# Remote State Estimation with Privacy Against Active Eavesdroppers

Matthew J. Crimson  Justin M. Kennedy  Daniel E. Quevedo  School of Electrical
Engineering and Robotics, Queensland University of Technology, Brisbane QLD,
4000 Australia. (e-mail: {m.crimson<EMAIL_ADDRESS>
###### Abstract

This paper considers a cyber-physical system under an active eavesdropping
attack. A remote legitimate user estimates the state of a linear plant from
the state information received from a sensor. Transmissions from the sensor
occur via an insecure and unreliable network. An active eavesdropper may
perform an attack during system operation. The eavesdropper intercepts
transmissions from the sensor, whilst simultaneously sabotaging the data
transfer from the sensor to the remote legitimate user to harm its estimation
performance. To maintain state confidentiality, we propose an encoding scheme
that is activated on the detection of an eavesdropper. Our scheme transmits
noise based on a pseudo-random indicator, pre-arranged at the legitimate user
and sensor. The transmission of noise harms the eavesdropper’s performance,
more than that of the legitimate user. Using the proposed encoding scheme, we
impair the eavesdropper’s expected estimation performance, whilst minimising
expected performance degradation at the legitimate user. We explore the trade-
off between state confidentiality and legitimate user performance degradation
through selecting the probability that the sensor transmits noise. Under
certain design choices, the trace of the expected estimation error covariance
of the eavesdropper is greater than that of the legitimate user. Numerical
examples are provided to illustrate the proposed encoding scheme.

###### keywords:

Network Security, Privacy, Eavesdropping Attacks, Remote Estimation.

††thanks: A preliminary version of this work was presented at IFAC World
Congress, 2023, Yokohama, Japan (see [5]). Corresponding author Matthew
Crimson.

##  1 Introduction

In recent years, Cyber-Physical Systems (CPS) have grown rapidly with the
ubiquitous rise of the Internet of Things. CPS integrate cyber-infrastructure
and physical systems. These systems encompass large-scale, geographically
dispersed systems, such as smart grids, intelligent transportation systems,
critical infrastructure and wearable medical devices [12]. Due to the tight
integration of cyber and physical components in such systems, CPS are
vulnerable to cyber-attacks. Cyber vulnerabilities can be exploited by
malicious adversaries, which seek to disrupt CPS operations. Attacks on CPS
may result in performance degradation or even system failure [11].

Security of a system is defined by availability, integrity, and
confidentiality. Targeting these security goals are; denial-of-service (DoS),
deception attacks, and eavesdropping. DoS attacks seek to compromise the
availability of information via jamming communication channels. Deception
attacks seek to target the integrity of a system by manipulating transmitted
information. Eavesdropping attacks seek to intercept transmitted information,
targeting confidentiality, [4]. In particular, the use of shared wireless
mediums makes it difficult to shield transmissions from unintended recipients.
Sophisticated attacks may combine elements to achieve goals, such as active
eavesdropping. Active eavesdropping is where the eavesdropper can not only
intercept transmissions but also invade the communication channel to degrade
the wireless performance of the system [7]. This performance degradation may
trigger the sensor to transmit data packets more frequently, or with higher
power which increases the packet arrival rate at the eavesdropper. Therefore,
estimation accuracy at the eavesdropper is increased. This style of attack
differs from a DoS attack, as jamming is utilised to improve the eavesdropper
performance as well, whereas a DoS attack seeks to degrade the legitimate
user’s performance, with no consideration to its own packet reception. In
practical applications, active eavesdropping is achieved through pilot signal
contamination [24].

There is a need to design estimation algorithms over wireless communication
networks that account for the reliability of the wireless network, while also
ensuring confidentiality of the transmitted information from an eavesdropper.
Guaranteeing confidentiality is crucial for sensor and control data, which
convey confidential information about the physical system state. This is
detrimental to system operation as it violates sensor and user privacy, which
may cause economic losses or even threaten human survival. For example, in
smart grids an eavesdropper can infer load taxonomy from the electricity
consumption data provided by smart meters. The malicious adversary can infer
daily schedules, and break into premises when no inhabitants are present [20].
In addition, with knowledge of how a system operates, an adversary may perform
an attack with greater sophistication.

To ensure privacy against eavesdropping attacks, cryptography-based tools [13]
are at times used in practice, however, they introduce computation and
communication overheads [15] and therefore are only of limited use. Several
works have developed encoding schemes [17, 22] and scheduling of transmissions
[16], which are lightweight and avoid significant computation. These encoding
techniques require knowledge of the legitimate estimator’s performance.
Performance is shared through acknowledgment of successful packet receipts to
the sensor. However, utilising an acknowledgement channel is a point of
vulnerability for eavesdropping and DoS attacks [7], as experimentally
demonstrated with the jamming of acknowledgments utilising commodity hardware
[14]. Therefore, the motivation for this work is to design a low-complexity
encoding scheme in a system without acknowledgments when an eavesdropper
attack is occurring, by adapting transmissions to ensure the confidentiality
of the state.

In this work, we consider a sensor transmitting the state estimate of a
stable, linear, dynamic system to a legitimate remote user over a random
packet-dropping wireless network. A malicious adversary may perform an active
eavesdropping attack during system operation and intercept the transmissions
from the sensor, compromising state confidentiality. We propose an encoding
scheme that is activated on the detection of an eavesdropper. We present a
Quickest Change Detection (QCD) scheme to detect the presence of an
eavesdropper through monitoring channel outcomes. QCD is a technique to detect
an abrupt change in statistical properties of an observed process as quickly
as possible after the change occurs, subject to a false alarm constraint [2].
Several formulations for QCD problems exist, which vary with respect to
assumptions about the change time and optimality criteria. Bayesian
formulations assume the change time is a random variable with a known
geometric prior with independent and identically distributed (i.i.d.)
observations [10, 23]. With these assumptions, Shiryaev established the
optimal stopping, which compared the change posterior probability to a
threshold.

In this work, we use QCD to detect the presence of an active eavesdropper,
which activates the encoding scheme. When an eavesdropper is detected, the
sensor transmits noise or its local state estimate, based on a pseudo-random
Bernoulli indicator that is pre-arranged at the legitimate user and sensor.
Under this encoding scheme, the eavesdropper’s estimation error covariance is
made larger than that of the legitimate user. This work extends the results in
[5]. We extend [5] from the scalar, first-order system to the vector case.
Additionally, no method was presented in [5] for the detection of the
eavesdropper.

##  2 Remote State Estimation with an Eavesdropper

We consider the following dynamic system that is modelled as a linear,
discrete-time invariant system

$x_{k+1}=Ax_{k}+w_{k},\quad y_{k}=Cx_{k}+v_{k}$ (1)

where $x_{k}\in\mathbb{R}^{n}$ is the system state vector and
$y_{k}\in\mathbb{R}^{m}$ is the measurement taken by the sensor at time $k\geq
0$. The dynamics is stable with $\rho(A)<1$, where $\rho(.)$ is the spectral
radius111The spectral radius is defined as the maximum absolute eigenvalue
$\rho(.)=\max|\beta_{i}(.)|$ where $\beta_{i}$ is the $i$th eigenvalue.. The
process noise $w_{k}\in\mathbb{R}^{n}$ and the measurement noise
$v_{k}\in\mathbb{R}^{m}$ are zero mean i.i.d. Gaussian random processes with
$\mathbb{E}[w_{k}w_{k}^{\intercal}]=Q$ with
$\mathbb{E}[w_{k}w_{l}^{\intercal}]=0$ and
$\mathbb{E}[v_{k}v_{k}^{\intercal}]=R$ with
$\mathbb{E}[v_{k}v_{l}^{\intercal}]=0$ and
$\mathbb{E}[w_{k}v_{l}^{\intercal}]=0\ \text{ for all }l\geq 0\text{ \ where\
}k\neq l$. The initial state of the process $x_{0}$ is a Gaussian random
variable with zero mean and covariance
$\mathbb{E}[x_{0}x_{0}^{\intercal}]=\Sigma_{0}$. The covariances
$Q,R,\Sigma_{0}$ are positive definite. The initial state is uncorrelated with
$w_{k}$ and $v_{k}$ with $\mathbb{E}[x_{0}w_{k}^{\intercal}]=0$ and
$\mathbb{E}[x_{0}v_{k}^{\intercal}]=0$. All system and noise parameters
$(A,C,Q,R,\Sigma_{0})$ are assumed to be public knowledge, available to the
sensor, legitimate user and eavesdropper. The pair $(A,C)$ is assumed to be
observable and $(A,\sqrt{Q})$ is controllable.

###  2.1 Local State Estimation and Open Loop Performance

In current CPS frameworks, sensors are equipped with on-board computation and
processes to improve the estimation performance of the system. Sensors process
the collected measurements by executing recursive algorithms, enabled by
advanced embedded systems-on-chip [6]. To emulate this sensor configuration,
we consider the sensor configuration is computationally capable of optimal
state estimation in real-time. The sensor locally estimates the system state
$x_{k}$, based upon all measurements it has collected up to time $k$ using
standard Kalman filtering recursions. The local estimate is transmitted
wirelessly to a remote legitimate user. We denote $\hat{x}_{k}^{s}$ as the
sensor’s local Minimum Mean-Squared Error (MMSE) estimate of the system state,
$x_{k}$, and $P_{k}^{s}$ as the corresponding estimation error covariance

$\begin{split}\hat{x}_{k}^{s}&=\mathbb{E}[x_{k}|y_{0},y_{1},...,y_{k}]\\\
{P}_{k}^{s}&=\mathbb{E}[(x_{k}-\hat{x}_{k}^{s})(x_{k}-\hat{x}_{k}^{s})^{\intercal}|y_{0},y_{1},...,y_{k}].\end{split}$

For any initial condition, the error covariance $P_{k}^{s}$ converges
exponentially fast to some unique value $\bar{P}$ [1]. Without loss of
generality, it is assumed the local state estimator has entered steady state
operation and that $P_{k}^{s}=\bar{P}$, where $\bar{P}$ is the unique solution
to the discrete-time algebraic Riccati equation

$\begin{split}\bar{P}=&A\bar{P}A^{\intercal}+Q-(A\bar{P}A^{\intercal}+Q)C^{\intercal}\\\
&(CA\bar{P}A^{\intercal}C+CQC^{\intercal}+R)^{-1}C(A\bar{P}A^{\intercal}+Q).\end{split}$
(2)

###  2.2 Communication Model

The legitimate user requires to reliably estimate the system state for the
purposes of remote monitoring or control. To obtain this estimate, the sensor
transmits an encoded packet $z_{k}\in\mathbb{R}^{n}$ of state information or
noise. We define the formation of $z_{k}$ in Section 4. The packet $z_{k}$ is
transmitted over a wireless network to the legitimate user. However, the
transmitted packets can also be received by an eavesdropper. We utilize a
standard packet-based transmission utilized in network control problems.
Denote the packet reception indicator outcome at the legitimate user by
$\lambda_{k}\in\\{0,1\\}$, where

$\lambda_{k}=\begin{cases}1,\quad\textrm{if a packet is received at the
legitimate user}\\\ 0,\quad\textrm{if a packet dropout occurs.}\end{cases}$

Denote the packet reception indicator outcome at the eavesdropper by
$\lambda_{k}^{e}\in\\{0,1\\}$, where

$\lambda_{k}^{e}=\begin{cases}1,\quad\textrm{if a packet is received at the
eavesdropper}\\\ 0,\quad\textrm{if a packet dropout occurs.}\end{cases}$

The channel outcomes for the legitimate user and eavesdropper are modelled as
i.i.d. Bernoulli processes, independent of the initial state of the process
and of the process noise.

###  2.3 Attack Model and Channel Qualities

An eavesdropper seeks to compromise the confidentiality of the system state by
eavesdropping on the transmitted state estimate from the sensor. At the same
time, the eavesdropper invades the communication channel between the sensor
and the legitimate user by attempting to block transmissions to degrade the
estimation accuracy of the legitimate user. At some time $k=\Lambda$ where
$\Lambda\geq 1$ the eavesdropper performs an attack.

Let the channel quality be defined as the probability of packet dropout. In
the absence of an eavesdropper attack, the channel outcomes at the legitimate
user $\lambda_{k}$ have a nominal packet dropout probability as
$\gamma\in(0,1)$, where

$\gamma=\mathbb{P}[\lambda_{k}=0]\quad k<\Lambda.$

We propose when the eavesdropper is performing an attack, the quality of the
legitimate user’s channel is degraded with a probability of packet dropout as
$\bar{\gamma}\in(0,1)$, where

$\bar{\gamma}=\mathbb{P}[\lambda_{k}=0]\quad k\geq\Lambda,$ (3)

where $\gamma<\bar{\gamma}$. When the eavesdropper is not performing an
attack, the eavesdropper does not receive any information with
$\lambda^{e}_{k}=0$ when $k<\Lambda$, i.e., the eavesdropper does not have a
communication channel. When the eavesdropper is performing an attack, the
probability of packet dropout for the eavesdropper’s channel is defined as
$\gamma^{e}\in(0,1)$, where

$\gamma^{e}=\mathbb{P}[\lambda_{k}^{e}=0]\quad k\geq\Lambda.$ (4)

In our detection method we utilise the legitimate user’s channel quality
distributions to detect when an eavesdropper attack is occuring, detailed in
Section 9.

##  3 Problem of Interest

As packet receipt acknowledgments may be vulnerable to adversaries, our
objective is to design an encoding scheme that is activated on the detection
of an eavesdropper, that does not rely on acknowledgements when the
eavesdropper is detected. To characterise performance bounds under the
encoding scheme for the eavesdropper’s and legitimate user’s estimation error
covariance, we introduce “open-loop performance”. We first define the solution
to the discrete-time Lyapunov equation, which is expressed as an infinite sum
in the case that $\rho(T)<1$

$V=\sum_{j=0}^{\infty}T^{j}U(T^{\intercal})^{j},$ (5)

where $V$ is the unique stabilising solution to

$V=TVT^{\intercal}+U.$ (6)

For convenience, we define the following notation to represent the solution to
the Lyapunov equation

$V=L(T,U).$ (7)

If no information is received from the sensor at the legitimate user or
eavesdropper, then open loop performance occurs. It is characterised by
$x^{OP}_{k}=\mathbb{E}[x_{k}]=0$ and
$P^{OP}_{k}=\mathbb{E}[x_{k}x_{k}^{\intercal}]$, where the asymptotic open
loop estimate and corresponding estimation error covariance satisfy [1]

$\begin{split}x^{OP}&\triangleq\lim_{k\to\infty}x_{k}^{OP}=0\\\
P^{OP}&\triangleq\lim_{k\to\infty}P^{OP}_{k}=L(A,Q).\end{split}$ (8)

The encoding scheme is designed with no information of the legitimate user’s
current estimate, hence we aim to ensure that the trace of the legitimate
user’s expected estimation error covariance is upper bounded by the trace of
open loop prediction. To ensure confidentiality, we desire that the trace of
the eavesdropper’s expected estimation error covariance is above the trace of
open loop prediction, i.e., the trace of the expected eavesdropper’s
estimation error covariance when using received packets is larger than if it
had not used any packets in its estimation process. These conditions guarantee
data confidentiality against an eavesdropper, whilst also ensuring the
legitimate user has a reliable estimate.

##  4 Encoding Scheme Design

Figure 1: The sensor transmits $z_{k}$, an encoded packet formed by the
indicator $u_{k}$ and alarm $\nu_{k}$. The alarm $\nu_{k}$ is formed at the
legitimate user, indicating the presence of an eavesdropper and shared to the
sensor. The indication to send noise $u_{k}$ is known to the legitimate user
and sensor, and unknown to the eavesdropper. The packet is transmitted over a
packet dropping network. The legitimate estimator and eavesdropper compute
state estimates $\hat{x}_{k}$ and $\hat{x}_{k}^{e}$.

The considered remote estimation architecture is shown in Fig. 1 and consists
of a sensor observing a dynamic system, a legitimate user, and an
eavesdropper. The sensor transmits packets of information over a wireless
channel. To degrade the eavesdropper’s estimation performance, we desire to
limit the amount of useful information it receives. To actively harm its
estimation performance, we propose to send noise intermittently.

###  4.1 Encoding Scheme

Our proposed strategy is to enact the encoding scheme, based upon the
detection of an eavesdropper. We first introduce the decision variable
$\nu_{k}\in\\{0,1\\}$, where

$\nu_{k}=\begin{cases}0,&\text{if eavesdropper has been detected}\\\
1,&\text{otherwise}.\\\ \end{cases}$ (9)

The decision variable $\nu_{k}$ is formed at the legitimate user and shared
with the sensor via an emergency acknowledgment channel. Once the sensor is
alerted that an active eavesdropping attack is occurring, the sensor enacts
the encoding scheme. We formulate eavesdropper detection and the design of
$\nu_{k}$ using QCD in Section 9.

When an eavesdropper is detected, our proposed encoding scheme is to send
either the sensor’s state estimate $\hat{x}_{k}^{s}$ or noise $n_{k}$ by using
a Bernoulli pseudo-random indicator $u_{k}\in\\{0,1\\}$. The transmission of
noise reduces transmissions of information to the legitimate user, reducing
its performance. We pre-arrange the indicator $u_{k}$ between the legitimate
estimator and sensor. This encoding scheme forms $z_{k}$ which is the
transmitted packet from the sensor

$z_{k}=\begin{cases}\hat{x}_{k}^{s},&\text{ if }\nu_{k}=1\text{ or
}(\nu_{k},u_{k})=(0,1)\\\ n_{k},&\text{ if }(\nu_{k},u_{k})=(0,0).\\\
\end{cases}$ (10)

The indicator $u_{k}$ informs what is transmitted at the sensor. In practical
systems, random numbers are generated through pseudo-random algorithms. By
agreeing upon an initial seed at the legitimate user and sensor, we pre-
arrange the indicator sequence $u_{k}$.

We choose the noise to have the same characteristics as the sensor’s state
estimate $\hat{x}_{k}^{s}$. The noise is designed such that a change in
transmission cannot be easily detected at the eavesdropper, as it exhibits the
same characteristics as the sensor’s state estimate $\hat{x}_{k}^{s}$ [18].
The noise $n_{k}$ is normally distributed, with zero mean
$\mathbb{E}[n_{k}]=0$ and covariance
$\mathbb{E}[n_{k}n_{k}^{\intercal}]=\bar{P}$ uncorrelated with the state
$x_{k}$ where $\mathbb{E}[x_{k}n_{k}]=0$.

We denote the probability of the sensor transmitting noise as $\mu\in(0,1)$,
where

$\mu\triangleq\mathbb{P}(u_{k}=0).$ (11)

This probability constitutes the main design variable. It can be adjusted
based on the dropout channel probabilities and the known process noise of the
dynamics to balance the impact between degrading the legitimate user’s
performance and ensuring data confidentiality. We explore this trade-off in
Section 7.

Under this formulation, the goal is to decide how often to send noise such
that data confidentiality is protected. The indicator is designed to be
pseudo-random, such that the eavesdropper cannot easily discern patterns in
transmission. Moreover, we require the sensor’s state estimate to be received
by the legitimate user intermittently to ensure its performance is not
degraded by too many losses of state estimates from the sensor. Due to the
pre-arrangement of the indicator $u_{k}$, the need for an acknowledgment
channel is omitted when an eavesdropper is performing an attack.

###  4.2 Remote State Estimation

We denote $\mathcal{I}_{k}$ as a collection of historical information at the
legitimate user up to time $k$, with
$\mathcal{I}_{k}\triangleq\\{u_{0},\lambda_{0},\lambda_{0}z_{0},...,u_{k},\lambda_{k},\lambda_{k}z_{k}\\}$,
noting that the legitimate user knows the encoding scheme variable $u_{k}$. We
define the legitimate user’s own state estimate and the corresponding
estimation error covariance as

$\begin{split}\hat{x}_{k}&\triangleq\mathbb{E}[x_{k}|\mathcal{I}_{k}]\\\
P_{k}&\triangleq\mathbb{E}[(x_{k}-\hat{x}_{k})(x_{k}-\hat{x}_{k})^{\intercal}|\mathcal{I}_{k}],\end{split}$
(12)

which is based on all sensor data packets received up to time step $k$. We
denote $\mathcal{I}^{e}_{k}$ as a collection of historical information at the
eavesdropper up to time $k$ with
$\mathcal{I}_{k}^{e}\triangleq\\{\lambda_{0}^{e},\lambda_{0}^{e}z_{0},...,\lambda_{k}^{e},\lambda_{k}^{e}z_{k}\\}$.
We define the eavesdropper’s own state estimate and the corresponding
estimation error covariance as

$\begin{split}\hat{x}^{e}_{k}&\triangleq\mathbb{E}[x_{k}|\mathcal{I}_{k}^{e}]\\\
P_{k}^{e}&\triangleq\mathbb{E}[(x_{k}-\hat{x}_{k}^{e})(x_{k}-\hat{x}_{k}^{e})^{\intercal}|\mathcal{I}_{k}^{e}],\end{split}$
(13)

which is based on all sensor data packets received up to time step $k$. Note
that the eavesdropper is unaware of the indication for the sensor to send
noise $u_{k}$, Hence, the eavesdropper is deceived to incorporate noise
$n_{k}$ into its state estimate.

##  5 Remote State Estimation with an Eavesdropper

In an unreliable, insecure network, the sensor’s state estimate may not be
received by the legitimate user, or the received packet may be noise due to
the employed encoding scheme. To estimate the state of the system with
intermittent observations, a second estimation scheme is employed at the
legitimate user.

The legitimate user obtains the state estimate as follows: Once the sensor’s
transmission arrives, and the sensor has not transmitted noise, the estimator
synchronizes $\hat{x}_{k}$ with the optimal state estimate of the sensor
$\hat{x}_{k}^{s}$. Otherwise, the legitimate user predicts $x_{k}$ based on
the previous estimate using the system model in (1). Thus, the state estimate
at the legitimate user $\hat{x}_{k}$ (12) satisfies

$\hat{x}_{k}=\begin{cases}\hat{x}_{k}^{s},\quad~{}~{}~{}~{}~{}\text{ if
}(\lambda_{k},u_{k})=(1,1)\\\ A\hat{x}_{k-1},~{}~{}~{}\text{ if
}\lambda_{k}=0\text{ or }(\lambda_{k},u_{k})=(1,0).\end{cases}$

The corresponding estimation error covariance $P_{k}$ satisfies

$P_{k}=\begin{cases}\bar{P},\quad~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{
if }(\lambda_{k},u_{k})=(1,1)\\\ AP_{k-1}A^{\intercal}+Q,~{}\text{ if
}\lambda_{k}=0\text{ or }(\lambda_{k},u_{k})=(1,0).\end{cases}$ (14)

It follows from the eavesdropper’s estimation error covariance in (14) that
when a packet arrives and it is the state estimate from the sensor, the
legitimate user’s estimation error covariance is that of the sensor’s
$\bar{P}$ (see (2)). When a packet of noise arrives, or a packet dropout
occurs, the legitimate user’s estimation error covariance grows.

The goal of the eavesdropper is to construct an estimate of the state of the
system $x_{k}$ based on all sensor data packets received up to time step $k$.
The eavesdropper is unaware of the detection variable $\nu_{k}$ (9). The
eavesdropper is additionally unaware of the randomised indicator to send noise
$u_{k}$ (10). As the noise is designed to exhibit the same statistical
characteristics as the sensor’s state estimate, and the encoding scheme is
unknown, the eavesdropper cannot distinguish $\hat{x}_{k}^{s}$ from $n_{k}$.
This feature can be used to deceive the eavesdropper to incorporate noise
$n_{k}$ into its state estimate. The eavesdropper obtains the state estimate
as follows: Once the sensor’s transmission arrives, and the sensor has not
transmitted noise, the eavesdropper synchronizes $\hat{x}^{e}_{k}$ with that
of the sensor. If the sensor’s transmission arrives, and the sensor has
transmitted noise, the eavesdropper synchronizes $\hat{x}^{e}_{k}$ with the
transmitted noise $n_{k}$. If a packet dropout occurs, the eavesdropper
predicts $x_{k}$ based on the previous estimate using the system model in (1),
such that the eavesdropper’s state estimate $\hat{x}_{k}^{e}$ (13) from the
sensor’s viewpoint satisfies

$\hat{x}_{k}^{e}=\begin{cases}\hat{x}_{k}^{s},\quad~{}~{}\text{ if
}(\lambda_{k}^{e},u_{k})=(1,1)\\\ n_{k},\quad~{}~{}\text{ if
}(\lambda_{k}^{e},u_{k})=(1,0)\\\ A\hat{x}_{k-1}^{e},\text{ if
}\lambda_{k}^{e}=0.\end{cases}$ (15)

We characterise the eavesdropper’s estimation error covariance in Lemma 1.

######  Lemma 1.

The estimation error covariance $P_{k}^{e}$ (13) satisfies

$P_{k}^{e}=\begin{cases}\bar{P},\quad~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{
if }(\lambda_{k}^{e},u_{k})=(1,1)\\\
P_{n},\quad~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}~{}\text{ if
}(\lambda_{k}^{e},u_{k})=(1,0)\\\ AP_{k-1}^{e}A^{\intercal}+Q,~{}\text{ if
}\lambda_{k}^{e}=0,\end{cases}$ (16)

where $P_{n}$ is

$P_{n}=P^{OP}+\bar{P},$ (17)

which is the estimation error covariance under receiving and utilising a
packet of noise, as the packet is mistaken for the state estimate, with
$P^{OP}$ defined in (8). Firstly, consider the scenario that the eavesdropper
successfully receives the state estimate from the sensor with
$(\lambda_{k}^{e},u_{k})=(1,1)$, the eavesdropper’s estimate is
$\hat{x}_{k}^{e}=\hat{x}_{k}^{s}$ and the estimation error covariance is that
of the sensor’s (see (2)) with $P_{k}^{e}=\bar{P}$. Secondly, in the scenario
that a packet dropout occurs with $\lambda_{k}^{e}=0$, the eavesdropper
predicts $\hat{x}_{k}^{e}$ with that of its previous estimate, with
$\hat{x}_{k}^{e}=A\hat{x}_{k-1}^{e}$. The corresponding estimation error
covariance is $P_{k}^{e}=AP_{k-1}^{e}A^{\intercal}+Q$. Thirdly, in the
scenario that the eavesdropper successfully receives a packet of noise
$(\lambda_{k}^{e},u_{k})=(1,0)$, the eavesdropper directly uses the packet as
the state estimate $\hat{x}_{k}^{e}=n_{k}$. The estimation error covariance is

$\begin{split}P_{n}&=\mathbb{E}[(x_{k}-n_{k})(x_{k}-n_{k})^{\intercal}|\mathcal{I}_{k}^{e}]\\\
&=\mathbb{E}[x_{k}x_{k}^{\intercal}]+2\mathbb{E}[x_{k}n_{k}]+\mathbb{E}[n_{k}n_{k}^{\intercal}].\end{split}$
(18)

These cases show the eavesdropper’s estimation error covariance (16). By
definition the transmitted noise is designed such that
$\mathbb{E}[n_{k}n_{k}^{\intercal}]=\bar{P}$, uncorrelated with $x_{k}$ such
that $\mathbb{E}[x_{k}n_{k}]=0$. As per (8), the open loop estimation error
covariance is $\mathbb{E}[x_{k}x_{k}^{\intercal}]=P^{OP}$. Then from (18)

$P_{n}=\mathbb{E}[x_{k}x_{k}^{\intercal}]+2\mathbb{E}[x_{k}n_{k}]+\mathbb{E}[n_{k}n_{k}^{\intercal}]=P^{OP}+0+\bar{P},$

which shows the estimation error covariance under receiving a packet of noise
(17).

Lemma 1 characterises the instantaneous worst-case performance for the
eavesdropper under the encoding scheme. It shows that if noise is transmitted
often, we can drive the eavesdropper’s estimation error covariance above open-
loop prediction. However, greater transmissions of noise degrade the
legitimate user’s performance as well. Therefore, the probability of the
sensor sending noise $\mu$ needs to be chosen to balance the impact between
degrading the legitimate user’s performance and protecting data
confidentiality.

To quantify the estimation performance of the legitimate user and eavesdropper
under the encoding scheme from a sensor viewpoint, we explore the mathematical
expectation of the estimation error covariance. The use of the expectation
allows us to compute the expected performance over the stochastic channel and
encoding scheme. Particularly, due to the absence of acknowledgments when the
eavesdropper is performing an attack, the channel outcomes are unknown at the
sensor. Thus we use the expected estimation performance to design the sensor’s
encoding variable $\mu$. We additionally explore the steady-state behaviour of
the expected estimation error covariance to quantify long-term performance
under the encoding scheme as $k\rightarrow\infty$.

We present an analytic expression of the steady state expectation of
estimation error covariance of the legitimate user in Lemma 2 and of the
eavesdropper in Lemma 3. Our expressions depend on the channel qualities (3),
(4), process dynamics (1), and the encoding design variable (11).

######  Lemma 2.

The steady-state expectation of the legitimate user’s estimation error
covariance satisfies

$\lim_{k\to\infty}\mathbb{E}[P_{k}]=(1-\bar{\gamma})(1-\mu){W}+(\bar{\gamma}+(1-\bar{\gamma})\mu){S},$
(19)

where $W$ and $S$ are solutions to the Lyapunov equation via (6), with
$W=L\left(\left(\sqrt{(\bar{\gamma}+(1-\bar{\gamma})\mu)}A\right),\bar{P}\right)$
and
$S=L\left(\left(\sqrt{(\bar{\gamma}+(1-\bar{\gamma})\mu)}A\right),Q\right)$.

The proof is included in Section 6.

######  Lemma 3.

The steady-state expectation of the eavesdropper’s estimation error covariance
satisfies

$\displaystyle\lim_{k\to\infty}\mathbb{E}[P_{k}^{e}]$ (20)
$\displaystyle=(1-\gamma^{e})(1-\mu)W^{e}+\gamma^{e}S^{e}+(1-\gamma^{e})\mu
H^{e}$

where $W^{e}$ and $S^{e}$ and $H^{e}$ are solutions to the Lyapunov equation
via (6), with $W^{e}=L(\sqrt{\gamma^{e}}A,\bar{P})$,
$S^{e}=L(\sqrt{\gamma^{e}}A,Q)$, and $H^{e}=L(\sqrt{\gamma^{e}}A,P_{n})$.

The proof is included in Section 6.

##  6 Markov Chain Model

The mathematical expectation of the legitimate user’s and eavesdropper’s
estimation error covariance at time $k$ can be found by taking the sum of all
possible channel and sensor transmission outcomes multiplied by the
corresponding probability of that channel and sensor transmission realisation
occurring. To find the mathematical expectation of the eavesdropper’s and
legitimate user’s estimation error covariance, we introduce the following
Markov chain models.

###  6.1 Legitimate User’s Markov Chain Model

Let $G_{k}$ be the state of a Markov chain taking values in the countably
infinite set $G_{k}\in\\{0,1,\dots\\}$. We define $G_{k}=0$ as the state when
the sensor’s state estimate is received, and $G_{k}=j$ for $j>0$ as the $j$th
dropout. By application of the legitimate user’s channel quality (3) and the
probability of the sensor sending noise (11), the transition probabilities
$p_{ij}\triangleq\mathbb{P}[G_{k+1}=i|G_{k}=j]$ is characterised by

$\mathbb{P}[G_{k+1}=i|G_{k}=j]=\begin{cases}(1-\bar{\gamma})(1-\mu),&\text{ if
$i=0$},\\\ \bar{\gamma}+(1-\bar{\gamma})\mu,&\text{ if $i=j+1$},\\\ 0,&\text{
otherwise}.\end{cases}$

The Markov chain and its transitions are depicted in Fig. 2.

Figure 2: Markov chain model for the legitimate user. State $G_{k}=0$
represents the legitimate user receiving the state estimate from the sensor.
This event can be followed by infinite packet dropouts.

We define the transition matrix of the Markov chain as $\mathbf{P}=[p_{ij}]$.
Since all states communicate, the Markov chain is irreducible, aperiodic, and
recurrent and has a stationary distribution $\pi$ [3]. By solving the relation
$\pi=\pi\mathbf{P}$ with $\sum_{j=0}^{\infty}\pi_{j}=1$, the stationary
distribution can be found as

$\displaystyle\pi_{j}$
$\displaystyle=(\bar{\gamma}+(1-\bar{\gamma})\mu)^{j}(1-\bar{\gamma})(1-\mu).$

Following the legitimate user’s estimation error covariance in (14), the
conditional expectation of the estimation error covariance $P_{k}$ at the
Markov chain state $G_{k}=j$ is

$\mathbb{E}[P_{k}|G_{k}=j]=A^{j}\bar{P}(A^{\intercal})^{j}+\sum_{i=0}^{j-1}A^{i}Q(A^{\intercal})^{i}.$

Taking the limit as $k\rightarrow\infty$, then the expected estimation error
covariance is

$\begin{split}&\lim_{k\to\infty}\mathbb{E}[P_{k}]=\lim_{k\to\infty}\sum_{j=0}^{k}\mathbb{E}[P_{k}|G_{k}=j]\pi_{j}\\\
&=(1-\bar{\gamma})(1-\mu)\\\
&\quad\times\left(\sum_{j=0}^{\infty}(\bar{\gamma}+(1-\bar{\gamma})\mu)^{j}(A^{j}\bar{P}(A^{\intercal})^{j}+\sum_{i=0}^{j-1}A^{i}Q(A^{\intercal})^{i}\right)\\\
&=(1-\bar{\gamma})(1-\mu)\\\
&\quad\times\sum_{j=0}^{\infty}(\sqrt{(\bar{\gamma}+(1-\bar{\gamma})\mu)}A)^{j}\bar{P}(A^{\intercal}\sqrt{(\bar{\gamma}+(1-\bar{\gamma})\mu)})^{j}\\\
&\quad+(\bar{\gamma}+(1-\bar{\gamma})\mu)\\\
&\quad\times\sum_{j=0}^{\infty}(\sqrt{(\bar{\gamma}+(1-\bar{\gamma})\mu}A)^{j}Q(A^{\intercal}\sqrt{(\bar{\gamma}+(1-\bar{\gamma})\mu})^{j}.\\\
\end{split}$

These infinite sums satisfy the solution to the discrete-time Lyapunov
equation (see (5)) with
$\rho(\sqrt{(\bar{\gamma}+(1-\bar{\gamma})\mu}A)<1$ and therefore can be
written in the form of (7), where
$W=L(\sqrt{(\bar{\gamma}+(1-\bar{\gamma})\mu)}A,\bar{P})$ and
$S=L(\sqrt{(\bar{\gamma}+(1-\bar{\gamma}))\mu}A,Q)$, and can be written in the
form of (19). If $\bar{\gamma}=1$ then no packets are received, or if $\mu=1$
when only noise is sent then the legitimate user receives no packets
containing the state estimate, as such the Markov chain does not return to
state $G_{k}=0$ and no longer has a stationary distribution, and the expected
steady-state estimation error covariance satisfies

$\lim_{k\to\infty}\mathbb{E}[P_{k}]=P^{OP},\text{ if $\bar{\gamma}=1$ or
$\mu=1$}.$

###  6.2 Eavesdropper’s Markov Chain Model

Let $G_{k}^{e}$ be the state of a countably infinite Markov chain taking
values in the set $G_{k}^{e}\in\\{0,1,\dots\\}$. We define two key events,
$G_{k}^{e}=0$ as the state when the sensor’s state estimate is received, and
$G_{k}^{e}=1$ as the state when noise is received by the eavesdropper. These
two events can be followed by an infinite number of packet dropouts. Let
$G_{k}^{e}=j$ for $j\geq 2$, be the dropouts after $G_{\ell}^{e}=0$ when $j$
is even, and $G_{\ell}^{e}=1$ when $j$ is odd, for some $\ell<k$. The states
$G_{k}^{e}=0$ and $G_{k}^{e}=1$ can be reached from any other state. By
application of the eavesdropper’s channel quality (4) and the probability of
the sensor sending noise (11), the transition probability
$p_{ij}^{e}\triangleq\mathbb{P}[G_{k+1}^{e}=i|G_{k}^{e}=j]$ is characterised
by

$\mathbb{P}[G_{k+1}^{e}=i|G_{k}^{e}=j]=\begin{cases}(1-\gamma^{e})(1-\mu),&\text{
if $i=0$,}\\\ (1-\gamma^{e})\mu,&\text{ if $i=1$,}\\\ \gamma^{e},&\text{ if
$i=j+2$},\\\ 0,&\text{ otherwise .}\end{cases}$

The Markov chain and its transitions are depicted in Fig. 3. We define the
transition matrix of the Markov chain as $\mathbf{P}^{e}=[p_{ij}^{e}]$. Since
all states communicate, the Markov chain is irreducible, aperiodic, and
recurrent and has a stationary distribution $\pi^{e}$ [3]. By solving the
relation $\pi^{e}=\pi^{e}\mathbf{P}^{e}$ with
$\sum_{j=0}^{\infty}\pi_{j}^{e}=1$, the stationary distribution for all states
are as follows

$\displaystyle\pi_{0}^{e}$ $\displaystyle=(1-\gamma^{e})(1-\mu),$
$\displaystyle\pi_{1}^{e}$ $\displaystyle=(1-\gamma^{e})\mu,$
$\displaystyle\pi_{j+2}^{e}$ $\displaystyle=(\gamma^{e})\pi_{j}^{e}.$

Following the eavesdropper’s estimation error covariance (16), the conditional
expectation of the estimation error covariance $P_{k}^{e}$ at the Markov chain
state $G_{k}^{e}=j$ is

$\displaystyle\mathcal{P}_{j}^{e}=\mathbb{E}[P_{k}^{e}|G_{k}^{e}=j]$
$\displaystyle=\begin{cases}A^{\frac{1}{2}j}\bar{P}(A^{\intercal})^{\frac{1}{2}j}+\sum_{i=0}^{\frac{1}{2}j-1}A^{i}Q(A^{\intercal})^{i},&\text{
if $j$ even},\\\
A^{\frac{1}{2}(j-1)}P_{n}(A^{\intercal})^{\frac{1}{2}(j-1)}\\\
+\sum_{i=0}^{\frac{1}{2}(j-1)-1}A^{i}Q(A^{\intercal})^{i},&\text{ if $j$
odd}.\end{cases}$

Figure 3: Markov chain model for the eavesdropper. State $G_{k}^{e}=0$
represents the eavesdropper receiving the state estimate from the sensor.
State $G_{k}^{e}=1$ represents the eavesdropper receiving noise from the
sensor. These two events can be followed by infinite packet dropouts.

Taking the limit as time $k\rightarrow\infty$, then the expected estimation
error covariance is

$\displaystyle\lim_{k\to\infty}\mathbb{E}[P_{k}^{e}]=\lim_{k\to\infty}\sum_{j=0}^{k}\mathbb{E}[P_{k}^{e}|G_{k}^{e}=j]\pi_{j}^{e}$
$\displaystyle=\sum_{j=0}^{\infty}(\gamma^{e})^{j}\pi_{0}^{e}\mathcal{P}_{2j}^{e}+\sum_{j=0}^{\infty}(\gamma^{e})^{j}\pi_{1}^{e}\mathcal{P}_{2j+1}^{e}$
$\displaystyle=(1-\gamma^{e})(1-\mu)\sum_{j=0}^{\infty}(\gamma^{e})^{j}\left(A^{j}\bar{P}(A^{\intercal})^{j}+\sum_{i=0}^{j-1}A^{i}Q(A^{\intercal})^{i}\right)$
$\displaystyle\quad+(1-\gamma^{e})\mu\sum_{j=0}^{\infty}(\gamma^{e})^{j}\left(A^{j}P_{n}(A^{\intercal})^{j}+\sum_{i=0}^{j-1}A^{i}Q(A^{\intercal})^{i}\right)$
$\displaystyle=(1-\gamma^{e})(1-\mu)\sum_{j=0}^{\infty}(\sqrt{\gamma^{e}}A)^{j}\bar{P}(A^{\intercal}\sqrt{\gamma^{e}})^{j}$
$\displaystyle\quad+(1-\mu)\gamma^{e}\sum_{j=0}^{\infty}(\sqrt{\gamma^{e}}A)^{j}Q(A^{\intercal}\sqrt{\gamma^{e}})^{j}$
$\displaystyle\quad+(1-\gamma^{e})\mu\sum_{j=0}^{\infty}(\sqrt{\gamma^{e}}A)^{j}P_{n}(A^{\intercal}\sqrt{\gamma^{e}})^{j}$
$\displaystyle\quad+\mu\gamma^{e}\sum_{j=0}^{\infty}(\sqrt{\gamma^{e}}A)^{j}Q(A^{\intercal}\sqrt{\gamma^{e}})^{j}.$

These infinite sums satisfy the solution to the discrete-time Lyapunov
equation (see (5)), with $\rho(\sqrt{\gamma^{e}}A)<1$ and therefore can be
written in the form of (7), where $W^{e}=L(\sqrt{\gamma^{e}}A,\bar{P})$,
$S^{e}=L(\sqrt{\gamma^{e}}A,Q)$, and $H^{e}=L(\sqrt{\gamma^{e}}A,P_{n})$. and
can be written in the form of (20). If $\gamma^{e}=1$, then there are only
dropouts and the Markov chain cannot return to state $G_{k}^{e}=0$ or
$G_{k}^{e}=1$ for any $k>0$ and no longer has a stationary distribution, and
the expected steady-state estimation error covariance satisfies

$\lim_{k\to\infty}\mathbb{E}[P_{k}^{e}]=P^{OP}\text{ if $\gamma^{e}=1$}.$

##  7 Encoding Design

To provide a range of the encoding design variable $\mu$ (11), we write the
expectations of the estimation error covariances as a function of the encoding
design variable. The expectation of the steady-state estimation error
covariance of the legitimate user from Lemma 2 can be written as

$\begin{split}J(\mu)&=\operatorname{trace}\lim_{k\to\infty}\mathbb{E}[P_{k}]\\\
&=(1-\bar{\gamma})(1-\mu)\operatorname{trace}{W}\\\ &\
+(\bar{\gamma}+(1-\bar{\gamma})\mu)\operatorname{trace}{S},\end{split}$ (21)

where $W$ and $S$ are functions of the encoding design variable $\mu$.
Similarly, the expectation of the steady-state estimation error covariance of
the eavesdropper from Lemma 3 can be written as

$\begin{split}J^{e}(\mu)&=\operatorname{trace}\lim_{k\to\infty}\mathbb{E}[P_{k}^{e}]\\\
&=(1-\gamma^{e})(1-\mu)\operatorname{trace}W^{e}\\\ &\
+\gamma^{e}\operatorname{trace}S^{e}+(1-\gamma^{e})\mu\operatorname{trace}H^{e}\end{split}$
(22)

Before providing a method to select the encoding design variable $\mu$, we
observe the following monotonicity result.

######  Lemma 4.

Consider any eavesdropper channel quality $\gamma^{e}$ in $0<\gamma^{e}<1$ and
consider the encoding design variable (11) in $0<\mu^{\star}<\mu<1$. The trace
of the expected steady-state estimation error covariance of the eavesdropper
(22) is monotonically increasing in $\mu$ such that
$J^{e}(\mu^{\star})<J^{e}(\mu)$.

Consider $0<\mu^{\star}<\mu<1$. From the trace of the eavesdropper’s expected
steady-state estimation error covariance (see (21)), the inequality
$J^{e}(\mu^{\star})<J^{e}(\mu)$ is written as

$\displaystyle(1-\gamma^{e})(1-\mu^{*})\operatorname{trace}W^{e}+$
$\displaystyle\quad\gamma^{e}\operatorname{trace}S^{e}+(1-\gamma^{e})\mu^{*}\operatorname{trace}H^{e}$
$\displaystyle<$
$\displaystyle(1-\gamma^{e})(1-\mu)\operatorname{trace}W^{e}+$
$\displaystyle\quad\gamma^{e}\operatorname{trace}S^{e}+(1-\gamma^{e})\mu\operatorname{trace}H^{e}.$

By algebraic manipulation

$\begin{split}&\mu^{*}(1-\gamma^{e})(\operatorname{trace}H^{e}-\operatorname{trace}W^{e})\\\
<&\mu(1-\gamma^{e})(\operatorname{trace}H^{e}-\operatorname{trace}W^{e}).\end{split}$
(23)

From [19, Lemma 1], let $P^{*}=W^{e}$, $\hat{P}=H^{e}$ with $Q^{*}=\bar{P}$
and $\hat{Q}=P_{n}$. As $P_{n}=P^{OP}+\bar{P}$, it shows that $P_{n}>\bar{P}$.
Therefore $H^{e}>W^{e}$. As
$\operatorname{trace}H^{e}>\operatorname{trace}W^{e}$, it shows that
$\operatorname{trace}H^{e}-\operatorname{trace}W^{e}>0$. Therefore from the
inequality (23), it is shown that

$\mu^{\star}<\mu,$

which completes the proof. Considering the choice of $\mu$ in our encoding
scheme, we desire to meet two conditions. Firstly, the trace of the legitimate
user’s expected steady-state estimation error covariance should be upper
bounded by that of the trace of open-loop prediction, with
$J(\mu)<\operatorname{trace}P^{OP}$. This condition ensures that transmissions
from the sensor remain informative to the legitimate user.

Secondly, the trace of the eavesdropper’s expected steady-state estimation
error covariance should be greater than the trace of open loop prediction,
with $J^{e}(\mu)>\operatorname{trace}P^{OP}$. This condition ensures that the
confidentiality of the state is protected. The following Lemma provides an
upper bound for the trace of the legitimate user’s steady-state estimation
error covariance.

######  Lemma 5.

Consider the encoding design variable $\mu<1$ (11) and the legitimate user’s
channel quality $\bar{\gamma}<1$ (3). The trace of the expected steady-state
estimation error covariance for the legitimate user (21) is less than the
trace of open-loop performance, with $J(\mu)<\operatorname{trace}P^{OP}$.

Consider the encoding design variable $\mu<1$ and legitimate user’s channel
quality $\bar{\gamma}<1$, and the eavesdropper has been detected with
$\nu_{k}=1$. Suppose that there is a non-zero probability that the sensor’s
state estimate is received at the legitimate user
$\mathbb{P}[\lambda_{k}=1,\nu_{k}=1,u_{k}=0]>0$. Then, from the Borel–Cantelli
Lemma [8], a packet containing the state estimate from the sensor will arrive
at the legitimate user with probability one for some time $k>0$, that is,
$\mathbb{P}[\lambda_{k}=1,\nu_{k}=1,u_{k}=0,\text{ for some }k>0]=1$. The
estimation error covariance of the legitimate user when the state estimate is
received is $\bar{P}$, noting that $\bar{P}<P^{OP}$. Therefore,
$\lim_{k\to\infty}\mathbb{E}[P_{k}]<P^{OP}$ if $\mu<1$ and $\bar{\gamma}<1$
which shows $J(\mu)<\operatorname{trace}P^{OP}$. This completes the proof. The
following theorem provides a range for $\mu$ that protects data
confidentiality, whilst keeping transmissions informative to the legitimate
user.

######  Theorem 6.

Consider the encoding design variable in (11) is chosen in the range
$\mu^{OP}<\mu<1$, with $0\leq(\bar{\gamma},\gamma^{e})<1$, given that

$\mu^{OP}=\frac{\gamma^{e}(\operatorname{trace}S^{e}-\operatorname{trace}W^{e})+\operatorname{trace}W^{e}-\operatorname{trace}P^{OP}}{(\gamma^{e}-1)(\operatorname{trace}H^{e}-\operatorname{trace}W^{e})}.,$

then the following conditions hold

$\begin{split}J(\mu)&<\operatorname{trace}P^{OP}\textrm{ and}\\\
\operatorname{trace}P^{OP}&<J^{e}(\mu).\end{split}$

To show the first case, consider the encoding design variable $\mu<1$, then by
Lemma 5, $J(\mu)<\operatorname{trace}P^{OP}$.

To show the second case, consider $\mu^{OP}$ as the probability of the sensor
sending noise that ensures that $J^{e}(\mu^{OP})=\operatorname{trace}P^{OP}$.
Then from the trace of the eavesdropper’s steady-state estimation error
covariance in (22),

$\begin{split}\operatorname{trace}P^{OP}&=(1-\gamma^{e})(1-\mu^{OP})\operatorname{trace}W^{e}\\\
&+\gamma^{e}\operatorname{trace}S^{e}+(1-\gamma^{e})\mu^{OP}\operatorname{trace}H^{e}.\end{split}$
(24)

where $\mu^{OP}$ is found by re-arranging (24) in terms of $\mu^{OP}$.
Consider the monotonicty result in Lemma 4, and let $\mu^{OP}=\mu^{\star}$,
then for any $\mu>\mu^{OP}$, $J^{e}(\mu)>J^{e}(\mu^{OP})$, which shows that
$J^{e}(\mu)>\operatorname{trace}P^{OP}$. This completes the proof.

In comparison with the state secrecy codes developed for stable systems [22],
the current method allows one to drive the eavesdropper’s estimation error
covariance above open-loop prediction through transmitting noise with
$\mu>\mu^{OP}$. Thus, interceptions of packets are not informative to the
eavesdropper’s estimation process. Moreover, the eavesdropper intercepting
packets is actively harming its own estimation performance. With the
eavesdropper’s expected estimation error covariance above open-loop
prediction, the eavesdropper’s performance is worse than if it did not utilise
any packets at all from the sensor in its estimation process, i.e., if the
eavesdropper only predicted the state estimate using the system model in (1).
Since sending noise harms the legitimate user’s performance as well, we choose
$\mu<1$ so the performance of the legitimate user remains upper-bounded by
open loop prediction. This ensures that transmitted packets from the sensor
remain informative to the legitimate user for use in estimation.

##  8 Relative Performance Simulation

In this section, we illustrate the performance of the encoding scheme using
some examples. Consider the following second-order system (1) with parameters
$A=\begin{bmatrix}0.5&0.1\\\ 0.4&0.6\end{bmatrix}$, $C=I_{2}$,
$Q=R=10^{-2}I_{2}$, where $I$ is the identity, and $\rho(A)=0.7562<1$. This
gives $\operatorname{trace}\bar{P}=0.0114$, $\operatorname{trace}P_{n}=0.0502$
and $\operatorname{trace}P^{OP}=0.0388$. Consider a legitimate user channel
quality of $\bar{\gamma}=0.5$ and four cases of eavesdropper channel
qualities, with $\gamma^{e}_{1}=0.7$, $\gamma^{e}_{2}=\bar{\gamma}$
$\gamma^{e}_{3}=0.3$. We also consider the case that
$\gamma^{e}=0,\bar{\gamma}=1$. Fig. 4 shows the difference of the traces of
the steady-state expected estimation error covariance between the eavesdropper
and legitimate user $(J^{e}(\mu)-J(\mu))/J(\mu)$ against the encoding design
variable $\mu$ for these four cases.

Figure 4: Comparison of the absolute difference in trace of the expected
estimation error covariance of the legitimate user and eavesdropper with four
cases of channel qualities (worse, equal, better, much better). Eavesdropper
with worse channel quality in dashed blue, equal channel quality in dotted
magenta, eavesdropper with better channel quality in solid red, and
eavesdropper with perfect channel, legitimate user with no channel in dot-
dashed black.

In the case that the eavesdropper’s channel is worse than the legitimate
user’s, with $\bar{\gamma}<\gamma^{e}_{1}$ then the trace of the
eavesdropper’s estimation covariance is greater than the legitimate user’s for
$\mu\geq 0$ with $(J^{e}(\mu)-J(\mu))/J(\mu)>0$. Using Theorem 6, $\mu$ is
chosen in the range $0.7178<\mu<1$. In the case that the eavesdropper’s
channel is equal to the legitimate user’s with $\bar{\gamma}=\gamma^{e}_{2}$,
then the trace of the eavesdropper’s estimation covariance is greater than the
legitimate user’s for $\mu>0$, with $(J^{e}(\mu)-J(\mu))/J(\mu)>0$ if $\mu>0$,
and $(J^{e}(\mu)-J(\mu))/J(\mu)=0$ if $\mu=0$. Using Theorem 6, $\mu$ is
chosen in the range $0.7125<\mu<1$. In the case that the legitimate user’s
channel is worse than the eavesdropper’s with $\gamma^{e}_{3}<\bar{\gamma}$,
then the trace of the eavesdropper’s estimation covariance is greater than the
legitimate user’s for $\mu>0.14$. For $\mu\leq 0.14$, the trace of the
legitimate user’s expected estimation error covariance is greater than that of
the eavesdropper’s. Using Theorem 6, $\mu$ is chosen in the range
$0.7074<\mu<1$.

We also present the case that $\bar{\gamma}=1$ and $\gamma^{e}=0$, with the
legitimate user not receiving any packets from the sensor, and the
eavesdropper receives all packets transmitted from the sensor. If $\mu$ is
chosen $\mu>0.705$, then the trace of the eavesdropper’s expected estimation
error covariance is greater than the legitimate user’s and above open loop
prediction, even when the legitimate user is not receiving any transmissions
from the sensor.

##  9 Quickest Change Detection Formulation

To enact the proposed encoding scheme, the problem is to quickly detect the
presence of an eavesdropper, utilising the channel outcomes $\lambda_{k}$ at
the legitimate user. We present a Bayesian Quickest Change Detection (QCD)
statistical test and a corresponding efficient computation method.

Let $\lambda_{[0,k]}\triangleq\\{\lambda_{0},...,\lambda_{k}\\}$ be shorthand
for the legitimate user’s channel outcome sequence. We model the instance of
intrusion with a geometric distribution with
$\epsilon_{k}=\mathbb{P}[\Lambda=k]=(1-\kappa)^{k-1}\kappa$ with
$\kappa\in(0,1)$, where $\kappa$ is the probability that an eavesdropper
attack can occur. This formulation is amenable in the framework of a standard
QCD problem, where the goal is to detect a time in which the statistical
properties of an observed process change [18], which are the legitimate user’s
channel outcomes $\lambda_{k}$. Under the change description of the legitimate
user’s channel outcomes given in Section 2.3, the joint probability mass
function of the observed legitimate user packet outcomes is given by
$p_{\Lambda}(\lambda_{[0,k]})=\Pi_{i=1}^{\Lambda-1}(1-\gamma)\Pi_{j=\Lambda}^{k}(1-\bar{\gamma})$,
given that $\Pi_{j=\Lambda}^{k}(1-\bar{\gamma})=1$ when $k<\Lambda$.

Let $\mathcal{F}_{k}$ denote the filtration generated by $\lambda_{[0,k]}$ for
$k\geq 0$. We consider the probability space
$(\Omega,\mathcal{F},\mathbb{P}_{\Lambda})$, where $\Omega$ is the sample
space of infinite sequences $\\{\mathcal{F}_{k}:k\geq 0\\}$ and
$\mathcal{F}\triangleq\cup_{k=0}^{\infty}\mathcal{F}_{k}$ with the convention
that $\mathcal{F}_{0}=\\{0,\Omega\\}$ and $\mathbb{P}_{\Lambda}$ is the
probability measure constructed from the joint probability density
$p_{\Lambda}(\lambda_{[0,k]})$ using Kolmogorov’s extension theorem [9]. Let
us now construct a new probability measure
$\mathbb{P}_{\epsilon}(G)=\sum_{k=1}^{\infty}\epsilon_{k}\mathbb{P}_{k}(G)$,
for all $G\in\mathcal{F}$. We will denote the expectation operator associated
with $\mathbb{P}_{\epsilon}$ as $\mathbb{E}_{\epsilon}$. Our QCD problem is to
quickly detect a change in the statistical properties of $\lambda_{k}$, which
indicates that an active eavesdropper attack is occurring by designing a
stopping time $\tau>0$ with respect to the filtration $\mathcal{F}_{k}$. The
stopping time minimises the following unconstrained (Bayes risk) optimisation
problem of [21], to trade-off average detection delay with a probability of
false alarm as

$J(\tau)\triangleq
c\mathbb{E}_{\epsilon}[(\tau-\Lambda)^{+}]+\mathbb{P}_{\epsilon}[\tau<\Lambda],$
(25)

where $(\tau-\Lambda)^{+}\triangleq\max(0,\tau-\Lambda)$ and $c$ is the
penalty of each time step that alert is not declared after $\Lambda$.

###  9.1 Eavesdropper Detection

To enact the proposed encoding scheme, the legitimate user detects when an
eavesdropper attack is occurring to form the decision variable $\nu_{k}$ (9).
Let $\mathbb{P}_{\epsilon}(k<\Lambda|\mathcal{F}_{k})$ be the no-change
posterior probability. Following [23], the optimal stopping rule for the
Shiryaev cost criterion (25) is

$\tau^{*}=\inf\\{k\geq 1:\mathbb{P}_{\epsilon}(k<\Lambda|\mathcal{F}_{k})\leq
h\\},$ (26)

given that $h\in(0,1)$ is a threshold chosen to control the probability of
false alarms. At each time $k$, the no-change posterior
$\mathbb{P}_{\epsilon}(k<\Lambda|\mathcal{F}_{k})$ is calculated at the
legitimate user using its channel outcome $\lambda_{k}$. An optimal stopping
time is achieved under (26) to declare that an eavesdropper attack is
occurring. This forms the decision variable $\nu_{k}$ (9), which enacts the
encoding scheme.

$\nu_{k}=\begin{cases}0&\text{if $\tau^{*}$ has been declared}\\\
1&\text{otherwise}.\\\ \end{cases}$

###  9.2 Efficient Calculation of the No-Change Posterior

We now present an efficient recursive method to compute the no-change
posterior $\mathbb{P}_{\epsilon}(k<\Lambda|\mathcal{F}_{k})$. We define
$\hat{M}_{k}^{1}\triangleq\mathbb{P}_{\epsilon}(k<\Lambda|\mathcal{F}_{k})$.
Following [10, Lemma 1], given the sequence of legitimate user’s channel
outcome $\lambda_{k}$, the no-change posterior is provided by the scalar
recursion

$\hat{M}_{k}^{1}=N_{k}(1-\kappa)(1-\gamma)\hat{M}_{k-1}^{1},$

with $\hat{M}_{0}^{1}=1$, and where $N_{k}$ is the normalisation factor

$N^{-1}_{k}=(1-\bar{\gamma})+(1-\kappa)(\bar{\gamma}-\gamma)\hat{M}^{1}_{k-1}.$

##  10 Numerical Illustration with QCD

We show a numerical example to illustrate the results of QCD. Consider the
following second-order system detailed in Section 8. The eavesdropper
intrusion time is $\Lambda=700$. The nominal probability of channel dropout
for the legitimate user is $\gamma=0.2$. When the eavesdropper is performing
an attack, the legitimate user’s channel quality is degraded with
$\bar{\gamma}=0.3$. The eavesdropper’s probability of channel dropout is
$\gamma^{e}=0.3$. To demonstrate the need for QCD in detecting the
eavesdropper, we propose a moving average test over a window of length $N=100$
computed on the channel outcomes at the legitimate user for a finite time
horizon of $2000$ time steps. The moving average is displayed in Fig. 5.

Figure 5: The moving average of the legitimate user’s channel outcomes. The
moving average is displayed with a blue solid line. The eavesdropper intrusion
time is marked with a vertical red line. The theoretical mean of the channel
outcomes at the legitimate user when an eavesdropper attack is not occurring
is marked with a horizontal dashed black line, and when an eavesdropper attack
is occurring is marked in a horizontal dot-dashed black line.

The theoretical mean of the channel outcomes at the legitimate user under no
attack is $1-\gamma=0.8$, and when an eavesdropper attack occurring is
$1-\bar{\gamma}=0.7$. While it is visually apparent that there is a decrease
in the moving average, it is challenging to quickly detect the presence of the
eavesdropper with certainty, where a naive user would declare a false alarm at
time $k=668$ in Fig. 5. To quickly detect the presence of an eavesdropper, the
proposed QCD test is applied. Fig. 6 displays the no change posterior
calculated at the legitimate user for each time-step $k$.

Figure 6: The no change posterior QCD test at the legitimate user against
packet outcomes $\lambda_{k}$. The posterior probability of no change is
marked with a blue line. The eavesdropper intrusion time is marked with a
vertical red line. The zero false alarm threshold is marked by a horizontal
black dashed line.

The threshold is chosen at the minimum value that achieves zero false alarms.
This threshold for the given channel characteristics is $h=3\times 10^{-3}$
and was determined numerically via Monte-Carlo methods. The probability that
an eavesdropper performs an eavesdropping attack is $\kappa=5\times 10^{-6}$.
The legitimate user detects the presence of an eavesdropper at time $k=779$,
which is a detection delay of $79$ time steps. The posterior probability of no
change reduces to a small value after the eavesdropper intrusion time. This
indicates a strong probability of the presence of an active eavesdropper.
Therefore at $k=779$, a stopping time $\tau^{*}$ has been declared and the
legitimate alarms the sensor, with $\nu_{k}=0$. The indicator $\nu_{k}$ is
shared with the sensor. The sensor then enacts the randomised scheme of
transmitting noise or the state estimate.

To show the performance over time with the detection scheme, we show the trace
of the estimation error covariance of the legitimate user
($\operatorname{trace}P_{k}$, see (14)) and the eavesdropper
($\operatorname{trace}P_{k}^{e}$, see (16)) between the time window $k=650$
and $k=850$ in Fig. 7. The probability of sending noise $\mu$ is chosen to be
$0.8$, which satisfies Theorem 6.

Figure 7: The trace of the estimation error covariance of the legitimate user,
marked in solid blue, and eavesdropper, marked in dot-dashed orange for each
time. The eavesdropper intrusion time is marked by a vertical solid red line.
The eavesdropper detection time is marked by a vertical dotted black line.

Before the eavesdropper intrusion, the legitimate user is forming a state
estimate and a corresponding estimation error covariance. The trace of the
estimation error covariance increases when there are packet dropouts. After
the intrusion time, the eavesdropper performs an attack on the legitimate
user’s channel. There are more packet dropouts for the legitimate user and the
trace of the legitimate user’s estimation error covariance increases more
often. The eavesdropper intercepts packets at this time and forms its own
estimate of state and corresponding estimation error covariance. It is at time
$k=779$ that the eavesdropper is detected. The sensor enacts the encoding
scheme at this time. Under successive packet dropouts or transmissions of
noise from the sensor, the trace of the legitimate user’s estimation error
covariance is upper bounded by $\operatorname{trace}P^{OP}$. Under receiving a
packet of noise, the trace of the eavesdropper’s estimation error covariance
is $\operatorname{trace}P_{n}$.

##  11 Conclusion

This paper has developed an encoding scheme that is activated on the detection
of an active eavesdropper. We proposed that when the active eavesdropper is
performing an attack, there is a statistical change in the dropout probability
of the legitimate user’s channel. We propose a QCD statistical test to quickly
detect the presence of an eavesdropper. When an eavesdropper is detected, the
sensor transmits either its state estimate or noise based upon a pseudo-random
indicator. The encoding scheme ensures the confidentiality of the state of a
dynamic system, in a network environment without packet receipt
acknowledgments when the eavesdropper is performing an attack. Under our
encoding design, the expected steady-state estimation error covariance of the
eavesdropper is larger than that of the legitimate user. For certain design
choices, the trace of the eavesdropper’s expected steady-state estimation
error covariance is larger than the trace of the open-loop estimation error
covariance, whilst the trace of the legitimate user’s expected steady-state
estimation error covariance is smaller than the trace of open-loop estimation
covariance.

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On recovering non-local perturbation of non-selfadjoint Sturm–Liouville
operator

Maria Kuznetsova111Department of Mathematics, Saratov State University,
e-mail<EMAIL_ADDRESS>
Abstract. Recently, there appeared a significant interest in inverse spectral
problems for non-local operators arising in numerous applications. In the
present work, we consider the operator with frozen argument
$ly=-y^{\prime\prime}(x)+p(x)y(x)+q(x)y(a),$ which is a non-local perturbation
of the non-selfadjoint Sturm–Liouville operator. We study the inverse problem
of recovering the potential $q\in L_{2}(0,\pi)$ by the spectrum when the
coefficient $p\in L_{2}(0,\pi)$ is known. While the previous works were
focused only on the case $p=0,$ here we investigate the more difficult non-
selfadjoint case, which requires consideration of eigenvalues multiplicities.
We develop an approach based on the relation between the characteristic
function and the coefficients $\\{\xi_{n}\\}_{n\geq 1}$ of the potential $q$
by a certain basis. We obtain necessary and sufficient conditions on the
spectrum being asymptotic formulae of a special form. They yield that a part
of the spectrum does not depend on $q,$ i.e. it is uninformative. For the
unique solvability of the inverse problem, one should supplement the spectrum
with a part of the coefficients $\xi_{n},$ being the minimal additional data.
For the inverse problem by the spectrum and the additional data, we obtain a
uniqueness theorem and an algorithm.

Key words: inverse spectral problems, frozen argument, Sturm–Liouville
operators, non-local operators, necessary and sufficient conditions

2010 Mathematics Subject Classification: 34K29, 34A55

## Introduction

Inverse spectral problems consist in recovering operators from their spectral
characteristics. The classical results in this field were obtained for the
differential operators [1, 4, 2, 3, 5], which are local. Recently, in
connection with numerous applications, there appeared a considerable interest
in inverse problems for non-local operators [10, 9, 7, 11, 6, 12, 8], and, in
particular, for the operators with frozen argument [13, 14, 15, 16, 17, 18,
19, 20, 21, 24, 25, 23, 22]. Their studying is complicated by the fact that
non-local operators require development of non-standard methods.

In this paper, we study the recovery of a complex-valued potential $q\in
L_{2}(0,\pi)$ by the spectrum $\\{\lambda_{n}\\}_{n\geq 1}$ of the boundary
value problem

$ly:=-y^{\prime\prime}(x)+p(x)y(x)+y(a)q(x)=\lambda y(x),\quad x\in(0,\pi),$
(1) $y^{(\alpha)}(0)=y^{(\beta)}(\pi)=0,\quad\alpha,\beta\in\\{0,1\\},$ (2)

where $a\in[0,\pi]$ and $p\in L_{2}(0,\pi)$ is complex-valued. Operators $ly$
are usually called Sturm–Liouville operators with frozen argument. They have a
close relation to the operators with integral boundary conditions [27, 28, 26,
12, 8], which arise in studying diffusion and heating processes and in the
theory of elasticity, see [31, 29, 30, 32]. In connection with this, in [33],
some spectral properties of the operator $ly$ were established in the case of
periodic boundary conditions. However, the mentioned work does not address an
inverse spectral problem.

The previous studies of inverse spectral problems for Sturm–Liouville
operators with frozen argument [13, 14, 15, 16, 17, 18, 19, 20, 21, 24, 25,
23, 22] were focused only on the case $p=0.$ In this particular case, a
comprehensive study of recovering $q$ by the spectrum required a series of the
works [15, 16, 17, 18, 20, 25, 22]. The most general approach to the operator
with frozen argument was developed in [22], which allowed us to obtain
necessary and sufficient conditions on the spectrum and, afterwards, a uniform
stability of the inverse problem [25].

In [23], there was suggested another approach to the operator with frozen
argument within the framework of perturbation theory. According to it,
operator $ly$ could be treated as a one-dimensional perturbation of the
Sturm–Liouville operator $Ay:=-y^{\prime\prime}+p(x)y.$ However, for studying
the spectral properties of $ly,$ this approach needs the selfadjointness of
the operator $A.$ By this reason, it is inapplicable to the case of complex-
valued $p$ considered here, since the unperturbed operator $A$ is non-
selfadjoint.

Here, we develop an approach to the general situation of arbitrary $p\in
L_{2}(0,\pi),$ relying on some ideas of [22]. We significantly extend the
mentioned ideas to take into account the non-selfadjointness of the
unperturbed operator, which requires consideration of eigenvalues
multiplicities.

Main results of the paper consist in necessary and sufficient conditions on
the spectrum, a uniqueness theorem and an algorithm. As in the previous work
[22], the necessary and sufficient conditions are asymptotic formulae of a
special form. They give the so-called degeneration condition that some part of
the eigenvalues is uninformative, i.e. it does not depend on $q.$ However,
compared to [22], the non-selfadjointness of the unperturbed operator leads to
a new effect that the degeneration condition includes a restriction on the
minimal possible multiplicity of each uninformative eigenvalue.

The paper is organized as follows. In Section 1, we introduce necessary
objects and provide auxiliary statements. As well we obtain a characteristic
function of boundary value problem (1), (2) and a main equation of the inverse
problem. In Section 2, by necessity, we establish the conditions on the
spectrum. The main results and their proves are given in Section 3.

##  1 Preliminaries. Main equation of the inverse problem

First, we consider the unperturbed boundary value problem for the classical
Sturm–Liouville equation

$-y^{\prime\prime}(x)+p(x)y(x)=\lambda y(x),\quad x\in(0,\pi),$ (3)

with boundary conditions (2). Denote by $S_{a}(x,\lambda)$ and
$C_{a}(x,\lambda)$ the solutions of equation (3) under the initial conditions

$C_{a}(a,\lambda)=1,\;C^{\prime}_{a}(a,\lambda)=0,\quad
S_{a}(a,\lambda)=0,\;S^{\prime}_{a}(a,\lambda)=1;$

here and below the prime symbol stands for the derivative with respect to the
first argument. Let us agree that $\int_{a}^{b}f(t)\,dt$ is understood as
$-\int_{b}^{a}f(t)\,dt$ when $b<a.$ For $x\in[0,\pi],$ using transformation
operators (see, e.g. [4]), we obtain the following representations:

$\left.\begin{array}[]{c}\displaystyle
S_{a}(x,\lambda)=\frac{\sin\rho(x-a)}{\rho}-\frac{\cos\rho(x-a)}{\rho^{2}}\omega_{a}(x)+\int_{a}^{x}K_{a,1}(x,t)\frac{\cos\rho(t-a)}{\rho^{2}}\,dt,\\\\[11.38109pt]
\displaystyle
S^{\prime}_{a}(x,\lambda)=\cos\rho(x-a)+\frac{\sin\rho(x-a)}{\rho}\omega_{a}(x)+\int_{a}^{x}K_{a,2}(x,t)\frac{\sin\rho(t-a)}{\rho}\,dt,\\\\[11.38109pt]
\displaystyle
C_{a}(x,\lambda)=\cos\rho(x-a)+\frac{\sin\rho(x-a)}{\rho}\omega_{a}(x)+\int_{a}^{x}K_{a,3}(x,t)\frac{\sin\rho(t-a)}{\rho}\,dt,\\\\[11.38109pt]
\displaystyle
C^{\prime}_{a}(x,\lambda)=-\rho\sin\rho(x-a)+\cos\rho(x-a)\omega_{a}(x)+\int_{a}^{x}K_{a,4}(x,t)\cos\rho(t-a)\,dt,\end{array}\right\\}$
(4)

where $\rho^{2}=\lambda$ and $\omega_{a}(x)=\frac{1}{2}\int_{a}^{x}p(t)\,dt.$
In (4), for each fixed $x,$ there determined $K_{a,j}(x,\cdot)\in
L_{2}(0,\pi),$ and $K_{a,j}\in L_{2}[0,\pi]^{2},$ $j=\overline{1,4}.$

Consider the entire function
$\Delta_{0}(\lambda)=C_{a}^{(\alpha)}(0,\lambda)S_{a}^{(\beta)}(\pi,\lambda)-S_{a}^{(\alpha)}(0,\lambda)C_{a}^{(\beta)}(\pi,\lambda),$
wherein $y^{(0)}:=y$ and $y^{(1)}:=y^{\prime}.$ It is easy to see that
$\Delta_{0}$ is a characteristic function of unperturbed boundary value
problem (2), (3), i.e. a number $\mu_{n}$ is its eigenvalue if and only if
$\Delta_{0}(\mu_{n})=0.$ By $\\{\mu_{n}\\}_{n\geq 1}$ we denote the spectrum
of (2), (3), being the sequence of the eigenvalues taken with the account of
algebraic multiplicities. The following asymptotics are known (see [4]):

$\mu_{n}=\theta_{n}^{2},\quad\theta_{n}=n-\frac{\alpha+\beta}{2}+\frac{\omega}{\pi
n}+\frac{\kappa_{n}}{n},\;n\geq
1,\quad\omega:=\frac{1}{2}\int_{0}^{\pi}p(t)\,dt,\quad\\{\kappa_{n}\\}_{n\geq
1}\in\ell_{2}.$ (5)

By $m_{n}$ we denote multiplicity of the eigenvalue $\mu_{n}.$ By asymptotics
(5), for a sufficiently large $n,$ we have $m_{n}=1.$ Without loss of
generality, we assume that equal values in the spectrum follow each other.
Then, we have

$\mu_{n}=\mu_{n+1}=\ldots=\mu_{n+m_{n}-1},\quad
n\in\mathcal{S},\quad{\mathcal{S}}:=\\{n\geq
2\colon\mu_{n}\neq\mu_{n-1}\\}\cup\\{1\\}.$

The index $n\in\mathcal{S}$ corresponds to the unique elements in
$\\{\mu_{n}\\}_{n\geq 1},$ while for $n\in\mathcal{S}$ and
$\nu\in\overline{0,m_{n}-1},$ the index $k=n+\nu$ runs through $\mathbb{N}.$

Now, we are ready to study boundary value problem (1), (2). Introduce the
solutions $S(x,\lambda)$ and $C(x,\lambda)$ of equation (1) under the initial
conditions

$S(a,\lambda)=0,\;S^{\prime}(a,\lambda)=1,\quad
C(a,\lambda)=1,\;C^{\prime}(a,\lambda)=0.$

Any other solution of (1) is a linear combination of $S(x,\lambda)$ and
$C(x,\lambda).$ It is easy to see that

$S(x,\lambda)=S_{a}(x,\lambda),\quad
C(x,\lambda)=C_{a}(x,\lambda)+\int_{a}^{x}W(x,t,\lambda)q(t)\,dt,$ (6)

where
$W(x,t,\lambda):=C_{a}(t,\lambda)S_{a}(x,\lambda)-C_{a}(x,\lambda)S_{a}(t,\lambda).$
We introduce the entire function

$\Delta(\lambda)=C^{(\alpha)}(0,\lambda)S^{(\beta)}(\pi,\lambda)-S^{(\alpha)}(0,\lambda)C^{(\beta)}(\pi,\lambda).$
(7)

Then, $\Delta(\lambda)$ is a characteristic function of boundary value problem
(1), (2), while $\\{\lambda_{n}\\}_{n\geq 1}$ is a sequence of its zeroes
taken with the account of multiplicities.

Substituting (6) into (7), we obtain

$\Delta(\lambda)=\Delta_{0}(\lambda)-S_{a}^{(\beta)}(\pi,\lambda)\int_{0}^{a}W^{(\alpha)}(0,t,\lambda)\,q(t)\,dt-
S_{a}^{(\alpha)}(0,\lambda)\int_{a}^{\pi}W^{(\beta)}(\pi,t,\lambda)\,q(t)\,dt.$
(8)

Following the approach in [22], we should substitute into (8) the values
$\lambda=\mu_{n},$ being the zeroes of the main part $\Delta_{0}(\lambda).$ In
the paper [22] corresponding to the case $p=0,$ we had
$\mu_{n}=\big{(}n-\frac{\alpha+\beta}{2}\big{)}^{2},$ being simple
eigenvalues. Here, we have to take into account that $\mu_{n}$ may be
multiple. For each $n\in\mathcal{S},$ we differentiate the both parts of
formula (8) $\nu=\overline{0,m_{n}-1}$ times and put $\lambda=\mu_{n}.$ Since

$\frac{\partial^{\nu}}{\partial\lambda^{\nu}}\big{[}C^{(\alpha)}_{a}(0,\lambda)S^{(\beta)}_{a}(\pi,\lambda)\big{]}=\frac{\partial^{\nu}}{\partial\lambda^{\nu}}[S^{(\alpha)}_{a}(0,\lambda)C^{(\beta)}_{a}(\pi,\lambda)],\quad
n\in{\cal S},\;\nu=\overline{0,m_{n}-1},$ (9)

in $\lambda=\mu_{n},$ we obtain

$\Delta^{(\nu)}(\lambda)=\left[S_{a}^{(\beta)}(\pi,\lambda)\int_{0}^{\pi}g(t,\lambda)q(t)\,dt\right]^{(\nu)},\quad\lambda=\mu_{n},\;n\in{\cal
S},\;\nu=\overline{0,m_{n}-1},$ (10)

where $g(t,\lambda):=-W^{(\alpha)}(0,t,\lambda).$ One can see that
$g(t,\lambda)=S_{0}(x,\lambda)$ if $\alpha=0,$ and
$g(t,\lambda)=-C_{0}(x,\lambda)$ if $\alpha=1.$ Moreover,
$g^{(\beta)}(\pi,\lambda)=\Delta_{0}(\lambda).$

For $n\in{\cal S}$ and $\nu=\overline{0,m_{n}-1},$ we introduce

$a_{n+\nu}=\frac{n^{1-\beta}}{\nu!}\left.\frac{\partial^{\nu}S_{a}^{(\beta)}(\pi,\lambda)}{\partial\lambda^{\nu}}\right|_{\lambda=\mu_{n}},\quad
g_{n+\nu}(t)=\frac{n^{1-\alpha}}{(m_{n}-\nu-1)!}\left.\frac{\partial^{m_{n}-\nu-1}g(t,\lambda)}{\partial\lambda^{m_{n}-\nu-1}}\right|_{\lambda=\mu_{n}}.$

Note that $a_{n}$ and $g_{n}$ are the objects constructed by unperturbed
boundary value problem (2), (3), and they are known. Formulae (4) and (5)
yield that $a_{n}=O(1)$ and $\\{g_{n}(t)\\}_{n\geq 1}$ is an almost normalized
system. At the same time, $\\{g_{n}(t)\\}_{n\geq 1}$ is constructed from
eigen- and associated functions of the operator $Ay=-y^{\prime\prime}+p(x)y$
considered under strongly regular conditions (2). By the mentioned reasons,
the following proposition holds (see, e.g., [34, 35]).

######  Proposition 1.

The functional sequence $\\{g_{n}(t)\\}_{n\geq 1}$ is a Riesz basis in
$L_{2}(0,\pi).$

Applying the general Leibniz rule in (10), we get

$n^{2-\alpha-\beta}\frac{\Delta^{(\nu)}(\mu_{n})}{\nu!}=\sum_{\eta=0}^{\nu}a_{n+\nu-\eta}\xi_{n+m_{n}-1-\eta},\quad
n\in{\cal S},\;\nu=\overline{0,m_{n}-1},$ (11)

where $\xi_{k}:=\int_{0}^{\pi}g_{k}(t)q(t)\,dt.$ Further, we consider (11) as
an equation with respect to the coefficients $\\{\xi_{n}\\}_{n\geq 1}.$ From
Proposition 1 it follows that $\\{\xi_{n}\\}_{n\geq 1}\in\ell_{2}$ and its
knowledge allows one to recover $q$ uniquely. Formula (11) is called main
equation of the inverse problem.

##  2 Necessary conditions

In this section, we obtain necessary conditions on the spectrum
$\\{\lambda_{n}\\}_{n\geq 1},$ which consist in asymptotic formulae (22). In
the next section, we prove that they are the sufficient conditions as well.

Remind that $\lambda_{n}=\rho_{n}^{2}$ and $\mu_{n}=\theta_{n}^{2}.$ Without
loss of generality, we assume that
$\arg\rho_{n},\,\arg\theta_{n}\in[-\frac{\pi}{2},\frac{\pi}{2}).$ We start by
obtaining the weakest asymptotics for $\rho_{n}.$

######  Lemma 1.

The following asymptotics hold: $\rho_{n}=\theta_{n}+o(1),$ $n\geq 1.$

Proof.  For definiteness, we provide computations in the case $\alpha=1$ and
$\beta=0$ (the other cases are proceeded analogously). By asymptotics (5), it
is sufficient to prove that $\rho_{n}=n-\frac{1}{2}+o(1).$ For
$\lambda=\rho^{2},$ we consider $\Delta(\lambda)$ as an entire function of
$\rho.$ Then, $\\{\rho_{n}\\}_{n\geq 1}\cup\\{-\rho_{n}\\}_{n\geq 1}$ is the
set of all its zeroes.

Denote $\tau=|\mathrm{Im}\,\rho|.$ Using the corresponding formulae in (4),
for $\rho\to\infty,$ we obtain

$W^{\prime}(0,t,\lambda)=-C_{0}(t,\lambda)=O\left(e^{\tau t}\right),\quad
W(\pi,t,\lambda)=-S_{\pi}(t,\lambda)=O\left(\frac{e^{\tau(\pi-t)}}{\rho}\right),$
$S_{a}(\pi,\lambda)=O\left(\frac{e^{\tau(\pi-a)}}{\rho}\right),\
S_{a}^{\prime}(0,\lambda)=O\left(e^{\tau a}\right),\
\Delta_{0}(\lambda)=-C_{0}(\pi,\lambda)=-\cos\rho\pi+O\left(\frac{e^{\tau\pi}}{\rho}\right).$

Substituting these relations into (8), we get

$\Delta(\lambda)=\Delta_{0}(\lambda)+O\left(\frac{e^{\tau\pi}}{\rho}\right)=\cos\rho\pi+O\left(\frac{e^{\tau\pi}}{\rho}\right).$
(12)

For any $\delta\in(0,\frac{1}{4}),$ we have the following estimate with
$C_{\delta}>0$ (see [4]):

$|\cos\rho\pi|>C_{\delta}\,e^{\tau\pi},\quad\rho\in
G_{\delta}:=\left\\{z\in\mathbb{C}\colon\left|z-n+\frac{1}{2}\right|\geq\delta,\;n\in{\mathbb{Z}}\right\\}.$
(13)

Then, by (12), there exists $N_{\delta}\in\mathbb{N}$ such that
$|\cos\rho\pi|\geq|\cos\rho\pi-\Delta(\rho^{2})|$ as soon as $\rho\in
G_{\delta}$ and $|\rho|\geq N_{\delta}.$ Applying Rouche’s theorem, we arrive
at that in the circle $|\rho|<N_{\delta},$ the functions $\cos\rho\pi$ and
$\Delta(\rho^{2})$ have the same number of zeroes. Analogously, in each circle
$|\rho-n+\frac{1}{2}|<\delta,$ where $n\in\mathbb{Z}$ is such that
$|n-\frac{1}{2}|>N_{\delta},$ there is exactly one zero of $\Delta(\rho^{2}).$
Taking into account that $\Delta(\rho^{2})$ and $\cos\rho\pi$ are even
functions of $\rho,$ and that $\delta$ can be arbitrarily small, we arrive at
the needed asymptotics. $\Box$

Further, we clarify the obtained in Lemma 1 necessary conditions on the
spectrum.

######  Theorem 1.

I. Let $K\in\mathbb{N}$ be such that for $n\geq K,$ all $m_{n}=1.$ Then, the
following asymptotics hold:

$\lambda_{n}=\mu_{n}+a_{n}\varkappa_{n},\;n\geq
K,\quad\\{\varkappa_{n}\\}_{n\geq K}\in\ell_{2}.$ (14)

II. For $n\in\mathcal{S},$ denote $k_{n}=\min(m_{n},r_{n}),$ where $r_{n}$ is
multiplicity of $\lambda=\mu_{n}$ as a zero of the entire function
$S_{a}^{(\beta)}(\pi,\lambda).$ Then, there exists such numeration of
$\\{\lambda_{n}\\}_{n\geq 1}$ that

$\lambda_{n}=\lambda_{n+1}=\ldots=\lambda_{n+k_{n}-1}=\mu_{n},\quad
n\in{\mathcal{S}}.$ (15)

Proof.

I. By Lemma 1, we have $\rho_{n}=\theta_{n}+\eta_{n},$ where $\eta_{n}=o(1).$
To prove (14), we substitute $\lambda=\rho_{n}^{2}$ into (8) and, using Taylor
series, obtain asymptotics for $\eta_{n},$ where $n\geq K.$

For definiteness, consider the case $\alpha=1$ and $\beta=0$ (the other cases
are proceeded analogously). Since $\Delta_{0}(\lambda)=-C_{0}(\pi,\lambda),$
substituting $\lambda=\rho_{n}^{2}$ into the corresponding formula in (4),
applying trigonometric formulae along with asymptotics (5) and
$\eta_{n}=o(1),$ we get

$\Delta_{0}(\lambda_{n})=-\cos\theta_{n}\pi-\frac{\omega}{\theta_{n}}\sin\theta_{n}\pi-\int_{0}^{\pi}K_{0,3}(\pi,t)\frac{\sin\theta_{n}t}{\theta_{n}}\,dt+(-1)^{n+1}\eta_{n}\pi+o(\eta_{n}).$

In this formula, the first three summands compose $\Delta_{0}(\mu_{n})=0,$ and
we arrive at

$\Delta_{0}(\lambda_{n})=\eta_{n}\big{(}(-1)^{n+1}+o(1)\big{)}.$ (16)

Proceeding analogously, based on (4), we also obtain the following
asymptotics:

$\left.\begin{array}[]{cc}S_{a}(t,\lambda_{n})=S_{a}(t,\mu_{n})+O(n^{-1}\eta_{n}),&S_{a}^{\prime}(t,\lambda_{n})=S_{a}^{\prime}(t,\mu_{n})+O(\eta_{n}),\\\\[11.38109pt]
C_{a}(t,\lambda_{n})=C_{a}(t,\mu_{n})+O(\eta_{n}),&C_{a}^{\prime}(t,\lambda_{n})=C_{a}^{\prime}(t,\mu_{n})+O(n\eta_{n})\end{array}\right\\}$
(17)

uniformly on $t\in[0,\pi].$ Using (17), we have

$A_{n}:=S_{a}(\pi,\lambda_{n})\int_{0}^{a}W^{\prime}(0,t,\lambda_{n})q(t)\,dt=\\\
=\left(S_{a}(\pi,\mu_{n})+O\left(\frac{\eta_{n}}{n}\right)\right)\left(-\int_{0}^{a}g_{n}(t)q(t)\,dt+O(\eta_{n})\right).$

Since $g_{n}(t)=-\cos\theta_{n}t+O(n^{-1}),$ by the Riemann–Lebesgue lemma,
$\int_{0}^{a}g_{n}(t)q(t)\,dt=o(1).$ Using also that
$S_{a}(\pi,\mu_{n})=O(n^{-1}),$ we obtain

$A_{n}=-S_{a}(\pi,\mu_{n})\int_{0}^{a}g_{n}(t)q(t)\,dt+o(\eta_{n}).$ (18)

Analogously, we have

$B_{n}:=S_{a}^{\prime}(0,\lambda_{n})\int_{a}^{\pi}W(\pi,t,\lambda_{n})q(t)\,dt=S_{a}^{\prime}(0,\mu_{n})\int_{a}^{\pi}W(\pi,t,\mu_{n})q(t)\,dt+o(\eta_{n}).$

Relation (9) yields that
$S_{a}^{\prime}(0,\mu_{n})W(\pi,t,\mu_{n})=-S_{a}(\pi,\mu_{n})g_{n}(t),$ and

$B_{n}=-S_{a}(\pi,\mu_{n})\int_{a}^{\pi}g_{n}(t)q(t)\,dt+o(\eta_{n}).$ (19)

Combining (16), (18), and (19) with (8) in $\lambda=\rho_{n}^{2},$ we obtain

$0=((-1)^{n}+o(1))\eta_{n}+S(\pi,\mu_{n})\xi_{n},\quad\\{\xi_{n}\\}_{n\geq
1}\in\ell_{2}.$

For $n\geq K,$ the value $\eta_{n}$ is the unique solution of this equation,
which leads to $\eta_{n}=a_{n}\nu_{n}n^{-1}$ with $\\{\nu_{n}\\}_{n\geq
K}\in\ell_{2},$ and to (14).

II. From the definition it follows that

$a_{n}=a_{n+1}=\ldots=a_{n+k_{n}-1}=0.$ (20)

Then, by formula (11), $\mu_{n}$ is a zero of $\Delta(\lambda)$ of
multiplicity not less then $k_{n}.$ This means that $\mu_{n}$ occurs in the
spectrum $\\{\lambda_{n}\\}_{n\geq 1}$ at least $k_{n}$ times, and (15) holds
up to a numeration. $\Box$

In what follows, we can assume that the numeration of
$\\{\lambda_{n}\\}_{n\geq 1}$ satisfies (15). Denote

$\Omega=\\{n+\nu\colon
n\in{\mathcal{S}},\;\nu=\overline{0,k_{n}-1}\\},\quad\overline{\Omega}={\mathbb{N}}\setminus\Omega.$
(21)

Formula (15) yields that the part of the spectrum
$\\{\lambda_{n}\\}_{n\in\Omega}$ does not depend on $q,$ i.e. we have the
degeneration condition. Each unique eigenvalue $\lambda_{n}=\mu_{n}$ in this
part occurs $k_{n}$ times, which restricts its multiplicity to be not less
than $k_{n}.$ Note that (15) follows from (14) when $n\geq K$ since $k_{n}\leq
1.$ In [22], condition (15) was not required because for $p=0,$ we can take
$K=1.$

Now, we unify conditions (14) and (15) into one formula. Introduce the values

$b_{n+\nu}=\left\\{\begin{array}[]{cc}a_{n+\nu},&\nu=\overline{0,p_{n}-1},\\\\[5.69054pt]
1,&\nu=\overline{p_{n},m_{n}-1},\end{array}\right.\;p_{n}:=\max(1,k_{n}),\quad
n\in{\mathcal{S}}.$

Then, formulae (14) and (15) are particular cases of the following relation:

$\lambda_{n}=\mu_{n}+b_{n}\varkappa_{n},\quad n\geq
1,\quad\\{\varkappa_{n}\\}_{n\geq 1}\in\ell_{2}.$ (22)

It differs from (14) and (15) only by a finite number of formulae for
$n=k+\nu<K,$ $k\in\mathcal{S},$ $\nu\in\overline{k_{n},m_{n}-1}$ with
$b_{n}\neq 0,$ being non-restrictive. By this reason, (22) is equivalent to
(14) along with (15). In particular, $n\in\Omega$ if and only if $b_{n}=0.$

Note that $b_{n}=O(1),$ then the following asymptotics is weaker than (22):

$\lambda_{n}=\rho_{n}^{2},\quad\rho_{n}=n-\frac{\alpha+\beta}{2}+\frac{\omega}{\pi
n}+\frac{\nu_{n}}{n},\quad\\{\nu_{n}\\}_{n\geq 1}\in\ell_{2}.$ (23)

By the standard approach involving Hadamard’s factorization theorem (see,
e.g., [4]), one can prove that

$\Delta(\lambda)=(-1)^{\alpha}\pi^{\delta_{\alpha,\beta}}\prod_{k=1}^{\infty}\frac{\lambda_{k}-\lambda}{\xi_{k}},\quad\xi_{k}:=\left\\{\begin{array}[]{cc}\left(k-\frac{\alpha+\beta}{2}\right)^{2},&k\geq
1\text{ or }\alpha+\beta<2,\\\\[5.69054pt]
1,&k=1,\;\alpha=\beta=1.\end{array}\right.$ (24)

Thus, by the spectrum we can uniquely reconstruct the characteristic function
$\Delta(\lambda).$

##  3 Main results

First, we obtain the necessary and sufficient conditions on the spectrum.

######  Theorem 2.

For an arbitrary sequence $\\{\lambda_{n}\\}_{n\geq 1}$ of complex numbers to
be the spectrum of boundary value problem (1), (2) with some $q\in
L_{2}(0,\pi),$ it is necessary and sufficient to satisfy (22).

For the proof, we need the following lemma.

######  Lemma 2.

Let $\Delta(\lambda)$ be constructed via (24), where arbitrary values
$\\{\lambda_{n}\\}_{n\geq 1}$ satisfy asymptotics (23). Then, the following
representation holds:

$\Delta(\lambda)=\left\\{\begin{array}[]{cc}\displaystyle\rho^{2\alpha}\left(\frac{\sin\rho\pi}{\rho}-\frac{\cos\rho\pi}{\rho^{2}}\omega+\int_{0}^{\pi}\frac{\cos\rho
t}{\rho^{2}}\,W(t)\,dt\right),&\alpha=\beta,\\\\[11.38109pt]
\displaystyle(-1)^{\alpha}\left(\cos\rho\pi+\frac{\sin\rho\pi}{\rho}\omega+\int_{0}^{\pi}\frac{\sin\rho
t}{\rho}\,W(t)\,dt\right),&\alpha\neq\beta,\end{array}\right.$ (25)

where $W\in L_{2}(0,\pi).$

For $\alpha=\beta=0,$ the statement of the lemma easily follows from Lemma 3.3
in [6] after integration in parts. For the other combinations of $\alpha$ and
$\beta,$ the needed statements are proved by analogous computations.

###### Proof of Theorem 2.

The necessity part was proved in the previous section. Let us prove the
sufficiency part. Construct the function $\Delta(\lambda)$ via formula (24)
using the given numbers $\\{\lambda_{n}\\}_{n\geq 1}.$ Condition (22) yields
asymptotics (23), and, by Lemma 2, $\Delta(\lambda)$ has the form (25).

Now, we should find a function $q$ such that its coefficients
$\xi_{k}=\int_{0}^{\pi}q(t)g_{k}(t)\,dt$ satisfy (11). For every
$n\in\mathcal{S},$ relation (11) can be considered as a system of $m_{n}$
linear equations with respect to the vector $[\xi_{n+\nu}]_{\nu=0}^{m_{n}-1}:$

$\left\\{\begin{array}[]{c}a_{n}\xi_{n+m_{n}-1}=n^{2-\alpha-\beta}\Delta(\mu_{n}),\\\\[8.53581pt]
\displaystyle
a_{n}\xi_{n+m_{n}-2}+a_{n+1}\xi_{n+m_{n}-1}=n^{2-\alpha-\beta}\frac{\Delta^{\prime}(\mu_{n})}{1!},\\\\[8.53581pt]
\ldots\\\ \displaystyle
a_{n}\xi_{n}+a_{n+1}\xi_{n+1}+\ldots+a_{n+m_{n}-1}\xi_{n+m_{n}-1}=n^{2-\alpha-\beta}\frac{\Delta^{(m_{n}-1)}(\mu_{n})}{(m_{n}-1)!}.\end{array}\right.$
(26)

By (22), we have
$\Delta(\mu_{n})=\Delta^{\prime}(\mu_{n})=\ldots=\Delta^{(k_{n}-1)}(\mu_{n})=0.$
This along with (15) yields that first $k_{n}$ rows in system (26) turn
trivial identities and that arbitrary values of $\xi_{n+\nu},$
$\nu=\overline{0,k_{n}-1},$ satisfy this system. If $k_{n}<m_{n},$ then
$a_{n+k_{n}}\neq 0,$ and the rest $\xi_{n+\nu}$ are uniquely determined by
subsequent application of the following formulae:

$\left.\begin{array}[]{c}\displaystyle\xi_{n+m_{n}-1}=n^{2-\alpha-\beta}\frac{\Delta^{(k_{n})}(\mu_{n})}{a_{n+k_{n}}k_{n}!},\\\\[8.53581pt]
\displaystyle\xi_{n+m_{n}-\nu}=\frac{1}{a_{n+k_{n}}}\left(n^{2-\alpha-\beta}\frac{\Delta^{(k_{n}+\nu-1)}(\mu_{n})}{(k_{n}+\nu-1)!}-\sum_{\eta=1}^{\nu-1}a_{n+k_{n}+\eta}\xi_{n+m_{n}-\nu+\eta}\right),\;\nu=\overline{2,m_{n}-k_{n}}.\end{array}\right\\}$
(27)

Remind that for a sufficiently large $n\geq K,$ we have $m_{n}=1,$ and
$\xi_{n}$ either can be arbitrary (if $k_{n}=1$) or it is computed via the
first formula in (27) (if $k_{n}=0$).

Thus, we arrive at that the part of the coefficients
$\\{\xi_{k}\\}_{k\in\overline{\Omega}}$ is uniquely determined by
$\\{\lambda_{n}\\}_{n\geq 1},$ while $\\{\xi_{k}\\}_{k\in\Omega}$ can be
arbitrary (for the definition of $\Omega$ and $\overline{\Omega},$ see (21)).
Applying the scheme from the proof of Theorem 1 in [22], using representation
(25), we obtain that
$\\{n^{2-\alpha-\beta}\Delta(\mu_{n})a^{-1}_{n}\\}_{n\in\overline{\Omega}}\in\ell_{2},$
and $\\{\xi_{k}\\}_{k\in\overline{\Omega}}\in\ell_{2}.$ Choose arbitrary
coefficients $\\{\xi_{k}\\}_{k\in\Omega}\in l_{2}.$ Then, there exists $q\in
L_{2}(0,\pi)$ such that its coefficients with respect to the basis
$\\{g_{k}(t)\\}_{k\geq 1}$ are $\\{\xi_{k}\\}_{k\geq 1}.$

Consider boundary value problem (1), (2) with such potential $q.$ Let
$\Delta_{*}(\lambda)$ be the characteristic function of this boundary value
problem. Then, by construction,

$F(\lambda)=\frac{\Delta_{*}(\lambda)-\Delta(\lambda)}{\Delta_{0}(\lambda)}$

is an entire function. Representations (8) and (25) along with (4) yield
asymptotics

$\Delta_{*}(\lambda)-\Delta(\lambda)=O\left(\rho^{\alpha+\beta-2}e^{\tau\pi}\right),\quad\rho^{2}=\lambda,\;\tau=|{\mathrm{I}m}\rho|.$
(28)

Using (4), we also arrive at

$\Delta_{0}(\lambda)=\left\\{\begin{array}[]{cc}\displaystyle(-1)^{\alpha}\left(\cos\rho\pi+O\Big{(}\frac{e^{\tau\pi}}{\rho}\Big{)}\right),&\alpha\neq\beta,\\\\[8.53581pt]
\displaystyle\rho^{2\alpha-1}\left(\sin\rho\pi+O\Big{(}\frac{e^{\tau\pi}}{\rho}\Big{)}\right),&\alpha=\beta.\end{array}\right.$

Consider arbitrary $\delta\in(0,\frac{1}{4}).$ For a sufficiently large
$|\lambda|\geq N_{\delta},$ analogously to (13), one can prove that

$|\Delta_{0}(\lambda)|\geq
C_{\delta}|\rho|^{\alpha+\beta-1}e^{\tau\pi},\quad\rho\in
G_{\delta}=\left\\{z\in\mathbb{C}\colon\Big{|}z-n+\frac{\alpha+\beta}{2}\Big{|}\geq\delta,\;n\in\mathbb{Z}\right\\},$
(29)

where $C_{\delta}>0.$ Using (28) and (29), we arrive at $F(\lambda)=o(1)$ in
$G_{\delta}.$ By the maximum modulus principle and Liouville’s theorem,
$F(\lambda)=0.$ Thus, the function $\Delta(\lambda)$ is the characteristic
function of the boundary value problem (1), (2) with the considered potential
$q,$ and $\\{\lambda_{n}\\}_{n\geq 1}$ is its spectrum. ∎

From the proof of Theorem 2, it follows that the potentials $q$ corresponding
to one and the same spectrum have the same coefficients
$\\{\xi_{n}\\}_{n\in\overline{\Omega}},$ while for $n\in\Omega,$ the
coefficients $\xi_{n}$ may differ. At the same time, by Proposition 1, the
mapping $q\mapsto\\{\xi_{n}\\}_{n\geq 1}$ is a one-to-one correspondence
between $L_{2}(0,\pi)$ and $\ell_{2}.$ Thus, for a fixed spectrum
$\\{\lambda_{n}\\}_{n\geq 1},$ one can construct the set of all iso-spectral
potentials $q$ varying $\\{\xi_{n}\\}_{n\in\Omega}\in\ell_{2}$ or find a
unique $q$ setting additionally $\\{\xi_{n}\\}_{n\in\Omega}\in\ell_{2}.$ In
the latter case, we obtain a uniqueness theorem.

######  Theorem 3.

Let $\\{\tilde{\lambda}_{n}\\}_{n\geq 1}$ be the spectrum of boundary value
problem (1), (2) with a potential $\tilde{q}\in L_{2}(0,\pi),$ while
$\tilde{\xi}_{n}=\int_{0}^{\pi}\tilde{q}(t)g_{n}(t)\,dt,$ $n\in\Omega.$ If
$\\{\lambda_{n}\\}_{n\geq 1}=\\{\tilde{\lambda}_{n}\\}_{n\geq 1}$ and
$\xi_{n}=\tilde{\xi}_{n}$ for $n\in\Omega,$ then $q=\tilde{q}.$

Since the proof of Theorem 2 is constructive, we have the following algorithm
for recovering $q$ given $\\{\lambda_{n}\\}_{n\geq 1}$ and
$\\{\xi_{n}\\}_{n\in\Omega}.$

Algorithm 1. To recover the potential $q,$ one should:

1\. Construct $\Delta(\lambda)$ via formula (24).

2\. For $n\in\mathcal{S},$ by formula (27), compute the unknown coefficients
$\xi_{n+k_{n}},\ldots,\xi_{n+m_{n}-1}.$

3\. Find $q=\sum_{n=1}^{\infty}\xi_{n}f_{n},$ where $\\{f_{n}\\}_{n\geq 1}$ is
the basis biorthonormal to $\\{\overline{g}_{n}\\}_{n\geq 1}$ in
$L_{2}(0,\pi).$

Acknowledgements. This research was supported by grant No. 22-21-00509 of the
Russian Science Foundation, https://rscf.ru/project/22-21-00509/.

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