimize the performance by optimizing the DNN parameters. The
optimization we are solving is the following.

$\displaystyle\min_{\bm{\theta_{1}},\bm{\theta_{2}}}$
$\displaystyle\int_{t=0}^{T}l_{t}(\bm{z_{1}},\bm{u})dt$ (14)
$\displaystyle\qquad+\text{ReLu}\left(\lambda_{\max}\left(\alpha\bm{I}-\left.\frac{\partial^{2}\bm{\Phi}}{\partial\bm{z_{1}}^{2}}\right|_{\bm{z_{1}}=0}\right)\right)$
$\displaystyle\mathrm{s.t.}$ $\displaystyle\text{dynamics
}\eqref{eq.manipulator},\quad\text{errors }\eqref{eq.errors},$
$\displaystyle\text{controller }\eqref{eq.trcaking controller},$
$\displaystyle\text{initial state }\bm{q}(0),\bm{\dot{q}}(0),$

where $l_{t}$ denotes the stage cost at time $t$. The above optimization
problem can be solved by discretizing the dynamics and the cost function first
and then numerically solving the discrete-time counterpart.

##  IV LNNs based Model Approximation and Controller Design

In Section III, the inertia matrix $\bm{M}$, the Coriolis matrix $\bm{C}$, and
the gravitational force vector $\bm{G}$ of the model are used to design the
NBS tracking controller. However, these terms are difficult to obtain for
complex systems. In this section, we use LNN
$\bm{\mathcal{L}}(\bm{q},\bm{\dot{q}};\bm{\gamma})$ to learn the Lagrangian of
the system from data and then construct approximated inertia matrix, Coriolis
matrix and gravitational force vectors to design the controller. To ensure a
better approximation of the Lagrangian function, we proposed a modified LNN
structure, which can guarantee that Property 1 and Property 2 holds for the
approximated inertia matrix and the Coriolis matrix.

###  IV-A Modified LNN for Learning Dynamics

In LNNs, FCNNs are used to represent the Lagrangian function. In doing so, the
approximated inertia matrix
$\bm{\hat{M}}=\frac{\partial^{2}\mathcal{L}(\bm{q},\bm{\dot{q}};\bm{\gamma^{*}})}{\partial\bm{\dot{q}}\partial\bm{\dot{q}}}$
and the Coriolis matrix
$\bm{\hat{C}}=\bm{\dot{\hat{M}}}(\bm{q})-\frac{1}{2}\bm{\dot{\hat{M}}}(\bm{q})^{\top}$may
not satisfy the Property 1 and Property 2. To solve this problem, we use an
ICNN $\mathcal{L}_{T}(\bm{q},\bm{\dot{q}};\bm{\gamma_{1}})$ with the parameter
$\bm{\gamma_{1}}$ to approximate the kinetic energy $T(\bm{q},\bm{\dot{q}})$
in $\mathcal{L}$, which is convex w.r.t. $\bm{\dot{q}}$. Moreover, we use an
FCNN $\mathcal{L}_{V}(\bm{q};\bm{\gamma_{2}})$ with the parameter
$\bm{\gamma_{2}}$ and input $\bm{q}$ to learn $V(\bm{q})$ in $\mathcal{L}$.
Then we construct the Lagrangian neural network as follows.

$\displaystyle\mathcal{L}(\bm{q},\bm{\dot{q}};\bm{\gamma})=\mathcal{L}_{T}(\bm{q},\bm{\dot{q}};\bm{\gamma_{1}})-\mathcal{L}_{V}(\bm{q};\bm{\gamma_{2}}),$

where $\bm{\gamma}=\\{\bm{\gamma_{1}},\bm{\gamma_{2}}\\}$, the activation
function we choose is softplus. Through this design, we can ensure that
$\bm{\hat{M}}=\frac{\partial^{2}\mathcal{L}(\bm{q},\bm{\dot{q}};\bm{\gamma^{*}})}{\partial\bm{\dot{q}}\partial\bm{\dot{q}}}=\frac{\partial^{2}\mathcal{L}_{T}(\bm{q},\bm{\dot{q}};\bm{\gamma^{*}})}{\partial\bm{\dot{q}}\partial\bm{\dot{q}}}$
is positive definite and $\bm{\dot{\hat{M}}}-2\bm{\hat{C}}$ is skew-symmetric.

Using LNNs to approximate the dynamic system model may introduce approximation
errors. With $\hat{M},\hat{C},\hat{G}$, the system dynamics can be modeled as

$\displaystyle\bm{\hat{M}}\bm{\ddot{q}}+\bm{\hat{C}}\bm{\dot{q}}+\bm{\hat{G}}+\delta(\bm{q},\bm{\dot{q}})=\bm{u}+\bm{\tau^{d}},$
(15)

where
$\delta(\bm{q},\bm{\dot{q}})=-\bm{\hat{M}}\bm{M}^{-1}\\{\bm{u}+\bm{\tau^{d}}-\bm{C}\bm{\dot{q}}-\bm{G}\\}+\\{\bm{u}+\bm{\tau^{d}}-\bm{\hat{C}}\bm{\dot{q}}-\bm{\hat{G}}\\}$
is the model mismatch.

Since $\delta$ is a function of $\bm{q}$ and $\bm{\dot{q}}$, we can employ the
first-order Taylor expansion and assume that the uncertain term satisfies the
inequality

$\displaystyle\|\delta(\bm{q},\bm{\dot{q}})\|\leq
a\|\bm{q}\|+b\|\bm{\dot{q}}\|+c,$ (16)

where $a,b,c$ are positive constants.

###  IV-B Controller Design and Performance Analysis

With the approximate model parameters, the NBS tracking controller (9) is
modified to

$\displaystyle\bm{u}=$
$\displaystyle\bm{\hat{G}}+\bm{\hat{M}}\bm{\dot{\phi}}+\bm{\hat{C}}\bm{\phi}-\frac{\partial\bm{\Phi}(\bm{z_{1}};\bm{\theta_{1}},\bm{S})}{\partial\bm{z_{1}}}$
(17)
$\displaystyle\qquad\qquad\qquad\qquad\qquad-\bm{D}(\bm{z_{2}};\bm{\theta_{2}})\bm{z_{2}},$
$\displaystyle\bm{\phi}=$
$\displaystyle\bm{\dot{q}^{d}}-\frac{\partial\bm{\Phi}(\bm{z_{1}};\bm{\theta_{1}},\bm{S})}{\partial\bm{z_{1}}}.$

In the following theorem, we demonstrate that the modified NBS tracking
controller can achieve bounded tracking error in the presence of modeling
uncertainties.

######  Theorem 4.

Consider the system (15) with a bounded disturbance $\|\bm{\tau^{d}}\|^{2}\leq
d$, and the model uncertainty satisfies (16). If the controller is designed as
(17), with
$\bm{D}\geq(b+\frac{b^{2}}{2}+\frac{a}{2}+1)\bm{I},\left.\frac{\partial^{2}\bm{\Phi}}{\partial\bm{z_{1}}^{2}}\right|_{\bm{z_{1}}=0}\geq\alpha\bm{I}$
and $\alpha>\sqrt{a}$, then the tracking error $z_{1}$ will converge to the
set $\\{\bm{z}_{1}|\|\bm{z_{1}}\|^{2}\leq\frac{k^{2}+d}{\alpha^{2}-a}\\}$,
where $k=\max(c+a\|\bm{q^{d}}\|+b\|\bm{\dot{q}^{d}}\|)$.

###### Proof.

From (8) we have

$\displaystyle\|\bm{q}\|$ $\displaystyle\leq\|\bm{z_{1}}\|+\|\bm{q^{d}}\|,$
$\displaystyle\|\bm{\dot{q}}\|$
$\displaystyle\leq\|\bm{z_{2}}\|+\|\bm{\dot{q}^{d}}\|+\|\frac{\partial\bm{\Phi}(\bm{z_{1}})}{\partial\bm{z_{1}}}\|.$

According to (16), one can obtain

$\displaystyle\|\delta(\bm{q},\bm{\dot{q}})\|\leq
a\|\bm{z_{1}}\|+b\|\bm{z_{2}}\|$
$\displaystyle+b\|\frac{\partial\bm{\Phi}(\bm{z_{1}})}{\partial\bm{z_{1}}}\|$
$\displaystyle+c+a\|\bm{q^{d}}\|+b\|\bm{\dot{q}^{d}}\|.$

Since $\bm{q^{d}},\bm{\dot{q}^{d}}$ are bounded reference states, we can
define $k=\max(c+a\|\bm{q^{d}}\|+b\|\bm{\dot{q}^{d}}\|)$. Then we have the
following.

$\displaystyle\|\delta(\bm{q},\bm{\dot{q}})\|\leq
a\|\bm{z_{1}}\|+b\|\bm{z_{2}}\|+b\|\frac{\partial\bm{\Phi}(\bm{z_{1}})}{\partial\bm{z_{1}}}\|+k.$

Consider the following Lyapunov candidate

$\displaystyle
V(\bm{z_{1}},\bm{z_{2}})=\Phi(\bm{z_{1}})+\frac{1}{2}\bm{z_{2}}^{\top}\bm{\hat{M}}\bm{z_{2}}$

The time derivative of $V$ is

$\displaystyle\dot{V}=$
$\displaystyle\bm{z_{2}}^{\top}\left(-\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}-\bm{D}\bm{z_{2}}+\bm{\tau^{d}}-\delta\right)+\bm{\dot{z}_{1}}^{\top}\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}$
$\displaystyle=$
$\displaystyle-\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}^{\top}\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}-\bm{z_{2}}^{\top}\bm{D}\bm{z_{2}}+\bm{z_{2}}^{\top}(\bm{\tau^{d}}-\delta)$
$\displaystyle\leq$
$\displaystyle-\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}^{\top}\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}-\lambda_{\min}(\bm{D})\|\bm{z_{2}}\|^{2}+\|\bm{\tau^{d}}\|\|\bm{z_{2}}\|+b\|\bm{z_{2}}\|^{2}$
$\displaystyle+a\|\bm{z_{1}}\|\|\bm{z_{2}}\|+k\|\bm{z_{2}}\|+b\|\bm{z_{2}}\|\|\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}\|$
$\displaystyle\leq$
$\displaystyle-\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}^{\top}\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}+\frac{1}{2}\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}^{\top}\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}+\frac{a}{2}\|\bm{z_{1}}\|^{2}+\frac{k^{2}}{2}+\frac{1}{2}\|\bm{\tau^{d}}\|^{2}$
$\displaystyle-(\lambda_{\min}(\bm{D})-b-\frac{b^{2}}{2}-\frac{a}{2}-1)\|\bm{z_{2}}\|^{2}$
$\displaystyle=$
$\displaystyle-\frac{1}{2}\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}^{\top}\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}+\frac{a}{2}\|\bm{z_{1}}\|^{2}+\frac{k^{2}+d}{2}$
$\displaystyle-(\lambda_{\min}(\bm{D})-b-\frac{b^{2}}{2}-\frac{a}{2}-1)\|\bm{z_{2}}\|^{2}$

Since $\bm{D}\geq(b+\frac{b^{2}}{2}+\frac{a}{2}+1)\bm{I}$, we have

$\displaystyle h$
$\displaystyle=\lambda_{\min}(\bm{D})-b-\frac{b^{2}}{2}-\frac{a}{2}-1\geq 0$

Therefore,

$\displaystyle\dot{V}$
$\displaystyle\leq-\frac{1}{2}\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}^{\top}\frac{\partial\bm{\Phi}}{\partial\bm{z_{1}}}+\frac{a}{2}\|\bm{z_{1}}\|^{2}+\frac{k^{2}+d}{2}.$

Similar to the proof of Theorem 3, if
$\left.\frac{\partial^{2}\bm{\Phi}}{\partial\bm{z_{1}}^{2}}\right|_{\bm{z_{1}}=0}\geq\alpha\bm{I}$,
we further have

$\displaystyle\dot{V}\leq$
$\displaystyle-\frac{\alpha}{2}\|\bm{z_{1}}\|^{2}+\frac{a^{2}}{2}\|\bm{z_{1}}\|^{2}+\frac{k^{2}+d}{2}$
$\displaystyle=$
$\displaystyle-\frac{\alpha^{2}-a}{2}\|\bm{z_{1}}\|^{2}+\frac{k^{2}+d}{2}.$

Then we can conclude that the tracking error will converge to the set
$\\{\bm{z_{1}}|\|\bm{z_{1}}\|^{2}\leq\frac{k^{2}+d}{\alpha^{2}-a}\\}$. ∎

In the preceding proof, we show that the NBS tracking controller can achieve
bounded tracking errors and the explicit tracking error bounds can be further
obtained by carefully designing $\Phi$. In view of Theorem 4, we can set the
hyperparameter $m\geq\sqrt{b+\frac{b^{2}}{2}+\frac{a}{2}+1}$ in $\bm{D}$ to
guarantee $\bm{D}\geq(b+\frac{b^{2}}{2}+\frac{a}{2}+1)\bm{I}$. Moreover, we
can set the hyperparameter $\bm{S}\geq\alpha\bm{I}$ in $\bm{\Phi}$ or add the
regularizer
$\text{ReLu}\left(\lambda_{\max}\left(\alpha\bm{I}-\left.\frac{\partial^{2}\bm{\Phi}}{\partial\bm{z_{1}}^{2}}\right|_{\bm{z_{1}}=0}\right)\right)$
during training to ensure that
$\left.\frac{\partial^{2}\bm{\Phi}}{\partial\bm{z_{1}}^{2}}\right|_{\bm{z_{1}}=0}\geq\alpha\bm{I}$.
Similarly as in Section III-C, we can further optimize the neural network
parameters to achieve better control performance.

##  V Simulations

In this section, in order to validate our method, we apply the NBS controller
to the tracking control of manipulators. We first perform the sin and cos
signal tracking task for the two-link planar robot arm with known model
information. Our experiments encompass scenarios with and without disturbances
and verify the controller performance. Subsequently, using MuJoCo [31], we
extend our experimentation to a three-link system, employing LNN to learn the
dynamic model of the system and execute trajectory tracking control. We use
pytorch with Adam [32] optimizer throughout the experiments. The code is
available from: https://github.com/jiajqian/Neural-Backstepping-tracking-
controller.

###  V-A Two-Link Planar Robot Arm with Known Model Information

We consider the two-link planar robot arm model as shown in Fig. 2, where the
link masses are concentrated at the ends of the links [28]. The two-link
planar robot arm has $2$ control inputs $[u_{1},u_{2}]$ denoting the torque
applied to each link and $4$ state variables
$[\beta_{1},\beta_{2},\bm{\dot{\beta_{1}}},\bm{\dot{\beta_{2}}}]$,
representing the angle and angular velocity of the links. Each link has mass
$m_{i}=1\mathrm{kg}$ and length $l_{i}=1\mathrm{m}$, where $i=1,2$. The
dynamic model of the two-link planar robot arm can be described by (1) with
$\bm{q}=[\beta_{1},\beta_{2}],\bm{\dot{q}}=[\bm{\dot{\beta}_{1}},\bm{\dot{\beta}_{2}}]$,

$\displaystyle T(\bm{\dot{q}})=$
$\displaystyle\frac{1}{2}(3+2\cos{\beta_{2}})\bm{\dot{\beta_{1}}}^{2}$
$\displaystyle\quad+(1+\cos{\beta_{2}})\bm{\dot{\beta_{1}}}\bm{\dot{\beta_{2}}}+\frac{1}{2}\bm{\dot{\beta_{2}}}^{2},$
$\displaystyle V(\bm{q})=$ $\displaystyle
2g\sin{\beta_{1}}+g\sin(\beta_{1}+\beta_{2}),$

where $g=9.8\mathrm{N/kg}$. The control target is to let $\beta_{1}$ track the
signal $sin(0.1t)$ and $\beta_{2}$ track the signal $cos(0.1t)$. The desired
trajectory is
$\bm{q^{d}}=[\beta_{1}^{d},\beta_{2}^{d}]=[sin(0.1t),cos(0.1t)],\bm{\dot{q}^{d}}=[\bm{\dot{\beta}_{1}^{d}},\bm{\dot{\beta}_{2}^{d}}]=[cos(0.1t),-sin(0.1t)]$

$x$$y$$\bm{\beta}_{2}(t)$$\bm{\beta}_{1}(t)$$l_{1}$$l_{2}$$m_{1}$$m_{2}$$g$
Figure 2: The model of a 2-link planar robot arm

We define the errors as (8). Therefore, the control target can be expressed as
stabilizing the errors, that is, ensuring
$\underset{t\rightarrow\infty}{\lim}\bm{z_{1}}=0$.

The NBS tracking controller is designed as (9). In the controller, $\bm{\psi}$
has $3$ hidden layers and each hidden layer has $32$ neurons, $\bm{S}=\bm{I}$.
Moreover, each FCNN in $\bm{D}$ has $2$ hidden layers and each hidden layer
has $32$ neurons, $m=0.001$.

To demonstrate the unconditional stability of the NBS tracking controller, we
conduct tracking experiments without any prior training in the parameters of
the neural network. The simulations are performed with a time step of $0.01$
seconds, and the initial state is set to $[0,0,0,0]$. Fig. 3 depicts the joint
angles of the robot $\bm{q}$ and the tracking errors $\bm{z_{1}}$,
representing the deviation between $\bm{q}$ and the desired trajectory
$\bm{q^{d}}$. Remarkably, the NBS tracking controller can successfully track
the sin and cos signals, respectively, even without prior training.

Figure 3: The time evolution of angles of the two-link planar robot arm using
the NBS-tracking controller without training.

We can improve the performance of the NBS tracking controller by solving the
optimization problem (14) to achieve better performance. We discrete the
optimization problem (14) with a step size of $0.01s$, the time horizon
$T=1s$, and choose the stage cost as $l_{t}=\bm{z_{1}}^{\top}\bm{z_{1}}$. We
use a decaying learning rate that starts at $1e^{-3}$ for $200$ epochs. Fig. 4
shows the trajectory and tracking error of the angles using the trained
controller. As depicted in Fig. 4, it is evident that the NBS tracking
controller effectively steers the angles, rapidly aligning them with the
desired trajectory. This outcome signifies a noticeable improvement in
performance.

Figure 4: The time evolution of angles of the two-link planar robot arm using
the NBS tracking controller after training.

To demonstrate that the NBS tracking controller can guarantee a bounded
tracking error in the presence of disturbances, we consider the constant
disturbance case $\bm{\tau^{d}}=[1.0,1.0]$. We set the hyperparameter $m$ in
$\bm{D}$ to $1.0$ to ensure $\bm{D}(\bm{z_{2}})\geq\frac{1}{2}\bm{I}$. We
choose the stage cost as $l_{t}=\bm{z_{1}}^{\top}\bm{z_{1}}$, discrete the
optimization problem (14) with a step size of $0.01s$ and the time horizon
$T=1s$. We solve the optimization problem (14) to get
$\theta_{1}^{*},\theta_{2}^{*}$. To investigate the impact of the parameter
$\alpha$ of the regularization term on the stable tracking error bound, we
uniformly sample $40$ values for $\alpha$ from $[0,2]$. We use a learning rate
$1e^{-3}$ for each $\alpha$ over a span of $200$ epochs. According to Thereom
2, the tracking error should converge to
$\|z_{1}\|^{2}\leq\frac{1}{2\alpha^{2}}\times 2$. Fig. 5 shows that the NBS
tracking controller can be trained to ensure a small steady-state tracking
error, and the error is bounded in the presence of disturbances. It can be
seen that increasing the value $\alpha$ can reduce the maximum steady-state
error. However, it should be noted that $\alpha$ should not be set too large;
otherwise, this will make learning the optimal controller difficult.

Figure 5: The steady state tracking error under different $\alpha$ and the
corresponding upper bounds in Theorem 3

###  V-B Three-link Planar Robot Arm with Unknown Model Information

The mathematical model of a three-link planar robot arm is difficult to
obtain. Therefore, we propose to use the MuJoCo physics simulator [31] to
sample data and then learn the Lagrangian function of the three-link planar
robot arm, which gives an approximated model of the three-link planar robot
arm. The model we built in MuJoCo has $3$ control inputs $[u_{1},u_{2},u_{3}]$
denoting the torque applied to each link, and $6$ state variables
$[\beta_{1},\beta_{2},\beta_{3},\bm{\dot{\beta_{1}}},\bm{\dot{\beta_{2}}},\bm{\dot{\beta_{3}}}]$,
representing links’ angle and angular velocity. The desired tracking
trajectory is
$[\beta_{1}^{d}=sin(0.1t),\beta_{2}^{d}=cos(0.1t),\beta_{3}^{d}=sin(0.1t)]$ .

When building the LNN $\mathcal{L}$, $\mathcal{L}_{T}$ is a PICNN in the form
of (7). $\mathcal{L}_{T}$ and $\mathcal{L}_{V}$ have $3$ hidden layers, each
layer has $32$ neurons, and the activation function is softplus. First, we let
the MuJoCo simulator run without imposing any control input on the robot arm,
and obtain the state information of the model, gathering the dataset for
training $\mathcal{L}$. We choose the simulation step size to be $0.001s$, the
initial state to be $[0,0,0,0,0,0]$, the control inputs $\bm{u}=0$, and sample
$10,000$ points. Then we solve (4) and obtain $\bm{\gamma^{*}}$. We train
$\mathcal{L}$ for $200$ epochs, a batch size of $10$ and a decaying learning
rate start at $1e^{-3}$.

After training $\mathcal{L}$, we use the LNN-based NBS tracking controller as
(17). $\bm{\psi}$ has $3$ hidden layers, and each hidden layer has $32$
neurons, $\bm{S}=\bm{I}$. Each FCNN in $\bm{D}$ has $2$ hidden layers and each
hidden layer has $32$ neurons, $m=1.0$. We discrete the optimization problem
(4) with a step size of $0.01s$, the time horizon $T=1s$ and choose the stage
cost as $l_{t}=\bm{z_{1}}^{\top}\bm{z_{1}}$. We still use a decaying learning
rate starting at $1e^{-3}$ for $200$ epochs. Fig. 6 shows the angular
trajectory and tracking errors. The NBS tracking controller effectively
achieves high-precision tracking, maintaining a steady-state tracking error
with $\bm{\|}{z_{1}}\|^{2}\leq 1.5e^{-3}$. This experiment further
demonstrates the efficacy of the LNNs-based NBS tracking controller,
particularly in scenarios where the system model remains unknown or difficult
to obtain.

Figure 6: The time evolution of angles of the three-link planar robot arm
using LNNs-based NBS-tracking controller under MuJoCo simulator.

##  VI Conclusions

In this paper, we propose a structured DNN controller based on backstepping
methods. With properly designed DNN structures, the controller has
unconditional stability guarantees. In addition, its parameters can be
optimized to achieve better performance. We further prove that the tracking
error is bounded in the presence of disturbances. When the model information
is unknown, we use ICNN to improve the LNN structure and use the improved LNNs
to learn system dynamics. The controller is then designed on the basis of the
learned system dynamics. We can also prove that the tracking error is bounded
in the presence of model uncertainties and external disturbances. In the
future, we plan to generalize the method to general nonlinear systems.

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