# Chetaev Instability Framework for Kinetostatic Compliance-Based Protein
Unfolding

Alireza Mohammadi and Mark W. Spong This work is supported by the National
Science Foundation (NSF) under Grants 2153744 and 2153901. A. Mohammadi is
with the Department of Electrical & Computer Engineering, University of
Michigan-Dearborn, Dearborn, MI 48128 USA. Mark W. Spong is with the
University of Texas at Dallas, Richardson, TX 75080 USA. Emails:
<EMAIL_ADDRESS><EMAIL_ADDRESS>∗Corresponding Author: A.
Mohammadi

###### Abstract

Understanding the process of protein unfolding plays a crucial role in various
applications such as design of folding-based protein engines. Using the well-
established kinetostatic compliance (KCM)-based method for modeling of protein
conformation dynamics and a recent nonlinear control theoretic approach to
KCM-based protein folding, this paper formulates protein unfolding as a
destabilizing control analysis/synthesis problem. In light of this
formulation, it is shown that the Chetaev instability framework can be used to
investigate the KCM-based unfolding dynamics. In particular, a Chetaev
function for analysis of unfolding dynamics under the effect of optical
tweezers and a class of control Chetaev functions for synthesizing control
inputs that elongate protein strands from their folded conformations are
presented. Based on the presented control Chetaev function, an unfolding input
is derived from the Artstein-Sontag universal formula and the results are
compared against optical tweezer-based unfolding.

##  I Introduction

Protein molecules make conformational transitions in-between two or more
native conformations to perform a plethora of crucial biological functions
[1]. These transitions usually involve two distinct processes of folding and
unfolding. During the folding/unfolding process, a polypeptide chain
folds/unfolds into/from a biologically active molecule (see Supplementary
Material for an energy landscape schematic of folding/unfolding). In addition
to its role for understanding the protein structures, unfolding is used for
engineering materials such as hydrogels and anti-viral drugs as well as
designing protein-based nanomachines [2, 3, 4].

Algorithmic prediction of pathways through which a folded protein molecule
unfolds to its denatured state has been among the main tools for investigating
the biochemical process of unfolding. The AI-based approaches to predicting
the folded structure of proteins such as Google AlphaFold [5], which are
rooted in pattern recognition, cannot address issues such as folding/unfolding
pathway prediction. Indeed, AlphaFold’s main task has been to predict the most
likely structures of folded proteins while ignoring the kinetics and stability
of folding/unfolding processes (see, e.g., [6] for further details).
Alternatively, dynamics-based approaches rooted in physical first-principles
provide a more preferred way for predicting the folding/unfolding pathways
[7].

To address the high computational burden of physics-based approaches, the
framework of kinetostatic compliance method (KCM), which is based on modeling
protein molecules as mechanisms with a large number of rigid nano-linkages
that fold under the nonlinear effect of electrostatic and van der Waals
interatomic forces, has been developed in the literature [8, 9, 10, 11, 12].
The KCM framework has been successful in investigating the effect of hydrogen
bond formation on protein kinematic mobility [13] and synthesizing molecular
nano-linkages [14, 15]. Furthermore, as demonstrated by [16], this framework
lends itself to a nonlinear control theoretic interpretation, where entropy-
loss constraints can be encoded in KCM-based folding via proper quadratic
optimization-based nonlinear control techniques. Despite its success for
investigating the structure of proteins, the KCM framework except for a
simulation study in [17] has not been systematically utilized for studying
unfolding.

In this paper, we formulate the KCM-based protein unfolding as a destabilizing
control analysis/synthesis problem by utilizing the Chetaev instability
framework. A systemic development of Chetaev functions (CFs) and control
Chetaev functions (CCFs), which are rooted in the seminal work of N. G.
Chetaev [18], has been recently initiated by Efimov and collaborators [19, 20,
21] and Kellett and collaborators [22]. CFs and CCFs are not only useful for
studying instability of nonlinear control systems but also for synthesis of
destabilizing control inputs to induce oscillations via static feedback and
safety critical control applications. CFs and CCFs in this paper are employed
to investigate the instability properties of the folded protein conformations
under study.

Contributions of the paper. This paper has the following contributions. First,
this paper bridges the two fields of protein unfolding in biochemistry and the
Chetaev instability framework in nonlinear control theory. Second, this paper
provides a nonlinear control theoretic foundation for single-molecule force-
based protein denaturation. The single-molecule protein unfolding using
devices such as optical tweezers [23] and atomic force microscopes (AFMs) [24]
has proven to be useful in nano-/molecular-robotics [25, 26]. Third, this
paper adds to the body of knowledge on KCM-based modeling for further
investigation of structural properties of protein molecules.

The rest of this paper is organized as follows. First, we review the KCM
framework for modeling protein molecules, the control theoretic interpretation
of it, and some basics from the Chetaev instability theory in Section II.
Thereafter, we model the dynamics of protein unfolding under optical tweezers
in Section III. Next, a CF for analysis of optical tweezer-based unfolding and
a class of CCFs for synthesizing unfolding control inputs, which elongate
protein molecules, are presented in Section IV. After presenting the
simulation results that compare optical tweezer-based unfolding against a
synthesized Artstein-Sontag destabilizing feedback in Section V, we conclude
the paper in Section VI.

Notation. We let $\mathbb{R}_{+}$ denote the set of all non-negative real
numbers. We let $(\cdot)^{\top}$ denote the vector/matrix transpose operator.
Given $\mathbf{x}\in\mathbb{R}^{M}$, we let
$|\mathbf{x}|:=\sqrt{\mathbf{x}^{\top}\mathbf{x}}$ denote the Euclidean norm
of $\mathbf{x}$. Given $\varepsilon\in\mathbb{R}_{+}$, we let
$\mathcal{B}_{\mathbf{x}}(\varepsilon)$ denote the open ball centered at
$\mathbf{x}$ with radius $\varepsilon$. Given two vectors
$\mathbf{v}_{1},\,\mathbf{v}_{2}\in\mathbb{R}^{3}$, we let
$\mathbf{v}_{1}\times\mathbf{v}_{2}$ denote their cross product.

##  II Preliminaries

Figure 1: The protein molecule kinematic structure.

###  II-A Kinematic linkage representation of protein molecules

In this section, we review the KCM-based modeling of the main chain _in vacuo_
(see [11, 12] for further details). Proteins essentially consist of peptide
planes that are rigidly joined together in a long chain (see Figure 1).
Central carbon atoms denoted by $\text{C}_{\alpha}$ play the role of hinges
in-between the amino acids. The red line segments in Figure 1 represent
covalent atomic bonds. The first $\text{C}_{\alpha}$ is connected to the
N-terminus, and one other peptide plane. Similarly, the last
$\text{C}_{\alpha}$ hinges to the C-terminus, and one other peptide plane.

The backbone conformation of protein molecules can be described by using a
collection of bond lengths and two sets of rotation angles around
$\text{N}-\text{C}_{\alpha}$ and $\text{C}_{\alpha}-\text{C}$, which are known
as the dihedral angles. Hence, the vector of dihedral angles
$\boldsymbol{\theta}=\big{[}\theta_{1},\cdots,\theta_{2N}]^{\top}\in\mathbb{R}^{2N}$
represents the protein configuration with $N-1$ peptide planes. Along the
rotation axis corresponding to each DOF of the protein molecule, a unit vector
denoted by $\mathbf{u}_{j}$, $1\leq j\leq 2N$, can be considered.

In addition to the unit vectors $\mathbf{u}_{j}$, Kazerounian and
collaborators [9, 10, 11] utilize the so-called body vectors to represent the
spatial orientation of the peptide planes. These body vectors are denoted by
$\mathbf{b}_{j}$, $1\leq j\leq 2N$, and describe the relative position of any
two atoms in the chain. Using the vectors $\mathbf{u}_{j}$ and
$\mathbf{b}_{j}$, the conformation of a protein molecule can be completely
described. In particular, after designating the zero position conformation
with $\boldsymbol{\theta}=\mathbf{0}$ corresponding to the biological
reference position of the chain (see [9]), the transformations

$\mathbf{u}_{j}(\boldsymbol{\theta})=\Xi(\boldsymbol{\theta},\mathbf{u}_{j}^{0})\mathbf{u}_{j}^{0},\,\mathbf{b}_{j}(\boldsymbol{\theta})=\Xi(\boldsymbol{\theta},\mathbf{u}_{j}^{0})\mathbf{b}_{j}^{0},$
(1)

where

$\Xi(\boldsymbol{\theta},\mathbf{u}_{j}^{0})=\prod_{r=1}^{j}R(\theta_{j},\mathbf{u}_{j}^{0}),$
(2)

describe the molecule kinematic configuration. In (2), the matrix
$R(\theta_{j},\mathbf{u}_{j}^{0})\in\text{SO}(3)$ represents the rotation
about the unit vector $\mathbf{u}_{j}^{0}$ with angle $\theta_{j}$. Having
obtained the body vectors $\mathbf{b}_{j}(\boldsymbol{\theta})$ from (1) and
under the assumption that the N-terminus nitrogen atom is fixed at the origin,
the coordinates of the backbone chain atoms in the $k$-th peptide plane are
found from
$\mathbf{r}_{i}(\boldsymbol{\theta})=\sum_{j=1}^{i}\mathbf{b}_{j}(\boldsymbol{\theta})$.

###  II-B Control theoretic interpretation of KCM-based dynamics

The KCM framework in [10, 11, 12] builds upon the experimental fact that the
inertial forces during the folding of a protein chain can be neglected in the
presence of the interatomic forces (see, e.g., [27, 10]). Instead, the
dihedral angles rotate under the kinetostatic influence of these forces.

To present the KCM-based protein dynamics, let us consider a polypeptide chain
with $N-1$ peptide planes and conformation vector
$\boldsymbol{\theta}\in\mathbb{R}^{2N}$. The molecule aggregated free energy
is given by (see [12] for detailed derivations)

$\mathcal{G}(\boldsymbol{\theta}):=\mathcal{G}^{\text{elec}}(\boldsymbol{\theta})+\mathcal{G}^{\text{vdw}}(\boldsymbol{\theta}),$
(3)

where $\mathcal{G}^{\text{elec}}(\boldsymbol{\theta})$ and
$\mathcal{G}^{\text{vdw}}(\boldsymbol{\theta})$ represent the the
electrostatic and van der Waals potential energies, respectively.

The resultant forces and torques on each of the $N-1$ rigid peptide links can
be computed from the molecule free energy in (3) and the protein chain
kinematics. These forces and torques can then be appended in the generalized
force vector $\mathcal{F}(\boldsymbol{\theta})\in\mathbb{R}^{6N}$. Following
the steps of derivation in [11], one can utilize a proper mapping to relate
$\mathcal{F}(\boldsymbol{\theta})$ to the equivalent torque vector
$\boldsymbol{\tau}(\boldsymbol{\theta})\in\mathbb{R}^{2N}$ acting on the
dihedral angles at conformation $\boldsymbol{\theta}$, which is given by

$\boldsymbol{\tau}(\boldsymbol{\theta})=\mathcal{J}^{\top}(\boldsymbol{\theta})\mathcal{F}(\boldsymbol{\theta}).$
(4)

In (4), the matrix $\mathcal{J}(\boldsymbol{\theta})\in\mathbb{R}^{6(2N)\times
2N}$ represents the Jacobian of the protein chain at conformation
$\boldsymbol{\theta}$. The vector $\mathcal{F}(\boldsymbol{\theta})$ is due to
the torques and forces, which arise from the interatomic electrostatic and van
der Waals effects, at conformation $\boldsymbol{\theta}$. As it is
demonstrated in [11, 10], for a protein molecule with $N-1$ peptide planes,
the matrix $\mathcal{J}^{\top}(\boldsymbol{\theta})\in\,\mathbb{R}^{2N\times
6(2N)}$ has the structure

$\mathcal{J}^{\top}(\boldsymbol{\theta})=\begin{bmatrix}\mathbf{J}_{1}^{\top}(\boldsymbol{\theta})&\mathbf{J}_{1}^{\top}(\boldsymbol{\theta})&\cdots&\mathbf{J}_{1}^{\top}(\boldsymbol{\theta})\\\
\textbf{0}&\mathbf{J}_{2}^{\top}(\boldsymbol{\theta})&\cdots&\mathbf{J}_{2}^{\top}(\boldsymbol{\theta})\\\
\vdots&\vdots&\vdots&\vdots\\\
\textbf{0}&\textbf{0}&\cdots&\mathbf{J}^{\top}_{2N}(\boldsymbol{\theta})\end{bmatrix},$
(5)

where

$\mathbf{J}_{k}(\boldsymbol{\theta})=\begin{bmatrix}\mathbf{u}_{k}(\boldsymbol{\theta})\\\
-\mathbf{u}_{k}(\boldsymbol{\theta})\times\mathbf{r}_{k}(\boldsymbol{\theta})\end{bmatrix}\in\,\mathbb{R}^{6},\,1\leq
k\leq 2N,$ (6)

in which $\mathbf{u}_{k}(\boldsymbol{\theta})$ denotes the unit vector
introduced in Section II-A and $\mathbf{r}_{k}(\boldsymbol{\theta})$ connects
the N-terminus to either $\text{C}_{\alpha}$ atom (even $k$) or $N$ atom (odd
$k$) on the $k$-th peptide plane.

Using the torque vector in (4) and starting from an initial conformation
$\boldsymbol{\theta}_{0}$, the KCM-based iteration is given by
$\boldsymbol{\theta}_{i+1}=\boldsymbol{\theta}_{i}+h\textbf{f}_{\text{KCM}}\big{(}\boldsymbol{\tau}(\boldsymbol{\theta}_{i})\big{)},$
where $i$ is a non-negative integer, $h$ is a positive real constant that
tunes the maximum dihedral angle rotation magnitude in each step, and
$\textbf{f}_{\text{KCM}}(\cdot)$ is a proper mapping, which can be chosen to
be the identity map in its simplest form (see [11] for further details). The
successive iteration of the KCM-based folding is performed until all of the
kinetostatic torques converge to a minimum corresponding to a local minimum of
the free energy. Furthermore,
$\mathcal{J}^{\top}(\boldsymbol{\theta})\mathcal{F}(\boldsymbol{\theta})$ is a
locally Lipschitz continuous function in a neighborhood of a given folded
conformation.

In [16], it has been demonstrated that the KCM-based iteration can be
interpreted as a control synthesis problem. In particular, the work in [16]
considered the following nonlinear control-affine system

$\dot{\boldsymbol{\theta}}=\mathcal{J}^{\top}(\boldsymbol{\theta})\mathcal{F}(\boldsymbol{\theta})+\mathbf{u}_{c},$
(7)

where the Jacobian matrix $\mathcal{J}(\boldsymbol{\theta})$ and the
generalized force vector $\mathcal{F}(\boldsymbol{\theta})$ are the same as in
(4). It was demonstrated that forward Euler discretization of the closed-loop
dynamics in (7) under a proper closed-loop feedback control input
$\mathbf{u}_{c}(\boldsymbol{\theta})$ restores the KCM-based iteration for
protein folding.

###  II-C Chetaev instability theory preliminaries

Consider the nonlinear dynamical system

$\dot{\mathbf{x}}=\mathbf{F}(\mathbf{x}),$ (8)

where $\mathbf{F}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ is a locally Lipschitz
continuous function. We say that the equilibrium
$\mathbf{x}^{\ast}\in\mathbb{R}^{n}$ is unstable for (8) if for any $\delta>0$
there exists $\mathbf{x}_{0}\in\mathcal{B}_{\mathbf{x}^{\ast}}(\delta)$ such
that
$\mathbf{x}(T_{\mathbf{x}_{0}},\mathbf{x}_{0})\notin\mathcal{B}_{\mathbf{x}^{\ast}}(\delta)$
for some $T_{\mathbf{x}_{0}}\in\mathbb{R}_{+}$, where the solution of (8) with
$\mathbf{x}(0)=\mathbf{x}_{0}$ is given by $\mathbf{x}(\cdot,\mathbf{x}_{0})$
on the solution maximal interval of existence.

######  Theorem II.1 ([18])

Consider the system in (8) and the equilibrium $\mathbf{x}^{\ast}$. Let
$V:\mathbb{R}^{n}\to\mathbb{R}$ be a locally Lipschitz continuous function
such that $V(\mathbf{x}^{\ast})=0$. Suppose that there exists $\epsilon_{0}>0$
such that
$\mathcal{V}^{+}\cap\mathcal{B}_{\mathbf{x}^{\ast}}(\epsilon)\neq\emptyset$
for any $\epsilon\in(0,\epsilon_{0}]$, where
$\mathcal{V}^{+}:=\\{\mathbf{x}\in\mathcal{B}_{\mathbf{x}^{\ast}}(\epsilon):V(\mathbf{x})>0\\}$.
If the lower Dini derivative of $V$ along $\mathbf{F}$ satisfies
$D^{-}V(\mathbf{x})\mathbf{F}(\mathbf{x})>0$ for all
$\mathbf{x}\in\mathcal{V}^{+}$, then the equilibrium $\mathbf{x}^{\ast}$ is
unstable for (8).

A function $V(\cdot)$ that satisfies the conditions in Proposition II.1 is
called a CF for system (8). While the classical theorem by Chetaev [18] only
provided sufficient conditions for instability, Efimov and collaborators
provided the necessary part of the Chetaev’s theorem in [21]. Furthermore,
they developed the CCF concept as a counterpart of the Control Lyapunov
function (CLF) theory [21]. Kellett and collaborators [22] extended the work
in [21] to differential inclusions. See Supplementary Material for the
required background on CCFs.

##  III KCM-Based Modeling of Protein Unfolding under Optical Tweezers

In this section we present the dynamics of protein unfolding under the effect
of an optical tweezer. We remark that the derivation of KCM-based dynamical
model of protein unfolding under optical tweezers has not been presented
elsewhere in the literature.

Optical tweezers can be utilized for unfolding protein molecules by chemically
attaching a latex bead (microsphere) to the protein under study, where the
bead can be trapped and used as a handle for stretching the molecule. The
reader is referred to Supplementary Material for further details. The optical
tweezer tension force on the molecule strand can be modeled by the Hookean
force [23]

$\textbf{F}_{\text{twz}}(\boldsymbol{\theta})=\kappa(\boldsymbol{\theta})\mathbf{x}_{\text{twz}},$
(9)

where $\mathbf{x}_{\text{twz}}$ represents the displacement of the microsphere
from the center of the beam and $\kappa(\boldsymbol{\theta})$ is the optical
trap stiffness modulated by the protein conformation using a proper technique
such as linear polarization [28, 29]. In particular, we assume that the trap
stiffness can be modulated as a smooth function of $\boldsymbol{\theta}$ in a
way that at a given folded conformation $\boldsymbol{\theta}^{\ast}$,
$\kappa(\boldsymbol{\theta}^{\ast})=0$.

###### Remark III.1

For a given positive integer $m$ and a positive constant $\kappa_{0}$, a
candidate optical trap modulated stiffness is

$\kappa(\boldsymbol{\theta})=\kappa_{0}|\mathbf{r}_{\text{NC}}(\boldsymbol{\theta}^{\ast})-\mathbf{r}_{\text{NC}}(\boldsymbol{\theta})|^{m},$
(10)

where at any conformation $\boldsymbol{\theta}$, the vector
$\mathbf{r}_{\text{NC}}(\boldsymbol{\theta})\in\mathbb{R}^{3}$ connects the
nitrogen atom in the N-terminus to the carbon atom in the C-terminus (see
Figure 1).

Let us consider the optical tweezer force in (9) and a protein molecule chain
with $N-1$ peptide planes. Without loss of generality, we assume that this
force is being exerted on the C-terminus of the protein molecule. Accordingly,
the resultant torque applied to the last link $\text{AA}_{\text{last}}$ of the
molecule at conformation $\boldsymbol{\theta}$, is given by
$\mathbf{T}_{\text{twz}}(\boldsymbol{\theta})=\sum_{a_{i}\in\text{AA}_{\text{last}}}\mathbf{r}_{i}(\boldsymbol{\theta})\times\kappa(\boldsymbol{\theta})\mathbf{x}_{\text{twz}},$
where $\mathbf{r}_{i}(\boldsymbol{\theta})$ denotes the position of atom
$a_{i}$, which is the $i$-th atom located in peptide plane
$\text{AA}_{\text{last}}$. Therefore, the force-torque couple due to the
effect of the optical tweezer is
$\mathbf{TF}_{\text{twz}}(\boldsymbol{\theta}):=\begin{bmatrix}\mathbf{T}_{\text{twz}}(\boldsymbol{\theta})^{\top},\mathbf{F}_{\text{twz}}(\boldsymbol{\theta})^{\top}\end{bmatrix}^{\top}\in\mathbb{R}^{6}.$

Consequently, the optical tweezer-based force-torque couple, which is exerted
on the last peptide plane of the protein molecule, gives rise to the
equivalent joint torque vector

$\mathbf{u}_{\text{unfold}}(\boldsymbol{\theta})=\mathcal{J}^{\top}(\boldsymbol{\theta})\mathcal{F}_{\text{twz}}(\boldsymbol{\theta}),$
(11)

where $\mathcal{F}_{\text{twz}}(\boldsymbol{\theta})$ is the generalized force
vector

$\mathcal{F}_{\text{twz}}(\boldsymbol{\theta})=\begin{bmatrix}\mathbf{0}^{\top},\cdots,\mathbf{0}^{\top},\mathbf{TF}_{\text{twz}}(\boldsymbol{\theta})^{\top}\end{bmatrix}^{\top}\in\;\mathbb{R}^{6(2N)},$
(12)

which gets mapped through (11) to the vector
$\mathbf{u}_{\text{unfold}}(\boldsymbol{\theta})\in\mathbb{R}^{2N}$ affecting
the dihedral angles in the molecule.

##  IV Destabilizing control analysis/synthesis for studying the KCM-based
protein unfolding

In this section we first formulate the KCM-based protein unfolding as a
destabilizing control analysis/synthesis problem for (7). Next, we present a
CF for analysis of a folded conformation instability under optical tweezers.
Thereafter, we introduce a class of CCFs for synthesis of destabilizing inputs
that elongate protein strands from their folded conformations.

###  IV-A Control problem formulation and a trivial solution

It is known that the KCM-based iteration when initiated in a vicinity of a
stable conformation $\boldsymbol{\theta}^{\ast}$ converges to it [10, 11]. The
conformation $\boldsymbol{\theta}^{\ast}$ corresponds to a local minimum of
the aggregated free energy function $\mathcal{G}(\cdot)$ in (3), where the
equivalent torque in (4) becomes
$\boldsymbol{\tau}(\boldsymbol{\theta}^{\ast})=\mathcal{J}^{\top}(\boldsymbol{\theta}^{\ast})\mathcal{F}(\boldsymbol{\theta}^{\ast})=\mathbf{0}$.
Indeed, the conformation $\boldsymbol{\theta}^{\ast}$ is an isolated and
locally asymptotically stable equilibrium point for
$\dot{\boldsymbol{\theta}}=\mathcal{J}^{\top}(\boldsymbol{\theta})\mathcal{F}(\boldsymbol{\theta})$
(see, e.g., [11, 10, 16]). In light of the properties of a folded conformation
$\boldsymbol{\theta}^{\ast}$, we can state our control objective for unfolding
as follows.

Protein Unfolding Control Problem (PUCP). Consider the KCM-based protein
conformation dynamics given by (7) and a stable folded conformation given by
$\boldsymbol{\theta}^{\ast}$. Find a closed-loop feedback control input
$\mathbf{u}_{c}=\mathbf{u}_{\text{unfold}}(\boldsymbol{\theta})$ for (7) such
that the folded conformation $\boldsymbol{\theta}^{\ast}$ becomes an unstable
equilibrium for

$\dot{\boldsymbol{\theta}}=\mathbf{f}^{\text{unfold}}(\boldsymbol{\theta}):=\mathcal{J}^{\top}(\boldsymbol{\theta})\mathcal{F}(\boldsymbol{\theta})+\mathbf{u}_{\text{unfold}}(\boldsymbol{\theta}).$
(13)

A _trivial solution_ to PUCP is given as follows.

###### Proposition IV.1

Consider
$u_{\text{unfold}}(\boldsymbol{\theta})=-2\mathcal{J}^{\top}(\boldsymbol{\theta})\mathcal{F}(\boldsymbol{\theta})$
for (13). Under this control input the folded conformation
$\boldsymbol{\theta}^{\ast}$ will become a purely repulsing equilibrium for
the resulting closed-loop dynamics given by
$\dot{\boldsymbol{\theta}}=-\mathcal{J}^{\top}(\boldsymbol{\theta})\mathcal{F}(\boldsymbol{\theta})$.

###### Proof:

The folded conformation $\boldsymbol{\theta}^{\ast}$ is an isolated and
locally asymptotically stable equilibrium point for
$\dot{\boldsymbol{\theta}}=\mathcal{J}^{\top}(\boldsymbol{\theta})\mathcal{F}(\boldsymbol{\theta})$.
Therefore, we can invoke a converse argument [30] to conclude that there
exists a Lyapunov function

$V^{\text{fold}}:\mathbb{R}^{2N}\to\mathbb{R}_{+},\,\boldsymbol{\theta}\mapsto
V^{\text{fold}}(\boldsymbol{\theta})$ (14)

for these dynamics from which the local asymptotic stability of
$\boldsymbol{\theta}^{\ast}$ follows. It can be easily seen that
$V^{\text{fold}}(\cdot)$ is indeed an anti-control Lyapunov function (ALF)
[20] for
$\dot{\boldsymbol{\theta}}=-\mathcal{J}^{\top}(\boldsymbol{\theta})\mathcal{F}(\boldsymbol{\theta})$,
where $V^{\text{fold}}(\boldsymbol{\theta}^{\ast})=0$,
$V^{\text{fold}}(\boldsymbol{\theta})>0$ on
$\mathcal{N}_{\boldsymbol{\theta}^{\ast}}\backslash\\{\boldsymbol{\theta}^{\ast}\\}$,
where $\mathcal{N}_{\boldsymbol{\theta}^{\ast}}$ is an open neighborhood of
$\boldsymbol{\theta}^{\ast}$, and $-\tfrac{\partial
V^{\text{fold}}}{\partial\boldsymbol{\theta}}\mathcal{J}^{\top}(\boldsymbol{\theta})\mathcal{F}(\boldsymbol{\theta})>0$
for all $\boldsymbol{\theta}\neq\boldsymbol{\theta}^{\ast}$ on an open
neighborhood of $\boldsymbol{\theta}^{\ast}$ that is a subset of
$\mathcal{N}_{\boldsymbol{\theta}^{\ast}}$. ∎

###### Remark IV.2

There are some drawbacks associated with considering $V^{\text{fold}}(\cdot)$
as a candidate Chetaev function for protein unfolding. First, no closed-form
expression for $V^{\text{fold}}(\cdot)$ is readily available. Second,
$V^{\text{fold}}(\cdot)$ does not inherently possess the notion of direction
of the unfolding forces.

###  IV-B Instability of folded conformations under optical tweezers

In this section, we present an analysis to guarantee that the forces due to
optical tweezers can be considered as solutions to PUCP formulated in Section
IV-A. Such an analysis provides numerical conditions for real-time control of
optical tweezer-based protein unfolding. Using the torque in (11), the
unfolding conformation dynamics in (13) become

$\dot{\boldsymbol{\theta}}=\mathbf{f}^{\text{unfold}}(\boldsymbol{\theta}),$
(15)

where
$\mathbf{f}^{\text{unfold}}(\boldsymbol{\theta}):=\mathcal{J}^{\top}(\boldsymbol{\theta})\big{\\{}\mathcal{F}(\boldsymbol{\theta})+\mathcal{F}_{\text{twz}}(\boldsymbol{\theta})\big{\\}}$.
Accordingly, if the protein molecule is under the effect of the forces
generated by an optical tweezer as in (9), then the unfolding dynamics are
given by (13), where the unfolding control torque vector
$\mathbf{u}_{\text{unfold}}(\boldsymbol{\theta})$ is given by (11) and (12).
Since the trap stiffness is modulated such that
$\kappa(\boldsymbol{\theta}^{\ast})=0$ (see Remark 10), the conformation
$\boldsymbol{\theta}^{\ast}$ is an equilibrium of (15).

To analyze a folded conformation instability under the effect of optical
tweezers, we are interested in deriving numerical conditions guaranteeing that
PUCP objectives are met for the unfolding dynamics in (15). It is natural to
expect that these conditions should depend on the direction of the applied
force in (9). Considering the fact that a protein molecule extends under an
optical tweezer-based force from its folded conformation
$\boldsymbol{\theta}^{\ast}$, we consider the candidate CF

$\displaystyle
C_{\text{twz}}(\boldsymbol{\theta})=|\Delta\mathbf{r}_{\text{NC}}(\boldsymbol{\theta},\boldsymbol{\theta}^{\ast})|^{2}|\mathbf{r}_{\text{NC}}(\boldsymbol{\theta}^{\ast})|^{2}\cos^{2}(\alpha_{C})-$
(16)
$\displaystyle\big{(}\Delta\mathbf{r}_{\text{NC}}(\boldsymbol{\theta},\boldsymbol{\theta}^{\ast})^{\top}\mathbf{r}_{\text{NC}}(\boldsymbol{\theta}^{\ast})\big{)}^{2},$

for the unfolding dynamics in (15). At any conformation $\boldsymbol{\theta}$,
the vector $\mathbf{r}_{\text{NC}}(\boldsymbol{\theta})\in\mathbb{R}^{3}$ in
(16) connects the nitrogen atom in the N-terminus to the carbon atom in the
C-terminus. Furthermore, the difference vector is given by
$\Delta\mathbf{r}_{\text{NC}}(\boldsymbol{\theta},\boldsymbol{\theta}^{\ast}):=\mathbf{r}_{\text{NC}}(\boldsymbol{\theta})-\mathbf{r}_{\text{NC}}(\boldsymbol{\theta}^{\ast})$.
Finally, the constant $\alpha_{C}$ is any angle in-between $0$ and
$\tfrac{\pi}{2}$. Details on the geometric interpretation of this candidate CF
is provided in Supplementary Material. As it will be seen in the next section
(Propositions IV.7 and IV.8), this candidate CF is also a CCF. The proof of
the next lemma is given in Supplementary Material.

###### Lemma IV.3

Consider the candidate CF $C_{\text{twz}}:\mathbb{R}^{2N}\to\mathbb{R}$ in
(16). We have $C_{\text{twz}}(\boldsymbol{\theta}^{\ast})=0$. Moreover, there
exists $\epsilon_{0}>0$ such that
$\mathcal{C}_{\text{twz}}^{+}\cap\mathcal{B}_{\boldsymbol{\theta}^{\ast}}(\epsilon)\neq\emptyset$,
where
$\mathcal{C}_{\text{twz}}^{+}:=\big{\\{}\boldsymbol{\theta}\big{|}\,C_{\text{twz}}(\boldsymbol{\theta})>0\big{\\}}$,
for all $\epsilon\in(0,\epsilon_{0}]$.

Lemma IV.3 is used in the following proposition whose proof is given in
Supplementary Material.

###### Proposition IV.4

Consider the optical tweezer-based unfolding dynamics in (15). The folded
conformation $\boldsymbol{\theta}^{\ast}$ is an unstable equilibrium for (15)
if

$P(\boldsymbol{\theta}):=\frac{\partial
C_{\text{twz}}}{\partial\boldsymbol{\theta}}\mathbf{f}^{\text{unfold}}(\boldsymbol{\theta})>0\text{
for all }\boldsymbol{\theta}\in\mathcal{C}_{\text{twz}}^{+},$ (17)

where $\mathcal{C}_{\text{twz}}^{+}$ is defined in Lemma IV.3, and

$\displaystyle\frac{\partial
C_{\text{twz}}}{\partial\boldsymbol{\theta}}=2\Delta\mathbf{r}_{{}_{\text{NC}}}(\boldsymbol{\theta},\boldsymbol{\theta}^{\ast})^{\top}\mathbf{M}(\boldsymbol{\theta}^{\ast})\frac{\partial\mathbf{r}_{\text{NC}}}{\partial\boldsymbol{\theta}},$
(18)

where
$\mathbf{M}(\boldsymbol{\theta}^{\ast}):=|\mathbf{r}_{{}_{\text{NC}}}(\boldsymbol{\theta}^{\ast})|^{2}\cos^{2}(\alpha_{c})\mathbf{I}_{3}-\mathbf{r}_{{}_{\text{NC}}}(\boldsymbol{\theta}^{\ast})\mathbf{r}_{{}_{\text{NC}}}(\boldsymbol{\theta}^{\ast})^{\top}$.

The condition given by (17) can be numerically checked by investigating the
Hessian of the function $P(\cdot)$ at the folded conformation under study as
follows.

###### Proposition IV.5

Consider the folded protein molecule conformation $\boldsymbol{\theta}^{\ast}$
for the closed-loop dynamics in (15) and the set
$\mathcal{C}_{\text{twz}}^{+}$ in Lemma IV.3. Considering the Hessian matrix
$\text{Hess}(P)|_{\boldsymbol{\theta}^{\ast}}$, the condition given by (17)
holds if and only if
$\delta\boldsymbol{\theta}^{\top}\text{Hess}(P)|_{\boldsymbol{\theta}^{\ast}}\delta\boldsymbol{\theta}>0$
for all $\delta\boldsymbol{\theta}$ such that
$\boldsymbol{\theta}^{\ast}+\delta\boldsymbol{\theta}\in\mathcal{C}_{\text{twz}}^{+}\cap\mathcal{B}_{\boldsymbol{\theta}^{\ast}}(\epsilon)$
for some $\epsilon>0$.

###### Remark IV.6

The condition in Proposition IV.5, which relies on numerical computation of
$\text{Hess}(P)$ at the folded conformation $\boldsymbol{\theta}^{\ast}$, can
be used for real-time control of optical tweezers and AFMs in protein
unfolding applications. In particular, to approximate each entry
$\tfrac{\partial^{2}P}{\partial\theta_{i}\partial\theta_{j}}$, one can use

$\tfrac{P\big{(}\boldsymbol{\theta}^{\ast}+\varepsilon_{i}\mathbf{1}_{i}+\varepsilon_{j}\mathbf{1}_{j}\big{)}-P\big{(}\boldsymbol{\theta}^{\ast}+\varepsilon\mathbf{1}_{i}\big{)}-P\big{(}\boldsymbol{\theta}^{\ast}+\varepsilon\mathbf{1}_{j}\big{)}+P(\boldsymbol{\theta}^{\ast})}{\varepsilon_{i}\varepsilon_{j}},$
(19)

where $P(\boldsymbol{\theta}^{\ast})=0$ because
$\mathbf{f}^{\text{unfold}}(\boldsymbol{\theta}^{\ast})=\mathbf{0}$ (see
Supplementary Material for analytical expression of these entries). Moreover,
$\varepsilon_{i}$, $\varepsilon_{j}$ are some small positive constants and the
vectors $\mathbf{1}_{i}$, $\mathbf{1}_{j}$ are the $i$-th, $j$-th columns of
the identity matrix $\mathbf{I}_{2N}$, respectively (see, e.g., [31]). For
real-time implementation, which is beyond the scope of the current article,
there is a need for high-speed scanning and high-bandwidth control at
nanoscale, where specialized DSP/FPGA-based embedded hardware have been
developed (see, e.g., [32, 24]).

###  IV-C Synthesis of destabilizing control inputs for unfolding

In this section we present a class of CCFs for the KCM-based molecular
dynamics. We then utilize these CCFs in the Artstein-Sontag universal formula,
which was first presented in [21] in the context of CCFs, to synthesize
destabilizing state feedback control torques for folded protein conformations.
The following propositions provide two types of CCFs for the KCM-based
dynamics of protein molecules, where the CCF in the second proposition is the
candidate CF given by (16).

###### Proposition IV.7

Consider the KCM-based dynamics in (7) and a folded conformation
$\boldsymbol{\theta}^{\ast}$. Consider the function
$C:\mathcal{B}_{\boldsymbol{\theta}^{\ast}}(\epsilon_{0})\to\mathbb{R}$ given
by

$\displaystyle
C(\boldsymbol{\theta})=g(|\Delta\mathbf{r}_{\text{NC}}(\boldsymbol{\theta},\boldsymbol{\theta}^{\ast})|^{2}),$
(20)

where $g:\mathbb{R}_{+}\to\mathbb{R}_{+}$ is any smooth strictly monotonic
function such that $g(0)=0$. The function $C(\cdot)$ is a CCF for the
nonlinear control-affine system in (7).

###### Proof:

Consider the set
$\mathcal{C}^{+}:=\\{\boldsymbol{\theta}\in\mathcal{B}_{\boldsymbol{\theta}^{\ast}}(\epsilon_{0}):C(\boldsymbol{\theta})>0\\}$.
From (20), we have $C(\boldsymbol{\theta}^{\ast})=0$. Also,
$\mathcal{C}^{+}\cap\mathcal{B}_{\boldsymbol{\theta}^{\ast}}(\epsilon)\neq\emptyset$
for all $\epsilon\in(0,\epsilon_{0}]$. Furthermore, $\tfrac{\partial
C}{\partial\boldsymbol{\theta}}=g^{\prime}(|\Delta\mathbf{r}_{\text{NC}}(\boldsymbol{\theta},\boldsymbol{\theta}^{\ast})|^{2})\Delta\mathbf{r}_{\text{NC}}(\boldsymbol{\theta},\boldsymbol{\theta}^{\ast})^{\top}\tfrac{\partial\mathbf{r}_{\text{NC}}}{\partial\boldsymbol{\theta}},$
where $g^{\prime}(.)$ is the derivative of $g(\cdot)$. We claim that

$\partial\mathcal{C}^{+}\cap\mathcal{C}^{+}\cap\mathcal{B}_{\boldsymbol{\theta}^{\ast}}(\epsilon)=\emptyset,$
(21)

where $\partial\mathcal{C}^{+}:=\\{\boldsymbol{\theta}:\big{|}\tfrac{\partial
C}{\partial\boldsymbol{\theta}}\big{|}=0\\}$. To see this, note that

$\boldsymbol{\theta}\in\partial\mathcal{C}^{+}\text{ if and only if
}\Delta\mathbf{r}_{\text{NC}}(\boldsymbol{\theta},\boldsymbol{\theta}^{\ast})^{\top}\tfrac{\partial\mathbf{r}_{\text{NC}}}{\partial\boldsymbol{\theta}}=\mathbf{0}.$
(22)

On the set $\mathcal{C}^{+}$,
$\Delta\mathbf{r}_{\text{NC}}(\boldsymbol{\theta},\boldsymbol{\theta}^{\ast})\neq\mathbf{0}$.
Therefore, the relation in (22) holds if and only if
$\Delta\mathbf{r}_{\text{NC}}(\boldsymbol{\theta},\boldsymbol{\theta}^{\ast})^{\top}\tfrac{\partial\mathbf{r}_{\text{NC}}}{\partial\theta_{j}}=\mathbf{0}$
for all $1\leq j\leq 2N$. Due to the kinematic structure of the protein
molecule, which can be classified as a manipulator with hyper degrees of
freedom (see [33]), we have

$\displaystyle\frac{\partial\mathbf{r}_{\text{NC}}}{\partial\theta_{j}}=\Xi(\boldsymbol{\theta},\mathbf{u}_{j}^{0})\mathbf{u}_{j}^{0}\times(\mathbf{r}_{\text{NC}}(\boldsymbol{\theta})-\mathbf{r}_{j}(\boldsymbol{\theta})),\,1\leq
j\leq 2N,$ (23)

where $\Xi$ is given by (2) and $\mathbf{r}_{j}(\boldsymbol{\theta})$ is the
position vector of the backbone chain atom ($\text{C}_{\alpha}$ atom for even
$j$ and $N$ atom for odd $j$) in the $j$-th peptide plane. Therefore, the
relation in (22) holds if and only if the nonzero vector
$\Delta\mathbf{r}_{\text{NC}}(\boldsymbol{\theta},\boldsymbol{\theta}^{\ast})$
is perpendicular to the collection of $2N$ three-dimensional vectors
$\frac{\partial\mathbf{r}_{\text{NC}}}{\partial\theta_{j}}$. This geometric
relation holds only if the protein molecule conformation $\boldsymbol{\theta}$
is such that all the amino-acid linkages are co-located on the same two-
dimensional plane corresponding to an unfolded structure, which is impossible
in a neighborhood of a folded conformation $\boldsymbol{\theta}^{\ast}$. Due
to (21), we conclude that
$\underset{\mathbf{u}_{\text{c}}\in\mathbb{R}^{2N}}{\sup}\\{\tfrac{\partial
C}{\partial\boldsymbol{\theta}}(\mathcal{J}^{\top}(\boldsymbol{\theta})\mathcal{F}(\boldsymbol{\theta})+\mathbf{u}_{\text{c}})\\}>0$
for all $\boldsymbol{\theta}\in\mathcal{C}^{+}$. Therefore, $C(\cdot)$ is a
CCF for (7). ∎

###### Proposition IV.8

Consider the KCM-based dynamics in (7) and a folded conformation
$\boldsymbol{\theta}^{\ast}$. The function $C_{\text{twz}}(\cdot)$ in (16) is
a CCF for (7).

Having a family of CCFs afforded by Propositions IV.7 and IV.8, we can use the
following CCF-based Artstein-Sontag universal formula from [21]

$\mathbf{u}_{\text{c}}(\boldsymbol{\theta})=-\phi\big{[}a(\boldsymbol{\theta}),|B(\boldsymbol{\theta})|\big{]}B(\boldsymbol{\theta}),$
(24)

to synthesize unfolding control inputs that solve the PUCP formulated in
Section IV-A, where $a(\boldsymbol{\theta}):=\tfrac{\partial
C}{\partial\boldsymbol{\theta}}\mathcal{J}^{\top}(\boldsymbol{\theta})\mathcal{F}(\boldsymbol{\theta})$,
$B^{\top}(\boldsymbol{\theta}):=\tfrac{\partial
C}{\partial\boldsymbol{\theta}}$, and
$\phi(a,b):=\tfrac{a-\sqrt[p]{|a|^{p}+b^{2q}}}{b^{2}}$ if $b\neq 0$, and
$\phi(a,b):=0$ if $b=0$, with any $2q\geq p>1$, and $q>1$.

##  V Simulation Results

In this section we present simulation results to validate our proposed
approach to protein unfolding using Chetaev instability framework. We consider
a protein molecule chain consisting of $N-1=10$ peptide planes with a
$22$-dimensional dihedral angle space in (7). Our implementation follows the
PROTOFOLD I guidelines [10, 12].

We have performed the unfolding simulations on the folded protein conformation
$\boldsymbol{\theta}^{\ast}$111$\boldsymbol{\theta}^{\ast}=$ [1.34 1.37 1.26
1.29 1.42 1.73 1.65 1.65 1.46 1.62 1.49 2.01 1.31 0.99 1.98 1.9 1.59 1.56 1.57
0.93 1.22 1.29]⊤ in radians. that has been obtained using the KCM-based
iteration in [10]. The candidate Chetaev function
$C_{\text{twz}}:\mathbb{R}^{22}\to\mathbb{R}_{+}$ is chosen according to (16)
with $\alpha_{C}=\tfrac{\pi}{4}$. To visualize this function, we have plotted
it using the dihedral angle vectors of the form
$\boldsymbol{\theta}=\boldsymbol{\theta}^{\ast}+[\delta\theta_{1},\delta\theta_{2},\delta\theta_{3},0,\cdots,0]^{\top}$,
where the triplet $(\delta\theta_{1},\delta\theta_{2},\delta\theta_{3})$
belong to the three-dimensional sphere of radius $0.1$ radians and centered at
the origin. Figure 2 depicts both the folded conformation at
$\boldsymbol{\theta}^{\ast}$ and the visualization of the candidate Chetaev
function when translated to the origin.

Using the setup above, we conduct two numerical simulations. The first one
entails the optical tweezer-based unfolding while the second one entails an
unfolding simulation using the Artstein-Sontag universal formula in (24) with
$p=q=2$ and $C_{\text{twz}}$ in Proposition IV.8 as the CCF. In the first
simulation, where the numerical condition of Proposition IV.5 is verified, we
let the optical tweezer displacement be
$\mathbf{x}_{\text{twz}}=x_{0}\tfrac{\mathbf{r}_{\text{NC}}(\boldsymbol{\theta}^{\ast})}{\|\mathbf{r}_{\text{NC}}(\boldsymbol{\theta}^{\ast})\|}$,
where $x_{0}$ is chosen to be $51$ nm. In other words, we elongated the
protein molecule along the vector that connects the $N$-terminus to the
$C$-terminus. Furthermore, we utilized a modulated optical trap stiffness of
$\kappa(\boldsymbol{\theta})=\kappa_{0}|\mathbf{r}_{\text{NC}}(\boldsymbol{\theta}^{\ast})-\mathbf{r}_{\text{NC}}(\boldsymbol{\theta})|^{2}$,
where $\kappa_{0}=0.16\text{ pN/nm}$. Figure 3 depicts the free energy of the
molecule under the optical tweezer-based and the CCF-based unfolding control
inputs as well as several protein conformations along their respective
unfolding pathways (see Supplementary Material for further discussion).

Figure 2: Visualization of the candidate Chetaev function
$C_{\text{twz}}:\mathbb{R}^{22}\to\mathbb{R}$ based on the depicted folded
conformation at $\boldsymbol{\theta}^{\ast}$. Figure 3: Energy profiles from
unfolding numerical simulations under optical tweezer-based control inputs
(red curve; molecule labeled by TWZ) and a theoretically-motivated input
generated by the Artstein-Sontag universal formula (blue curve; molecule
labeled by CCF).

##  VI Conclusion

This paper formulated the important biochemical process of KCM-based protein
unfolding as a destabilizing control analysis/synthesis problem. In light of
this formulation and using the Chetaev instability framework, numerical
conditions on optical tweezer forces that lead to protein unfolding have been
derived. Furthermore, after presenting two types of CCFs, a theoretically-
motivated CCF-based Artstein-Sontag unfolding input has been derived. These
findings lead us to further research avenues such as encoding entropy-loss
constraints for protein unfolding using the obtained CCFs, investigating
folding pathways for two-state proteins, and studying protein
folding/unfolding in solutions.

## References

  * [1] A. V. Finkelstein and O. Ptitsyn, _Protein physics: a course of lectures_. Elsevier, 2016.
  * [2] F. Bergasa-Caceres and H. A. Rabitz, “Interdiction of protein folding for therapeutic drug development in SARS CoV-2,” _J. Phys. Chem. B_ , vol. 124, no. 38, pp. 8201–8208, 2020.
  * [3] J. Fang, A. Mehlich, N. Koga, J. Huang, R. Koga, X. Gao, C. Hu, C. Jin, M. Rief, J. Kast, D. Baker, and H. Li, “Forced protein unfolding leads to highly elastic and tough protein hydrogels,” _Nature Commun._ , vol. 4, no. 1, pp. 1–10, 2013.
  * [4] L. Fu, H. Wang, and H. Li, “Harvesting mechanical work from folding-based protein engines: from single-molecule mechanochemical cycles to macroscopic devices,” _CCS Chem._ , vol. 1, no. 1, pp. 138–147, 2019.
  * [5] M. AlQuraishi, “A watershed moment for protein structure prediction,” _Nature_ , vol. 577, pp. 627–628, 2020.
  * [6] A. R. Fersht, “AlphaFold–A personal perspective on the impact of machine learning,” _J. Mol. Biol._ , vol. 433, no. 20, p. 167088, 2021.
  * [7] L. Heo, C. F. Arbour, and M. Feig, “Driven to near-experimental accuracy by refinement via molecular dynamics simulations,” _Proteins: Struct. Funct. Bioinform._ , vol. 87, no. 12, pp. 1263–1275, 2019.
  * [8] C. Alvarado and K. Kazerounian, “On the rotational operators in protein structure simulations,” _Prot. Eng._ , vol. 16, no. 10, pp. 717–720, 2003\.
  * [9] K. Kazerounian, K. Latif, K. Rodriguez, and C. Alvarado, “Nano-kinematics for analysis of protein molecules,” _ASME J. Mech. Des._ , vol. 127, no. 4, pp. 699–711, 2005.
  * [10] K. Kazerounian, K. Latif, and C. Alvarado, “PROTOFOLD: A successive kinetostatic compliance method for protein conformation prediction,” _ASME J. Mech. Des._ , vol. 127, no. 4, pp. 712–717, 2005.
  * [11] P. Tavousi, M. Behandish, H. T. Ilieş, and K. Kazerounian, “PROTOFOLD II: Enhanced model and implementation for kinetostatic protein folding,” _ASME J. Nanotechnol. Eng. Med._ , vol. 6, no. 3, 2015.
  * [12] P. Tavousi, “On the systematic design and analysis of artificial molecular machines,” Ph.D. dissertation, University of Connecticut, 2016.
  * [13] Z. Shahbazi, H. T. Ilieş, and K. Kazerounian, “Hydrogen bonds and kinematic mobility of protein molecules,” _ASME J. Mech. Robot._ , vol. 2, no. 2, 2010.
  * [14] P. Tavousi, K. Kazerounian, and H. Ilieş, “Synthesizing functional mechanisms from a link soup,” _ASME J. Mech. Des._ , vol. 138, no. 6, 2016\.
  * [15] M. T. Chorsi, P. Tavousi, C. Mundrane, V. Gorbatyuk, K. Kazerounian, and H. Ilieş, “Kinematic design of functional nanoscale mechanisms from molecular primitives,” _ASME J. Micro-Nano-Manuf._ , vol. 9, no. 2, p. 021005, 2021.
  * [16] A. Mohammadi and M. W. Spong, “Quadratic optimization-based nonlinear control for protein conformation prediction,” _IEEE Control Syst. Lett._ , vol. 6, pp. 373–378, 2022, doi: 10.1109/LCSYS.2021.3076869.
  * [17] H.-J. Su, J. Parker, K. Kazerounian, and H. Ilieş, “A comparison of kinetostatic and multibody dynamics models for simulating protein structures,” in _ASME Int. Des. Eng. Tech. Conf. Comput. Inf. Eng. Conf. (IDETC-CIE)_ , vol. 48043, 2007, pp. 897–904.
  * [18] N. G. Chetaev, _The stability of motion_. New York: Pergamon Press, 1961.
  * [19] D. V. Efimov and A. L. Fradkov, “Oscillatority of nonlinear systems with static feedback,” _SIAM J. Contr. Optim._ , vol. 48, no. 2, pp. 618–640, 2009.
  * [20] D. Efimov and W. Perruquetti, “Oscillating system design applying universal formula for control,” in _2011 50 th IEEE Conf. Dec. Contr. Eur. Contr. Conf. (CDC-ECC)_, 2011, pp. 1747–1752.
  * [21] D. Efimov, W. Perruquetti, and M. Petreczky, “On necessary conditions of instability and design of destabilizing controls,” in _53 rd IEEE Conf. Dec. Contr. (CDC)_. IEEE, 2014, pp. 3915–3917.
  * [22] P. Braun, L. Grüne, and C. M. Kellett, “Complete instability of differential inclusions using Lyapunov methods,” in _57 th IEEE Conf. Dec. Contr. (CDC)_. IEEE, 2018, pp. 718–724.
  * [23] C. Bustamante, L. Alexander, K. Maciuba, and C. M. Kaiser, “Single-molecule studies of protein folding with optical tweezers,” _Annu. Rev. Biochem._ , vol. 89, pp. 443–470, 2020.
  * [24] M. R. Ragazzon, J. T. Gravdahl, and K. Y. Pettersen, “Model-based identification of nanomechanical properties in atomic force microscopy: Theory and experiments,” _IEEE Trans. Contr. Syst. Technol._ , vol. 27, no. 5, pp. 2045–2057, 2018.
  * [25] N. Thammawongsa, F. D. Zainol, S. Mitatha, J. Ali, and P. P. Yupapin, “Nanorobot controlled by optical tweezer spin for microsurgical use,” _IEEE Trans. Nanotechnol._ , vol. 12, no. 1, pp. 29–34, 2012.
  * [26] Y. Wen, H. Lu, Y. Shen, and H. Xie, “Nanorobotic manipulation system for 360o characterization atomic force microscopy,” _IEEE Trans. Ind. Electron._ , vol. 67, no. 4, pp. 2916–2924, 2019.
  * [27] D. B. Adolf and M. D. Ediger, “Brownian dynamics simulations of local motions in polyisoprene,” _Macromolecules_ , vol. 24, no. 21, pp. 5834–5842, 1991\.
  * [28] E. Schonbrun and K. Crozier, “Spring constant modulation in a zone plate tweezer using linear polarization,” _Opt. Lett._ , vol. 33, no. 17, pp. 2017–2019, 2008.
  * [29] X. Li and C. C. Cheah, “Tracking control for optical manipulation with adaptation of trapping stiffness,” _IEEE Trans. Contr. Syst. Technol._ , vol. 24, no. 4, pp. 1432–1440, 2015.
  * [30] Y. Lin, E. D. Sontag, and Y. Wang, “A smooth converse Lyapunov theorem for robust stability,” _SIAM J. Contr. Optim._ , vol. 34, no. 1, pp. 124–160, 1996.
  * [31] J. E. Dennis Jr and R. B. Schnabel, _Numerical methods for unconstrained optimization and nonlinear equations_. SIAM, 1996.
  * [32] Y. Sun, Y. Fang, Y. Zhang, and X. Dong, “Field programmable gate array (FPGA) based embedded system design for AFM real-time control,” in _2010 IEEE Int. Conf. Contr. Appl. (CCA)_. IEEE, 2010, pp. 245–250.
  * [33] H. Mochiyama and H. Kobayashi, “The shape Jacobian of a manipulator with hyper degrees of freedom,” in _Proc. 1999 IEEE Int. Conf. Robot. Automat. (ICRA)_ , vol. 4. IEEE, 1999, pp. 2837–2842.

## Supplementary Material To “Chetaev Instability Framework for Kinetostatic
Compliance-Based Protein Unfolding”

Notation. In these supplementary notes, we will number the equations using
(s#), the references using [S#], and the figures using SF#. Therefore, (s1)
refers to Equation 1 in the supplementary notes while (1) refers to Equation 1
in the original article. The same holds true for the references and the
figures.

###  VI-A Schematic of protein folding/unfolding against the free energy
landscape

Figure SF1 depicts a schematic of folding/unfolding against the free energy
landscape of the protein molecule.

Figure SF1: Protein folding/unfolding against the free energy landscape of the
molecule.

###  VI-B Principles of operation of optical tweezers

Optical tweezers, which were discovered and developed by Ashkin [S2], operate
based on the principle of force exertion by light on matter due to the change
in momentum that light carries. Figure SF2 depicts a schematic of optical
tweezer-based protein unfolding. Optical tweezers can be utilized for
unfolding protein molecules by chemically attaching a latex bead (microsphere)
to the protein under study. The bead could subsequently be trapped and
utilized both as a handle for manipulating the protein molecule and as a tool
for measuring displacements at a very small scale. Under the assumptions of a
nearly Gaussian beam and a locally harmonic potential energy for a given
optical tweezer, the optical trap can be modeled as a Hookean spring [23].

Figure SF2: Schematic of optical tweezer-based protein unfolding (recreated
from [S1]).

###  VI-C On the choice of the candidate CF

Figure SF3 represents the geometric interpretation associated with the three
dimensional vectors utilized for defining $C_{\text{twz}}(\cdot)$ in (16). The
vectors $\mathbf{r}_{\text{NC}}(\boldsymbol{\theta})$ and
$\mathbf{r}_{\text{NC}}(\boldsymbol{\theta}^{\ast})$ represent the “length” of
the molecule at a general conformation $\boldsymbol{\theta}$ and the folded
conformation $\boldsymbol{\theta}^{\ast}$, respectively. The positive level
sets of the candidate CF $C_{\text{twz}}(\cdot)$ in (16), which depends on
$\mathbf{r}_{\text{NC}}(\boldsymbol{\theta})$ and
$\mathbf{r}_{\text{NC}}(\boldsymbol{\theta}^{\ast})$, correspond to
conformations at which the protein molecule has been elongated from the folded
conformation along desired directions. Indeed, when $\dot{C}_{\text{twz}}>0$
along the trajectories of the unfolding dynamics on the positive level sets of
$C_{\text{twz}}(\cdot)$, the molecule is being elongated away from its folded
conformation.

Figure SF3: When $C_{\text{twz}}(\boldsymbol{\theta})=0$,
$\Delta\mathbf{r}_{\text{NC}}(\boldsymbol{\theta},\boldsymbol{\theta}^{\ast})$
belongs to a right circular cone with vertex at the origin, axis parallel to
$\mathbf{r}_{\text{NC}}(\boldsymbol{\theta}^{\ast})$, and aperture
$2\alpha_{C}$.

###  VI-D Proof of Technical Results

Proof of Lemma IV.3. Since
$\Delta\mathbf{r}_{\text{NC}}(\boldsymbol{\theta}^{\ast},\boldsymbol{\theta}^{\ast})=\mathbf{0}$,
we conclude that $C_{\text{twz}}(\boldsymbol{\theta}^{\ast})=0$. Let us define
the mapping

$\displaystyle
f:\mathbb{R}^{2N}\backslash\\{\boldsymbol{\theta}^{\ast}\\}\to\mathbb{R},\,\boldsymbol{\theta}\mapsto\cos^{-1}(\tfrac{\Delta\mathbf{r}_{\text{NC}}(\boldsymbol{\theta},\boldsymbol{\theta}^{\ast})^{\top}\mathbf{r}_{\text{NC}}(\boldsymbol{\theta}^{\ast})}{|\Delta\mathbf{r}_{\text{NC}}(\boldsymbol{\theta},\boldsymbol{\theta}^{\ast})||\mathbf{r}_{\text{NC}}(\boldsymbol{\theta}^{\ast})|}),$
(s1)

where $f(\boldsymbol{\theta})$ is the angle between
$\mathbf{r}_{\text{NC}}(\boldsymbol{\theta}^{\ast})$ and
$\Delta\mathbf{r}_{\text{NC}}(\boldsymbol{\theta},\boldsymbol{\theta}^{\ast})$.
We claim

$\mathcal{C}_{\text{twz}}^{+}=f^{-1}\big{(}(\alpha_{C},\tfrac{\pi}{2})\cap(-\tfrac{\pi}{2},-\alpha_{C})\big{)}.$
(s2)

Consider an arbitrary $\boldsymbol{\theta}_{0}\in\mathbb{R}^{2N}$ such that
$C_{\text{twz}}(\boldsymbol{\theta}_{0})>0$. From (16) and (s1), it can be
seen that

$C_{\text{twz}}(\boldsymbol{\theta}_{0})=|\Delta\mathbf{r}_{{}_{\text{NC}}}(\boldsymbol{\theta}_{0},\boldsymbol{\theta}^{\ast})|^{2}|\mathbf{r}_{{}_{\text{NC}}}(\boldsymbol{\theta}^{\ast})|^{2}(\cos^{2}(\alpha_{C})\text{-}\cos^{2}(f(\boldsymbol{\theta}_{0}))).$
(s3)

Therefore, $C_{\text{twz}}(\boldsymbol{\theta}_{0})>0$ if and only if
$\boldsymbol{\theta}_{0}\in\mathcal{C}_{\text{twz}}^{+}$. Since $f(\cdot)$ is
a continuous function, we conclude that $\mathcal{C}_{\text{twz}}^{+}$ in (s2)
is an open set and $\boldsymbol{\theta}^{\ast}$ belongs to the boundary of
$\mathcal{C}_{\text{twz}}^{+}$. Therefore,
$\mathcal{C}_{\text{twz}}^{+}\cap\mathcal{B}_{\boldsymbol{\theta}^{\ast}}(\epsilon)\neq\emptyset$
for all $\epsilon\in(0,\epsilon_{0}]$ for some $\epsilon_{0}>0$.
$\blacksquare$

Proof of Proposition IV.4. Considering the smooth candidate Chetaev function
in (16), the directional derivative
$DC_{\text{twz}}(\boldsymbol{\theta})\mathbf{f}^{\text{unfold}}(\boldsymbol{\theta})$
in the direction of $\mathbf{f}^{\text{unfold}}(\cdot)$ is given by
$\tfrac{\partial
C_{\text{twz}}}{\partial\boldsymbol{\theta}}\mathbf{f}^{\text{unfold}}(\boldsymbol{\theta})$,
where

$\displaystyle\frac{\partial
C_{\text{twz}}}{\partial\boldsymbol{\theta}}=2\bigg{\\{}|\mathbf{r}_{{}_{\text{NC}}}(\boldsymbol{\theta}^{\ast})|^{2}\cos^{2}(\alpha_{C})\Delta\mathbf{r}_{{}_{\text{NC}}}(\boldsymbol{\theta},\boldsymbol{\theta}^{\ast})^{\top}-$
(s4)
$\displaystyle\Delta\mathbf{r}_{{}_{\text{NC}}}(\boldsymbol{\theta},\boldsymbol{\theta}^{\ast})^{\top}\mathbf{r}_{{}_{\text{NC}}}(\boldsymbol{\theta}^{\ast})\mathbf{r}_{{}_{\text{NC}}}(\boldsymbol{\theta}^{\ast})^{\top}\bigg{\\}}\frac{\partial\mathbf{r}_{\text{NC}}}{\partial\boldsymbol{\theta}}.$

Therefore, due to Lemma IV.3, the statement of the proposition follows from
Theorem II.1. $\blacksquare$

Proof of Proposition IV.5. Given the $C^{\infty}$-smoothness of $P(\cdot)$ in
a neighborhood of $\boldsymbol{\theta}^{\ast}$, we can utilize the Taylor’s
theorem for multivariate functions to arrive at

$P(\boldsymbol{\theta})=P(\boldsymbol{\theta}^{\ast})+\frac{\partial
P}{\partial\boldsymbol{\theta}}\bigg{|}_{\theta^{\ast}}\delta\boldsymbol{\theta}+\delta\boldsymbol{\theta}^{\top}\text{Hess}(P)|_{\boldsymbol{\theta}^{\ast}}\delta\boldsymbol{\theta}+\text{
H.O.T},$ (s5)

where
$\boldsymbol{\theta}\in\mathcal{B}_{\boldsymbol{\theta}^{\ast}}(\epsilon)\cap\mathcal{V}^{+}$
for some $\epsilon>0$ and
$\delta\boldsymbol{\theta}:=\boldsymbol{\theta}-\boldsymbol{\theta}^{\ast}$.
It can be seen that $P(\boldsymbol{\theta}^{\ast})=0$ because
$\boldsymbol{\theta}^{\ast}$ is an equilibrium for the dynamics in (15).
Furthermore, $\tfrac{\partial
P}{\partial\boldsymbol{\theta}}|_{\boldsymbol{\theta}^{\ast}}=\mathbf{0}$
because $\mathbf{f}^{\text{unfold}}(\boldsymbol{\theta}^{\ast})=\mathbf{0}$
and $\tfrac{\partial
C_{\text{twz}}}{\partial\boldsymbol{\theta}}|_{\boldsymbol{\theta}^{\ast}}=\mathbf{0}$
according to (18). Therefore, $P(\boldsymbol{\theta})>0$ on
$\mathcal{B}_{\boldsymbol{\theta}^{\ast}}(\epsilon)\cap\mathcal{V}^{+}$ if and
only if the stated condition in the proposition holds. $\blacksquare$

Proof of Proposition IV.8. The proof is almost verbatim the proof of
Proposition IV.7 with the difference that the relation in (s6) should be
replaced with

$\boldsymbol{\theta}\in\partial\mathcal{C}^{+}\text{ if and only if
}\Delta\mathbf{r}_{\text{NC}}(\boldsymbol{\theta},\boldsymbol{\theta}^{\ast})^{\top}\mathbf{M}(\boldsymbol{\theta}^{\ast})\tfrac{\partial\mathbf{r}_{\text{NC}}}{\partial\boldsymbol{\theta}}=\mathbf{0}.$
(s6)

Since $\mathbf{M}(\boldsymbol{\theta}^{\ast})$ is a symmetric and non-singular
matrix with
$\det(\mathbf{M})=-\sin^{2}(\alpha_{c})|\mathbf{r}_{\text{NC}}(\boldsymbol{\theta}^{\ast})|^{6}$,
the relation in (s6) holds if and only if the nonzero vector
$\mathbf{M}(\boldsymbol{\theta}^{\ast})\Delta\mathbf{r}_{\text{NC}}(\boldsymbol{\theta},\boldsymbol{\theta}^{\ast})$
is perpendicular to the collection of $2N$ three-dimensional vectors
$\frac{\partial\mathbf{r}_{\text{NC}}}{\partial\theta_{j}}$. The rest of the
proof follows verbatim the proof of Proposition IV.7. $\blacksquare$

The Analytical Expression for the Entries of Hessian. The entries of the
Hessian of the function $P(\cdot)$ given by (17) at a folded conformation
$\boldsymbol{\theta}^{\ast}$ take the form

$\displaystyle\tfrac{\partial^{2}P}{\partial\theta_{j}\partial\theta_{k}}\big{|}_{\boldsymbol{\theta}^{\ast}}=$
$\displaystyle
2\sum_{i=1}^{2N}\tfrac{\partial\mathbf{r}_{\text{NC}}}{\partial\theta_{i}}(\boldsymbol{\theta}^{\ast})\mathbf{M}(\boldsymbol{\theta}^{\ast})\big{\\{}\tfrac{\partial\mathbf{r}_{\text{NC}}(\boldsymbol{\theta}^{\ast})}{\partial\theta_{j}}\tfrac{\partial
f_{i,\text{unfold}}(\boldsymbol{\theta}^{\ast})}{\partial\theta_{k}}+$ (s7)
$\displaystyle\tfrac{\partial\mathbf{r}_{\text{NC}}(\boldsymbol{\theta}^{\ast})}{\partial\theta_{k}}\tfrac{\partial
f_{i,\text{unfold}}(\boldsymbol{\theta}^{\ast})}{\partial\theta_{j}}\big{\\}},\;1\leq
j,\,k\leq 2N,$

where $\mathbf{M}(\boldsymbol{\theta}^{\ast})$ is defined in Proposition IV.4,
the vector $\mathbf{r}_{\text{NC}}(\boldsymbol{\theta})\in\mathbb{R}^{3}$
connects the nitrogen atom in the N-terminus to the carbon atom in the
C-terminus, and $f_{i,\text{unfold}}(\boldsymbol{\theta})$ is the $i$-th
element of the vector $\mathbf{f}^{\text{unfold}}(\boldsymbol{\theta})$ in
(15).

###  VI-E Control Chetaev Functions

In this section we provide some preliminaries on control Chetaev functions due
to Efimov and collaborators [S3]. The reader is referred to the line of work
by Efimov, Kellett, and collaborators [S3] and [S4] for further details.

Consider the nonlinear control-affine system

$\dot{\mathbf{x}}=\mathbf{F}(\mathbf{x})+\mathbf{G}(\mathbf{x})\mathbf{u},$
(s8)

where $\mathbf{x}\in\mathbb{R}^{n}$ is the state vector,
$\mathbf{u}\in\mathbb{R}^{m}$ is the control input,
$\mathbf{F}(\mathbf{x}^{\ast})=\mathbf{0}$, and
$\mathbf{F}:\mathbb{R}^{n}\to\mathbb{R}^{n}$ and
$\mathbf{G}:\mathbb{R}^{m}\to\mathbb{R}^{n}$ are locally Lipschitz continuous
functions. A smooth function
$V:\mathcal{B}_{\mathbf{x}^{\ast}}(\epsilon)\to\mathbb{R}$, such that
$V(\mathbf{x}^{\ast})=0$ and
$\mathcal{V}^{+}\cap\mathcal{B}_{\mathbf{x}^{\ast}}(\epsilon)\neq\emptyset$,
where $\mathcal{V}^{+}:=\\{\mathbf{x}:V(\mathbf{x})>0\\}$ and for any
$\epsilon\in(0,\epsilon_{0}]$, is called a CCF for the control system in (s8)
if

$\sup\limits_{\mathbf{u}\in\mathbb{R}^{m}}\\{D^{-}V(\mathbf{x})\mathbf{F}(\mathbf{x})+(D^{-}V(\mathbf{x})\mathbf{G}(\mathbf{x}))^{\top}\mathbf{u}\\}>0,$
(s9)

for all $\mathbf{x}\in\mathcal{V}^{+}$. As it is argued in [S3], if for all
$\mathbf{x}\in\mathcal{V}^{+}$ at which
$|D^{-}V(\mathbf{x})\mathbf{G}(\mathbf{x})|=0$, it holds that
$|D^{-}V(\mathbf{x})\mathbf{F}(\mathbf{x})|>0$, then $V(\cdot)$ is a CCF for
(s8). Furthermore, in the special case when
$\mathbf{G}(\mathbf{x})=\mathbf{I}_{n\times n}$ and
$\mathbf{u}\in\mathbb{R}^{n}$, if $|D^{-}V(\mathbf{x})|\neq 0$ for all
$\mathbf{x}\in\mathcal{V}^{+}$, then $V(\cdot)$ is a CCF for (s8).

## Supplementary References

[S1] J. L. Killian, F. Ye, and M. D. Wang, “Optical tweezers: a force to be
reckoned with,” _Cell_ , vol. 175, no. 6, pp. 1445–1448, 2018.

[S2] A. Ashkin, “Acceleration and trapping of particles by radiation
pressure,” _Phys. Rev. Lett._ , vol. 24, no. 4, p. 156, 1970.

[S3] D. Efimov, W. Perruquetti, and M. Petreczky, “On necessary conditions of
instability and design of destabilizing controls,” in _53rd IEEE Conf. Dec.
Contr. (CDC)._ IEEE, 2014, pp. 3915–3917.

[S4] P. Braun, L. Grüne, and C. M. Kellett, “Complete instability of ¨
differential inclusions using Lyapunov methods,” in _57th IEEE Conf. Dec.
Contr. (CDC)._ IEEE, 2018, pp. 718–724.


# On the partial sums of Vilenkin-Fourier series

George Tephnadze  G. Tephnadze, Department of Mathematics, Faculty of Exact
and Natural Sciences, Tbilisi State University, Chavchavadze str. 1, Tbilisi
0128, Georgia<EMAIL_ADDRESS>
###### Abstract.

The main aim of this paper is to investigate weighted maximal operators of
partial sums of Vilenkin-Fourier series. We also use our results to prove
approximation and strong convergence theorems on the martingale Hardy spaces
$H_{p},$ when $0<p\leq 1.$

The research was supported by Shota Rustaveli National Science Foundation
grant no.13/06 (Geometry of function spaces, interpolation and embedding
theorems

2010 Mathematics Subject Classification. 42C10.

Key words and phrases: Vilenkin system, partial sums, martingale Hardy space,
modulus of continuity, convergence.

##  1\.  Introduction

It is well-known that Vilenkin system forms not basis in the space
$L_{1}\left(G_{m}\right).$ Moreover, there is a function in the martingale
Hardy space $H_{1}\left(G_{m}\right),$ such that the partial sums of $f$ are
not bounded in $L_{1}\left(G_{m}\right)$-norm, but partial sums $S_{n}$ of the
Vilenkin-Fourier series of a function $f\in L_{1}\left(G_{m}\right)$
convergence in measure [12].

Uniform convergence and some approximation properties of partial sums in
$L_{1}\left(G_{m}\right)$ norms was investigate by Goginava [8] (see also
[9]). Fine [3] has obtained sufficient conditions for the uniform convergence
which are in complete analogy with the Dini-Lipschits conditions. Guličev [13]
has estimated the rate of uniform convergence of a Walsh-Fourier series using
Lebesgue constants and modulus of continuity. Uniform convergence of
subsequence of partial sums was investigate also in [7]. This problem has been
considered for Vilenkin group $G_{m}$ by Fridli [4], Blahota [2] and Gát [6].

It is also known that subsequence $S_{n_{k}}$ is bounded from
$L_{1}\left(G_{m}\right)$ to $L_{1}\left(G_{m}\right)$ if and only if $n_{k}$
has uniformly bounded variation and subsequence of partial sums $S_{M_{n}}$ is
bounded from the martingale Hardy space $H_{p}\left(G_{m}\right)$ to the
Lebesgue space $L_{p}\left(G_{m}\right),$ for all $p>0.$ In this paper we
shall prove very unexpected fact:

There exists a martingale $f\in H_{p}\left(G_{m}\right)\left(0<p<1\right),$
such that

$\underset{n\in\mathbb{N}}{\sup}\left\|S_{M_{n}+1}f\right\|_{L_{p,\infty}}=\infty.$

The reason of divergence of $S_{M_{n}+1}f$ is that when $0<p<1$ the Fourier
coefficients of $f\in H_{p}\left(G_{m}\right)$ are not bounded (See [17]).

In Gát [5] the following strong convergence result was obtained for all $f\in
H_{1}\left(G_{m}\right):$

$\underset{n\rightarrow\infty}{\lim}\frac{1}{\log
n}\overset{n}{\underset{k=1}{\sum}}\frac{\left\|S_{k}f-f\right\|_{1}}{k}=0,$

where $S_{k}f$ denotes the $k$-th partial sum of the Vilenkin-Fourier series
of $f.$ (For the trigonometric analogue see Smith [16], for the Walsh system
see Simon [14]). For the Vilenkin system Simon [15] proved that there is an
absolute constant $c_{p},$ depending only on $p,$ such that

(1)
$\overset{\infty}{\underset{k=1}{\sum}}\frac{\left\|S_{k}f\right\|_{p}^{p}}{k^{2-p}}\leq
c_{p}\left\|f\right\|_{H_{p}}^{p},$

for all $f\in H_{p}\left(G_{m}\right)$, where $0<p<1.$ The author [18] proved
that for any nondecreasing function $\Phi:\mathbb{N}\rightarrow[1,$ $\infty)$,
satisfying the condition
$\underset{n\rightarrow\infty}{\lim}\Phi\left(n\right)=+\infty,$ there exists
a martingale $f\in H_{p}\left(G_{m}\right),$ such that

(2) $\text{
}\underset{k=1}{\overset{\infty}{\sum}}\frac{\left\|S_{k}f\right\|_{L_{p,\infty}}^{p}\Phi\left(k\right)}{k^{2-p}}=\infty,\text{
for }0<p<1.$

Strong convergence theorems of two-dimensional partial sums was investigate by
Weisz [23], Goginava [10], Gogoladze [11], Tephnadze [19].

The main aim of this paper is to investigate weighted maximal operators of
partial sums of Vilenkin-Fourier series. We also use this results to prove
some approximation and strong convergence theorems on the martingale Hardy
spaces $H_{p}\left(G_{m}\right),$ when $0<p\leq 1.$

##  2\. Definitions and Notations

Let $\mathbb{N}_{+}$ denote the set of the positive integers,
$\mathbb{N}:=\mathbb{N}_{+}\cup\\{0\\}.$

Let $m:=(m_{0,}$ $m_{1},...)$ denote a sequence of the positive integers not
less than 2.

Denote by

$Z_{m_{k}}:=\\{0,1,...,m_{k}-1\\}$

the additive group of integers modulo $m_{k}.$

Define the group $G_{m}$ as the complete direct product of the group
$Z_{m_{j}}$ with the product of the discrete topologies of $Z_{m_{j}}$‘s.

The direct product $\mu$ of the measures

$\mu_{k}\left(\\{j\\}\right):=1/m_{k}\text{ \qquad}(j\in Z_{m_{k}})$

is the Haar measure on $G_{m_{\text{ }}}$with $\mu\left(G_{m}\right)=1.$

If the sequence $m:=(m_{0,}$ $m_{1},...)$ is bounded than $G_{m}$ is called a
bounded Vilenkin group, else it is called an unbounded one.

The elements of $G_{m}$ represented by sequences

$x:=(x_{0},x_{1},...,x_{j},...)\qquad\left(\text{ }x_{k}\in Z_{m_{k}}\right).$

It is easy to give a base for the neighborhood of $G_{m}$

$I_{0}\left(x\right):=G_{m},$ $I_{n}(x):=\\{y\in G_{m}\mid
y_{0}=x_{0},...,y_{n-1}=x_{n-1}\\}\text{ }(x\in G_{m},\text{
}n\in\mathbb{N}).$

Denote $I_{n}:=I_{n}\left(0\right)$ for $n\in\mathbb{N}$ and
$\overline{I_{n}}:=G_{m}$ $\backslash$ $I_{n}$.

It is evident

(3) $\overline{I_{N}}=\overset{N-1}{\underset{s=0}{\bigcup}}I_{s}\backslash
I_{s+1}.$

If we define the so-called generalized number system based on $m$ in the
following way

$M_{0}:=1,\text{ \qquad}M_{k+1}:=m_{k}M_{k\text{ }}\ \qquad(k\in\mathbb{N})$

then every $n\in\mathbb{N}$ can be uniquely expressed as
$n=\overset{\infty}{\underset{k=0}{\sum}}n_{j}M_{j},$ where $n_{j}\in
Z_{m_{j}}$ $~{}(j\in\mathbb{N})$ and only a finite number of $n_{j}$‘s differ
from zero. Let $\left|n\right|:=\max$ $\\{j\in\mathbb{N},$ $n_{j}\neq 0\\}.$

Denote by $L_{1}\left(G_{m}\right)$ the usual (one dimensional) Lebesgue
space.

Next, we introduce on $G_{m}$ an orthonormal system which is called the
Vilenkin system.

At first define the complex valued function
$r_{k}\left(x\right):G_{m}\rightarrow\mathbb{C},$ the generalized Rademacher
functions as

$r_{k}\left(x\right):=\exp\left(2\pi ix_{k}/m_{k}\right)\text{
\qquad}\left(\iota^{2}=-1,\text{ }x\in G_{m},\text{ }k\in\mathbb{N}\right).$

Now define the Vilenkin system $\psi:=(\psi_{n}:n\in\mathbb{N})$ on $G_{m}$
as:

$\psi_{n}(x):=\overset{\infty}{\underset{k=0}{\Pi}}r_{k}^{n_{k}}\left(x\right)\text{
\qquad}\left(n\in\mathbb{N}\right).$

Specifically, we call this system the Walsh-Paley one if $m\equiv 2$.

The Vilenkin system is orthonormal and complete in
$L_{2}\left(G_{m}\right)\,$[1, 20].

Now we introduce analogues of the usual definitions in Fourier-analysis.

If $f\in L_{1}\left(G_{m}\right)$ we can establish the Fourier coefficients,
the partial sums of the Fourier series, the Dirichlet kernels with respect to
the Vilenkin system $\psi$ in the usual manner:

$\displaystyle\widehat{f}\left(k\right)$ $\displaystyle:$
$\displaystyle=\int_{G_{m}}f\overline{\psi}_{k}d\mu,\text{\thinspace}\left(\text{
}k\in\mathbb{N}\right),$ $\displaystyle S_{n}f$ $\displaystyle:$
$\displaystyle=\sum_{k=0}^{n-1}\widehat{f}\left(k\right)\psi_{k},\left(\text{
}n\in\mathbb{N}_{+},\text{ }S_{0}f:=0\right),$ $\displaystyle D_{n}$
$\displaystyle:$ $\displaystyle=\sum_{k=0}^{n-1}\psi_{k\text{ }},\text{
}\left(\text{ }n\in\mathbb{N}_{+}\text{ }\right).$

Recall that (see [1])

(4) $\quad
D_{M_{n}}\left(x\right)=\left\\{\begin{array}[]{l}M_{n},\text{\thinspace\thinspace\thinspace\thinspace
if\thinspace\thinspace}x\in I_{n}\\\
0,\text{\thinspace\thinspace\thinspace\thinspace\thinspace if
\thinspace\thinspace}x\notin I_{n}\end{array}\right.$

and

(5)
$D_{n}\left(x\right)=\psi_{n}(x)\left(\underset{j=0}{\overset{\infty}{\sum}}D_{M_{j}}\left(x\right)\overset{m_{j}-1}{\underset{u=m_{j}-n_{j}}{\sum}}r_{j}^{u}\left(x\right)\right).$

The norm (or quasinorm) of the space $L_{p}(G_{m})$ is defined by

$\left\|f\right\|_{p}:=\left(\int_{G_{m}}\left|f\right|^{p}d\mu\right)^{1/p}\qquad\left(0<p<\infty\right).$

The space $L_{p,\infty}\left(G_{m}\right)$ consists of all measurable
functions $f$ for which

$\left\|f\right\|_{L_{p},\infty}:=\underset{\lambda>0}{\sup}\lambda\mu\left(f>\lambda\right)^{1/p}<+\infty.$

The $\sigma$-algebra generated by the intervals
$\left\\{I_{n}\left(x\right):x\in G_{m}\right\\}$ will be denoted by
$\digamma_{n}$ $\left(n\in\mathbb{N}\right).$ Denote by
$f=\left(f_{n},n\in\mathbb{N}\right)$ a martingale with respect to
$\digamma_{n}$ $\left(n\in\mathbb{N}\right).$ (for details see e.g. [21]). The
maximal function of a martingale $f$ is defend by

$f^{\ast}=\sup_{n\in\mathbb{N}}\left|f^{\left(n\right)}\right|.$

In case $f\in L_{1}\left(G_{m}\right),$ the maximal functions are also be
given by

$f^{\ast}\left(x\right)=\sup_{n\in\mathbb{N}}\frac{1}{\left|I_{n}\left(x\right)\right|}\left|\int_{I_{n}\left(x\right)}f\left(u\right)d\mu\left(u\right)\right|$

For $0<p<\infty$ the Hardy martingale spaces $H_{p}\left(G_{m}\right)$ consist
of all martingales, for which

$\left\|f\right\|_{H_{p}}:=\left\|f^{\ast}\right\|_{p}<\infty.$

The dyadic Hardy martingale spaces $H_{p}$ $\left(G_{m}\right)$ for $0<p\leq
1$ have an atomic characterization. Namely the following theorem is true (see
[24]):

Theorem W: A martingale $f=\left(f_{n},\text{ }n\in\mathbb{N}\right)$ is in
$H_{p}\left(G_{m}\right)\left(0<p\leq 1\right)$ if and only if there exists a
sequence $\left(a_{k},\text{ }k\in\mathbb{N}\right)$ of p-atoms and a sequence
$\left(\mu_{k},\text{ }k\in\mathbb{N}\right)$ of a real numbers such that for
every $n\in\mathbb{N}$

(6) $\qquad\sum_{k=0}^{\infty}\mu_{k}S_{M_{n}}a_{k}=f_{n}$

and

$\qquad\sum_{k=0}^{\infty}\left|\mu_{k}\right|^{p}<\infty.$

Moreover,
$\left\|f\right\|_{H_{p}}\backsim\inf\left(\sum_{k=0}^{\infty}\left|\mu_{k}\right|^{p}\right)^{1/p}$,
where the infimum is taken over all decomposition of $f$ of the form (6).

Let $X=X(G_{m})$ denote either the space $L_{1}(G_{m}),$ or the space of
continuous functions $C(G_{m})$. The corresponding norm is denoted by
$\|.\|_{X}$. The modulus of continuity, when $X=C\left(G_{m}\right)$ and the
integrated modulus of continuity, where $X=L_{1}\left(G_{m}\right)$ are
defined by

$\omega\left(1/M_{n},f\right)_{X}=\sup\limits_{h\in
I_{n}}\left\|f\left(\cdot+h\right)-f\left(\cdot\right)\right\|_{X}.$

The concept of modulus of continuity in $H_{p}\left(G_{m}\right)\left(0<p\leq
1\right)$ can be defined in following way

$\omega\left(1/M_{n},f\right)_{H_{p}\left(G_{m}\right)}:=\left\|f-S_{M_{n}}f\right\|_{H_{p}\left(G_{m}\right)}.$

If $f\in L_{1}\left(G_{m}\right),$ then it is easy to show that the sequence
$\left(S_{M_{n}}\left(f\right):n\in\mathbb{N}\right)$ is a martingale.

If $f=\left(f_{n},n\in\mathbb{N}\right)$ is martingale then the Vilenkin-
Fourier coefficients must be defined in a slightly different manner:

$\widehat{f}\left(i\right):=\lim_{k\rightarrow\infty}\int_{G_{m}}f_{k}\left(x\right)\overline{\Psi}_{i}\left(x\right)d\mu\left(x\right).$

The Vilenkin-Fourier coefficients of $f\in L_{1}\left(G_{m}\right)$ are the
same as the martingale $\left(S_{M_{n}}\left(f\right):n\in\mathbb{N}\right)$
obtained from $f$ .

For the martingale $f$ we consider maximal operators

$\displaystyle S^{\ast}f$ $\displaystyle:$
$\displaystyle=\sup_{n\in\mathbb{N}}\left|S_{n}f\right|,$
$\displaystyle\widetilde{S}_{p}^{\ast}f$ $\displaystyle:$
$\displaystyle=\sup_{n\in\mathbb{N}}\frac{\left|S_{n}f\right|}{\left(n+1\right)^{1/p-1}\log^{\left[p\right]}\left(n+1\right)},\text{
}0<p\leq 1,$

where $\left[p\right]$ denotes integer part of $p.$

A bounded measurable function $a$ is p-atom, if there exist a dyadic interval
$I$, such that

$\int_{I}ad\mu=0,\text{ \ \ \ \
}\left\|a\right\|_{\infty}\leq\mu\left(I\right)^{-1/p},\text{ \ \ \ \ \
supp}\left(a\right)\subset I.$

##  3\. Formulation of Main Results

######  Theorem 1.

a) Let $0<p\leq 1.$ Then the

maximal operator $\overset{\sim}{S}_{p}^{\ast}$ is bounded from the Hardy
space $H_{p}\left(G_{m}\right)$ to the space $L_{p}\left(G_{m}\right).$

b) Let $0<p\leq 1$ and $\varphi:\mathbb{N}_{+}\rightarrow[1,$ $\infty)$ be a
nondecreasing function satisfying the condition

(7)
$\overline{\lim_{n\rightarrow\infty}}\frac{\left(n+1\right)^{1/p-1}\log^{\left[p\right]}\left(n+1\right)}{\varphi\left(n\right)}=+\infty.$

Then

$\sup_{n\in\mathbb{N}}\left\|\frac{S_{n}f}{\varphi\left(n\right)}\right\|_{L_{p,\infty}\left(G_{m}\right)}=\infty,\text{
for }0<p<1$

and

$\sup_{n\in\mathbb{N}}\left\|\frac{S_{n}f}{\varphi\left(n\right)}\right\|_{1}=\infty.$

######  Corollary 1.

(Simon [15]) Let $0<p<1\ $and $f\in H_{p}\left(G_{m}\right).$ Then there is an
absolute constant $c_{p},$ depends only $p,$ such that

$\overset{\infty}{\underset{k=1}{\sum}}\frac{\left\|S_{k}f\right\|_{p}^{p}}{k^{2-p}}\leq
c_{p}\left\|f\right\|_{H_{p}}^{p}.$

######  Theorem 2.

Let $0<p\leq 1,$ $f\in H_{p}\left(G_{m}\right)$ and $M_{k}<n\leq M_{k+1}$.
Then there is an absolute constant $c_{p},$ depends only $p,$ such that

$\left\|S_{n}\left(f\right)-f\right\|_{H_{p}\left(G_{m}\right)}\leq
c_{p}n^{1/p-1}\lg^{\left[p\right]}n\omega\left(\frac{1}{M_{k}},f\right)_{H_{p}\left(G_{m}\right)}.$

######  Theorem 3.

a) Let $0<p<1,$ $f\in H_{p}\left(G_{m}\right)$ and

$\omega\left(\frac{1}{M_{n}},f\right)_{H_{p}\left(G_{m}\right)}=o\left(\frac{1}{M_{n}^{1/p-1}}\right),\text{
as }n\rightarrow\infty.$

Then

$\left\|S_{k}\left(f\right)-f\right\|_{L_{p,\infty}\left(G_{m}\right)}\rightarrow
0,\,\,\,\text{when\thinspace\thinspace\thinspace}k\rightarrow\infty.$

b) For every $p\in\left(0,1\right)$ there exists martingale $f\in
H_{p}(G_{m})$, for which

$\omega\left(\frac{1}{M_{2n}},f\right)_{H_{p}(G_{m})}=O\left(\frac{1}{M_{2n}^{1/p-1}}\right),\text{
as }n\rightarrow\infty$

and

$\left\|S_{k}\left(f\right)-f\right\|_{L_{p,\infty}(G_{m})}\nrightarrow
0,\,\,\,\text{when\thinspace\thinspace\thinspace}k\rightarrow\infty$

######  Theorem 4.

Let $f\in H_{1}(G_{m})\ $and

$\omega\left(\frac{1}{M_{n}},f\right)_{H_{1}(G_{m})}=o\left(\frac{1}{n}\right),\text{
as }n\rightarrow\infty.$

Then

$\left\|S_{k}\left(f\right)-f\right\|_{1}\rightarrow
0,\,\,\,\text{when\thinspace\thinspace\thinspace}k\rightarrow\infty.$

b) There exists martingale $f\in H_{1}(G_{m})$ for which

$\omega\left(\frac{1}{M_{2M_{n}}},f\right)_{H_{1}(G_{m})}=O\left(\frac{1}{M_{n}}\right),\text{
as }n\rightarrow\infty$

and

$\left\|S_{k}\left(f\right)-f\right\|_{1}\nrightarrow
0\,\,\,\text{when\thinspace\thinspace\thinspace}k\rightarrow\infty.$

##  4\. Auxiliary propositions

######  Lemma 1.

[22] Suppose that an operator $T$ is sublinear and for some $0<p\leq 1$

$\int\limits_{\overline{I}}\left|Ta\right|^{p}d\mu\leq c_{p}<\infty,$

for every $p$-atom $a$, where $I$ denote the support of the atom. If $T$ is
bounded from $L_{\infty\text{ }}$ to $L_{\infty}.$ Then

$\left\|Tf\right\|_{p}\leq c_{p}\left\|f\right\|_{H_{p}(G_{m})}.$

######  Lemma 2.

[17] Let $n\in\mathbb{N}$ and $x\in I_{s}\backslash I_{s+1},$ $0\leq s\leq
N-1.$ Then

$\int_{I_{N}}\left|D_{n}\left(x-t\right)\right|d\mu\left(t\right)\leq\,\frac{cM_{s}}{M_{N}}.$

##  5\. Proof of the Theorems

Proof of Theorem 1. Since $\overset{\sim}{S}_{p}^{\ast}$ is bounded from
$L_{\infty}(G_{m})$ to $L_{\infty}(G_{m})$ by Lemma 1 we obtain that the proof
of theorem 1 will be complete, if we show that

$\int\limits_{\overline{I}_{N}}\left|\overset{\sim}{S}_{p}^{\ast}a\left(x\right)\right|^{p}d\mu\left(x\right)\leq
c<\infty,\text{ when }0<p\leq 1,$

for every $p$-atom $a,$ where $I$ denotes the support of the atom.

Let $a$ be an arbitrary $p$-atom with support$\ I$ and
$\mu\left(I\right)=M_{N}.$ We may assume that $I=I_{N}.$ It is easy to see
that $S_{n}\left(a\right)=0$ when $n\leq M_{N}.$ Therefore we can suppose that
$n>M_{N}.$

Since $\left\|a\right\|_{\infty}\leq M_{N}^{1/p}$ we can write

$\displaystyle\left|S_{n}\left(a\right)\right|\leq\int_{I_{N}}\left|a\left(t\right)\right|\left|D_{n}\left(x-t\right)\right|d\mu\left(t\right)$
$\displaystyle\leq$
$\displaystyle\left\|a\right\|_{\infty}\int_{I_{N}}\left|D_{n}\left(x-t\right)\right|d\mu\left(t\right)\leq
M_{N}^{1/p}\int_{I_{N}}\left|D_{n}\left(x-t\right)\right|d\mu\left(t\right).$

Let $0<p<1$ and $x\in I_{s}\backslash I_{s+1}.$ From Lemma 2 we get

(9)
$\frac{\left|S_{n}a\left(x\right)\right|}{\log^{\left[p\right]}\left(n+1\right)\left(n+1\right)^{1/p-1}}\leq\frac{cM_{N}^{1/p-1}M_{s}}{\log^{\left[p\right]}\left(n+1\right)\left(n+1\right)^{1/p-1}}.$

Combining (3) and (9) we obtain

$\displaystyle\int_{\overline{I_{N}}}\left|\widetilde{S}_{p}^{\ast}a\left(x\right)\right|^{p}d\mu\left(x\right)=\overset{N-1}{\underset{s=0}{\sum}}\int_{I_{s}\backslash
I_{s+1}}\left|\widetilde{S}_{p}^{\ast}a\left(x\right)\right|^{p}d\mu\left(x\right)$
$\displaystyle\leq$
$\displaystyle\frac{cM_{N}^{1-p}}{\log^{\left[p\right]p}\left(n+1\right)\left(n+1\right)^{1-p}}\overset{N-1}{\underset{s=0}{\sum}}\frac{M_{s}^{p}}{M_{s}}\leq\frac{cM_{N}^{1-p}N^{\left[p\right]}}{\log^{p\left[p\right]}\left(n+1\right)\left(n+1\right)^{1-p}}<c_{p}<\infty.$

Let $0<p<1.$ Applying (5), (5) and Theorem W we have

$\displaystyle\overset{\infty}{\underset{k=M_{N}}{\sum}}\frac{\left\|S_{k}a\right\|_{p}^{p}}{k^{2-p}}\leq\overset{\infty}{\underset{k=M_{N}}{\sum}}\frac{1}{k}\int_{\overline{I_{N}}}\left|\frac{S_{k}a\left(x\right)}{k^{1/p-1}}\right|^{p}d\mu\left(x\right)$
$\displaystyle+\overset{\infty}{\underset{k=M_{N}}{\sum}}\frac{M_{N}}{k^{2-p}}\int_{I_{N}}\left(\int_{I_{N}}\left|D_{k}\left(x-t\right)\right|d\mu\left(t\right)\right)^{p}d\mu\left(x\right)$
$\displaystyle\leq$ $\displaystyle
c_{p}M_{N}^{1-p}\overset{\infty}{\underset{k=M_{N}}{\sum}}\frac{1}{k^{2-p}}+c_{p}M_{N}^{1-p}\overset{\infty}{\underset{k=M_{N}}{\sum}}\frac{\log^{p}k}{k^{2-p}}\leq
c_{p}<\infty.$

Which complete the proof of corollary 1.

Let prove second part of Theorem 1. Let

$f_{n_{k}}\left(x\right)=D_{M_{2n_{k}+1}}\left(x\right)-D_{M_{{}_{2n_{k}}}}\left(x\right).\text{
}$

It is evident

$\widehat{f}_{n_{k}}\left(i\right)=\left\\{\begin{array}[]{l}\text{ }1,\text{
if }i=M_{{}_{2n_{k}}},...,M_{2n_{k}+1}-1\\\ \text{
}0,\text{otherwise.}\end{array}\right.$

Then we can write

(12)
$S_{i}f_{n_{k}}\left(x\right)=\left\\{\begin{array}[]{l}D_{i}\left(x\right)-D_{M_{{}_{2n_{k}}}}\left(x\right),\text{
if }i=M_{{}_{2n_{k}}}+1,...,M_{2n_{k}+1}-1,\\\ \text{
}f_{n_{k}}\left(x\right),\text{ if }i\geq M_{2n_{k}+1},\\\ 0,\text{
otherwise.}\end{array}\right.$

From (4) we get

(13)
$\left\|f_{n_{k}}\right\|_{H_{p}(G_{m})}=\left\|\sup\limits_{n\in\mathbb{N}}S_{M_{n}}\left(f_{n_{k}}\right)\right\|_{p}=\left\|D_{M_{2n_{k}+1}}-D_{M_{{}_{2n_{k}}}}\right\|_{p}\leq
c_{p}M_{{}_{2n_{k}}}^{1-1/p}.$

Let $0<p<1.$ Under condition (7) there exists positive integers $n_{k}$ such
that

$\lim_{k\rightarrow\infty}\frac{\left(M_{2n_{k}}+2\right)^{1/p-1}}{\varphi\left(M_{2n_{k}}+2\right)}=\infty,\text{
\ \ }0<p<1.$

Applying (4), (5) and (12) we can write

$\frac{\left|S_{M_{2n_{k}}+1}f_{n_{k}}\right|}{\varphi\left(M_{2n_{k}}+2\right)}=\frac{\left|D_{M_{2n_{k}}+1}-D_{M_{2_{n_{k}}}}\right|}{\varphi\left(M_{2n_{k}}+2\right)}=\frac{\left|w_{M_{2_{n_{k}}}}\right|}{\varphi\left(M_{2n_{k}}+2\right)}=\frac{1}{\varphi\left(M_{2n_{k}}+2\right)}.$

Hence we can write:

(14) $\mu\left\\{x\in
G_{m}:\frac{\left|S_{M_{2n_{k}}+1}f_{n_{k}}\left(x\right)\right|}{\varphi\left(M_{2n_{k}}+2\right)}\geq\frac{1}{\varphi\left(M_{2n_{k}}+2\right)}\right\\}=1.$

Combining (13) and (14) we have

$\displaystyle\frac{\frac{1}{\varphi\left(M_{2n_{k}}+2\right)}\left(\mu\left\\{x\in
G_{m}:\frac{\left|S_{M_{2n_{k}}+1}f_{n_{k}}\left(x\right)\right|}{\varphi\left(M_{2n_{k}}+2\right)}\geq\frac{1}{\varphi\left(M_{2n_{k}}+2\right)}\right\\}\right)^{1/p}}{\left\|f_{n_{k}}\left(x\right)\right\|_{H_{p}}}$
$\displaystyle\geq$
$\displaystyle\frac{1}{\varphi\left(M_{2n_{k}}+2\right)M_{{}_{2n_{k}}}^{1-1/p}}=\frac{\left(M_{{}_{2n_{k}}}+2\right)^{1/p-1}}{\varphi\left(M_{2n_{k}}+2\right)}\rightarrow\infty,\text{
when \ }k\rightarrow\infty.$

Now consider the case when $p=1.$ Under condition (7) there exists
$\left\\{n_{k}:k\geq 1\right\\},$ such that

$\lim_{k\rightarrow\infty}\frac{\log
q_{n_{k}}}{\varphi\left(q_{n_{k}}\right)}=\infty.$

Let $q_{n_{k}}$

$=M_{2n_{k}}+M_{2n_{k}-2}+M_{2}+M_{0}$ and $x\in I_{2s}\backslash I_{2s+1},$
$s=0,...,n_{k}.$ Combining (4) and (5) we have

$\displaystyle\left|D_{q_{n_{k}}}\left(x\right)\right|\geq\left|D_{M_{2s}}\left(x\right)\right|-\left|\underset{l=0}{\overset{s-2}{\sum}}r_{2l}^{m_{2l}-1}\left(x\right)D_{M_{2l}}\left(x\right)\right|$
$\displaystyle\geq$ $\displaystyle
M_{2s}-\underset{l=0}{\overset{s-2}{\sum}}M_{2l}\geq
M_{2s}-M_{2s-1}\geq\frac{M_{2s}}{2}.$

Hence

(15)
$\int_{G_{m}}\left|D_{q_{n_{k}}}\left(x\right)\right|d\mu\left(x\right)\geq\frac{1}{2}\underset{s=0}{\overset{n_{k}}{\sum}}\int_{I_{2s}\backslash
I_{2s+1}}M_{2s}d\mu\left(x\right)\geq
c\underset{s=0}{\overset{n_{k}}{\sum}}1\geq cn_{k}.$

From (12), (13) and (15) we have

$\displaystyle\frac{1}{\left\|f_{n_{k}}\left(x\right)\right\|_{H_{1}(G_{m})}}\int_{G_{m}}\frac{\left|S_{q_{n_{k}}}f_{n_{k}}\left(x\right)\right|}{\varphi\left(q_{n_{k}}\right)}d\mu\left(x\right)$
$\displaystyle\geq$
$\displaystyle\frac{1}{\left\|f_{n_{k}}\left(x\right)\right\|_{H_{1}(G_{m})}}\left(\int_{G_{m}}\frac{\left|D_{q_{n_{k}}}\left(x\right)\right|}{\varphi\left(q_{n_{k}}\right)}d\mu\left(x\right)-\int_{G_{m}}\frac{\left|D_{M_{{}_{2n_{k}}}}\left(x\right)\right|}{\varphi\left(q_{n_{k}}\right)}d\mu\left(x\right)\right)$
$\displaystyle\geq$
$\displaystyle\frac{c}{\varphi\left(q_{n_{k}}\right)}\left(\log
q_{n_{k}}-1\right)\geq\frac{c\log
q_{n_{k}}}{\varphi\left(q_{n_{k}}\right)}\rightarrow\infty,\text{ \qquad when
}k\rightarrow\infty.$

Which complete the proof of theorem 1.

Proof of Theorem 2. Let $0<p\leq 1$ and $M_{k}<n\leq M_{k+1}.$ Using Theorem 1
we have

$\left\|S_{n}f\right\|_{p}\leq
c_{p}n^{1/p-1}\log^{\left[p\right]}n\left\|f\right\|_{H_{p}(G_{m})}.$

Hence

$\displaystyle\left\|S_{n}f-f\right\|_{p}^{p}\leq\left\|S_{n}f-S_{M_{k}}f\right\|_{p}^{p}+\left\|S_{M_{k}}f-f\right\|_{p}^{p}=\left\|S_{n}\left(S_{M_{k}}f-f\right)\right\|_{p}^{p}$
$\displaystyle+\left\|S_{M_{k}}f-f\right\|_{p}^{p}\leq
c_{p}\left(n^{1-p}+1\right)\log^{p\left[p\right]}n\omega^{p}\left(\frac{1}{M_{k}},f\right)_{H_{p}(G_{m})}$

and

(16) $\left\|S_{n}f-f\right\|_{p}\leq
c_{p}n^{1/p-1}\log^{\left[p\right]}n\omega\left(\frac{1}{M_{k}},f\right)_{H_{p}(G_{m})}.$

Proof of Theorem 3. Let $0<p<1,$ $f\in H_{p}(G_{m})$ and

$\omega\left(\frac{1}{M_{2n}},f\right)_{H_{p}(G_{m})}=o\left(\frac{1}{M_{2n}^{1/p-1}}\right),\text{
as }n\rightarrow\infty.$

Using (16) we immediately get

$\left\|S_{n}f-f\right\|_{p}\rightarrow\infty,\text{ when
}n\rightarrow\infty.$

Let proof of second part of theorem 3. We set

$a_{k}\left(x\right)=\frac{M_{2k}^{1/p-1}}{\lambda}\left(D_{M_{2k+1}}\left(x\right)-D_{M_{2k}}\left(x\right)\right),$

where $\lambda=\sup_{n\in\mathbb{N}}m_{n}$ and

$f_{A}\left(x\right)=\underset{i=0}{\overset{A}{\sum}}\frac{\lambda}{M_{2i}^{1/p-1}}a_{i}(x).$

Since

(17)
$S_{M_{A}}a_{k}\left(x\right)=\left\\{\begin{array}[]{l}a_{k}\left(x\right),\text{
\ }2k\leq A,\\\ 0,\text{ \ \ \ \ \ \ \ \ }2k>A,\end{array}\right.$

and

$\text{supp}(a_{k})=I_{2k},\text{ \ \ }\int_{I_{2k}}a_{k}d\mu=0,\text{ \ \
}\left\|a_{k}\right\|_{\infty}\leq M_{2k}^{1/p-1}\cdot
M_{2k}=M_{2k}^{1/p}=\left(\text{supp }a_{k}\right)^{-1/p},$

if we apply Theorem W we conclude that $f\in H_{p}.$

It is easy to show that

$\displaystyle f-S_{M_{n}}f$ $\displaystyle=$
$\displaystyle\left(f^{\left(1\right)}-S_{M_{n}}f^{\left(1\right)},...,f^{\left(n\right)}-S_{M_{n}}f^{\left(n\right)},...,f^{\left(n+k\right)}-S_{M_{n}}f^{\left(n+k\right)}\right)$
$\displaystyle=$
$\displaystyle\left(0,...,0,f^{\left(n+1\right)}-f^{\left(n\right)},...,f^{\left(n+k\right)}-f^{\left(n\right)},...\right)$
$\displaystyle=$
$\displaystyle\left(0,...,0,\underset{i=n}{\overset{k}{\sum}}\frac{a_{i}(x)}{M_{i}^{1/p-1}},...\right),\text{
\ }k\in\mathbb{N}_{+}$

is martingale. Using (5) we get

$\omega(\frac{1}{M_{n}},f)_{H_{p}}\leq\sum\limits_{i=\left[n/2\right]+1}^{\infty}\frac{1}{M_{2i}^{1/p-1}}=O\left(\frac{1}{M_{n}^{1/p-1}}\right).$

where $\left[n/2\right]$ denotes integer part of $n/2.$ It is easy to show
that

(19) $\widehat{f}(j)=\left\\{\begin{array}[]{l}1,\,\ \text{if
\thinspace\thinspace}j\in\left\\{M_{2i},...,M_{2i+1}-1\right\\},\text{
}i=0,1,...\\\ 0,\text{ \thinspace\thinspace\thinspace if
\thinspace\thinspace\thinspace}j\notin\bigcup\limits_{i=0}^{\infty}\left\\{M_{2i},...,M_{2i+1}-1\right\\}.\text{
}\end{array}\right.$

Using (19) we have

$\limsup\limits_{k\rightarrow\infty}\|f-S_{M_{2k+1}-1}(f)\|_{L_{p,\infty}(G_{m})}$
$\geq\limsup\limits_{k\rightarrow\infty}\left(\|w_{M_{2k+1}-1}\|_{L_{p,_{\infty}}(G_{m})}-\|\sum\limits_{i=k+1}^{\infty}\left(D_{M_{2i+1}}-D_{M_{2i}}\right)\|_{L_{p,\infty}(G_{m})}\right)$
$\geq\limsup\limits_{k\rightarrow\infty}\left(1-c/M_{2k}^{1/p-1}\right)c>0.$

Which complete the proof of Theorem 3.

Proof of Theorem 4. Analogously we can prove first part of Theorem 4. Let
proof it‘s second part. We set

$a_{i}(x)=D_{M_{2M_{i}+1}}(x)-D_{M_{2M_{i}}}(x)$

and

$f_{A}\left(x\right)=\overset{A}{\sum_{i=1}}\frac{a_{i}(x)}{M_{i}}.$

Since

(20)
$S_{M_{A}}a_{k}\left(x\right)=\left\\{\begin{array}[]{l}a_{k}\left(x\right),\text{
}2M_{k}\leq A,\\\ 0,\text{ \qquad}2M_{k}>A,\end{array}\right.$

and

$\text{supp}(a_{k})=I_{2M_{k}},\text{ \ }\int_{I_{2M_{k}}}a_{k}d\mu=0,\text{ \
}\left\|a_{k}\right\|_{\infty}\leq M_{2M_{k}}=\mu(\text{supp }a_{k}),$

if we apply Theorem W we conclude that $f\in H_{1}.$

It is easy to show that

(21) $\omega(\frac{1}{M_{n}},f)_{H_{1}(G_{m})}\leq\sum\limits_{i=\left[\lg
n/2\right]}^{\infty}\frac{1}{M_{i}}=O\left(\frac{1}{n}\right),$

where $\left[\lg n/2\right]$ denotes integer part of $\lg n/2.$ By simple
calculation we get

(22) $\widehat{f}(j)=\left\\{\begin{array}[]{l}\frac{1}{M_{2i}},\,\ \text{if
\thinspace\thinspace}j\in\left\\{M_{2M_{i}},...,M_{2M_{i}+1}-1\right\\},\text{
}i=0,1,...\\\ 0,\text{ \thinspace\thinspace\thinspace if
\thinspace\thinspace\thinspace}j\notin\bigcup\limits_{i=0}^{\infty}\left\\{M_{2M_{i}},...,M_{2M_{i}+1}-1\right\\}.\text{
}\end{array}\right.$

Combining (15) and (22) we have

$\displaystyle\limsup\limits_{k\rightarrow\infty}\|f-S_{q_{M_{k}}}(f)\|_{1}$
$\displaystyle\geq$
$\displaystyle\limsup\limits_{k\rightarrow\infty}\left(\frac{1}{M_{2k}}\|D_{q_{M_{k}}}\|_{1}-\frac{1}{M_{2k}}\|D_{M_{2M_{k}+1}}\|_{1}-\|\overset{\infty}{\sum_{i=k+1}}\frac{D_{M_{2M_{i}+1}}-D_{M_{2M_{i}}}}{M_{2i}}\|_{1}\right)$
$\displaystyle\geq$
$\displaystyle\limsup\limits_{k\rightarrow\infty}\left(c-\overset{\infty}{\sum_{i=k+1}}\frac{1}{M_{2i}}-\frac{1}{M_{2k}}\right)\geq
c>0.$

Theorem 4 is proved.

## References

  * [1] G. N. Agaev, N. Ya. Vilenkin, G. M. Dzafary and A. I. Rubinshtein, Multiplicative systems of functions and harmonic analysis on zero-dimensional groups, Baku, Ehim, 1981 (in Russian).
  * [2] I. Blahota, Acta Acad. Paed. Nyiregyhaziensis, Approximation by Vilenkin-Fourier sums in $L_{p}(G_{m})$. 13(1992), 35-39.
  * [3] N. I. Fine, On Walsh function, Trans. Amer. Math. Soc. 65 (1949), 372-414.
  * [4] S. Fridli, Approximation by Vilenkin-Fourier series, Acta Math. Hungarica 47 (1-2), (1986), 33-44.
  * [5] G. Gát, Inverstigations of certain operators with respect to the Vilenkin sistem, Acta Math. Hung., 61 (1993), 131-149.
  * [6] G. Gát, Best approximation by Vilenkin-Like systems, Acta Acad. Paed. Nyiregyhaziensis, 17(2001), 161-169.
  * [7] U. Goginava, G. Tkebuchava, Convergence of subsequence of partial sums and logarithmic means of Walsh-Fourier series, Acta Sci. Math (Szeged) 72 (2006), 159-177.
  * [8] U. Goginava, On Uniform convergence of Walsh-Fourier series, Acta Math. Hungar. 93 (1-2) (2001), 59-70.
  * [9] U. Goginava, On approximation properties of partial sums of Walsh-Fourier series, Acta Sci. Math. (Szeged), 72 (2006), 569-579.
  * [10] U. Goginava, L. D. Gogoladze, Strong Convergence of Cubic Partial Sums of Two-Dimensional Walsh-Fourier series, Constructive Theory of Functions, Sozopol 2010: In memory of Borislav Bojanov. Prof. Marin Drinov Academic Publishing House, Sofia, 2012, pp. 108-117.
  * [11] L. D. Gogoladze, On the strong summability of Fourier series, Bull of Acad. Scie. Georgian SSR, 52, 2 (1968), 287-292.
  * [12] B. I. Golubov, A. V. Efimov and V. A. Skvortsov, Walsh series and transforms. (Russian) Nauka, Moscow, 1987, English transl, Mathematics and its Applications (Soviet Series), 64. Kluwer Academic Publishers Group, Dordrecht, 1991.
  * [13] N. V. Guličev, Approximation to continuous functions by Walsh-Fourier series, Analisys Math. 6(1980), 269-280.
  * [14] P. Simon, Strong convergence of certain means with respect to the Walsh-Fourier series, Acta Math. Hung., 49 (1-2) (1987), 425-431.
  * [15] P. Simon, Strong convergence Theorem for Vilenkin-Fourier Series. Journal of Mathematical Analysis and Applications, 245, (2000), pp. 52-68 .
  * [16] B. Smith, A strong convergence theorem for $H_{1}\left(T\right),$ in Lecture Notes in Math., 995, Springer, Berlin, (1994), 169-173.
  * [17] G. Tephnadze, A note on the Vilenkin-Fourier coefficients, Georgian Mathematical Journal, (to appear).
  * [18] G. Tephnadze, A note on the Fourier coefficients and partial sums of Vilenkin-Fourier series, Acta Mathematica Academiae Paedagogicae Nyiregyhaziensis (AMAPN), (to appear).
  * [19] G. Tephnadze, Strong convergence of two-dimensional Walsh-Fourier series, Ukrainian Mathematical Journal (UMJ), (to appear).
  * [20] N. Ya. Vilenkin, A class of complate ortonormal systems, Izv. Akad. Nauk. U.S.S.R., Ser. Mat., 11 (1947), 363-400.
  * [21] F. Weisz, Martingale Hardy spaces and their applications in Fourier Analysis, Springer, Berlin-Heideiberg-New York, 1994.
  * [22] F. Weisz, Hardy spaces and Cesáro means of two-dimensional Fourier series, Bolyai Soc. math. Studies, (1996), 353-367.
  * [23] F.Weisz, Strong convergence theorems for two-parameter Walsh-Fourier and trigonometric-Fourier series. (English) Stud. Math. 117, No.2, (1996), 173-194.
  * [24] F.Weisz, Hardy spaces and Cesáro means of two-dimensional Fourier series, Bolyai Soc. math. Studies, (1996), 353-367.