# Input-to-State Stabilization of $1$-D Parabolic PDEs under Output Feedback
Control

Yongchun Bi1<EMAIL_ADDRESS>Jun Zheng1,2<EMAIL_ADDRESS>Guchuan Zhu2<EMAIL_ADDRESS>1School of Mathematics, Southwest Jiaotong
University, Chengdu 611756, Sichuan, China
2Department of Electrical Engineering, Polytechnique Montréal, P.O. Box 6079,
Station Centre-Ville, Montreal, QC, Canada H3T 1J4

###### Abstract

This paper addresses the problem of input-to-state stabilization for a class
of parabolic equations with time-varying coefficients, as well as Dirichlet
and Robin boundary disturbances. By using time-invariant kernel functions,
which can reduce the complexity in control design and implementation, an
observer-based output feedback controller is designed via backstepping. By
using the generalized Lyapunov method, which can be used to handle Dirichlet
boundary terms, the input-to-state stability of the closed-loop system under
output feedback control, as well as the state estimation error system, is
established in the spatial $L^{\infty}$-norm. Numerical simulations are
conducted to confirm the theoretical results and to illustrate the
effectiveness of the proposed control scheme.

###### keywords:

Backstepping, parabolic equation, Dirichlet boundary disturbance, observer,
output feedback, generalized Lyapunov method, input-to-state stability.

, ,

##  1 Introduction

In the past few decades, backstepping is proven to be a powerful tool for
designing state feedback controllers, as well as output feedback controllers,
in the problem of stabilization for partial differential equations (PDEs);
see, e.g. [1, 2]. The main idea of backstepping is to apply the Volterra
transformation to transfer the original system with destabilizing terms into a
target system, in which the destabilizing terms are eliminated, and hence a
boundary stabilizing control can be obtained from the Volterra transformation.
The standard backstepping control design suggested in the existing literature
is first to select a target system, and then to determine the corresponding
kernel function in the Volterra transformation. This procedure can make the
application of backstepping in a systematic manner, bringing convenience in
solving the stabilization problem for PDEs. Especially, the target system is
often chosen to be PDEs with constant coefficients so that the stability can
be easily assessed by, e.g., using the Lyapunov method.

However, as indicated in [3], selecting a target system and a kernel function
in sequence brings difficulties to the control design and implementation for
PDEs that have time-varying coefficients. Indeed, when the original system has
time-varying coefficients, if the target system is chosen to be with constant
coefficients, the kernel function will in general depend on the time variable
(see, e.g., [4, 5, 6, 7]), which leads to computation burdens and complexities
in controller implementation due to the fact that the kernel equation must be
solved on the fly of control execution.

In the context of the stabilization problem for PDEs with external
disturbances, the method of backstepping has been used to design boundary
state feedback controllers that can ensure the input-to-state stability (ISS)
of the closed-loop system; see, e.g., [8, 9, 10, 11, 12, 13]. It is noted that
most of the PDEs considered in the literature involve only distributed in-
domain disturbances or Robin/Neumann boundary disturbances, while problems
with Dirichlet boundary disturbances are much less addressed. The main
influencing factor for this situation lies in difficulties encountered in
stability analysis. Indeed, when using backstepping to design boundary
controllers for PDEs, the disturbances involved in the target system keep the
same type as those in the original system. As is well known, in-domain
disturbances or Robin/Neumann boundary disturbances do not pose difficulties
for establishing the ISS of PDEs by using the Lyapunov method (see, e.g., [14,
15]), while, as indicated in [14, 16], the appearance of Dirichlet boundary
disturbances represents an intrinsic obstacle in the application of the
Lyapunov method for ISS analysis.

Another notable feature of the application of backstepping is that, when not
all the state variables are available or measured for state feedback control,
it is possible to implement output feedback control by using backstepping
state observers [1, 2]. In the absence of disturbances, observer-based
backstepping output feedback controllers were designed to stabilize single
PDEs [1, 17, 18], and coupled PDEs or PDE-ODEs [19, 20, 21, 22], just to cite
a few. However, under the framework of ISS theory, observer-based output
feedback control design for PDEs in the presence of disturbances has not yet
been reported. As the presence of external disturbances is invertible in many
practical application scenarios, it is obviously imperative to address the
stabilization problem for PDEs with disturbances under the framework of ISS
theory, which provides a powerful tool for describing the influence of
disturbances on the stability of control systems [23, 14, 24, 25].

In this paper, we consider the observer-based output feedback control for a
class of $1$-D parabolic PDEs with time-varying coefficients and distributed
in-domain and Dirichlet-Robin boundary disturbances under the framework of ISS
theory. Unlike the approach usually adopted in the existing literature: (i)
For the backstepping control design, we apply time-invariant kernel functions
to design a state feedback controller and a state observer for output feedback
control. Thus, the complexity in control design and implementation can be
significantly reduced. A key point of this design procedure is that we first
select a time-invariant kernel function and then determine the target system
with time-varying coefficients and Dirichlet boundary disturbances, which is
first applied in [3] for backstepping state feedback control. (ii) For
stability analysis, to deal with Dirichlet boundary terms in the target
system, we apply the so-called generalized Lyapunov functionals, which was
proposed in [16], to assess the ISS of the involved closed-loop systems.
Overall, the main contributions of this paper are threefold:

  1. (i)

An observer-based output feedback controller is designed for a class of
parabolic PDEs with disturbances under the framework of ISS theory.

  2. (ii)

The state and output feedback controllers are designed by using time-invariant
kernels functions, which can reduce the complexity in control design and
implementation.

  3. (iii)

For the involved parabolic PDEs in the presence of various disturbances,
especially, the Dirichlet boundary disturbances, the ISS is established in the
spatial $L^{\infty}$-norm by using the generalized Lyapunov method.

In the rest of the paper, we introduce first some basic notations used in this
paper. In Section 2, we present the problem formulation. In Section 3, we show
how to design a state feedback controller by using time-invariant kernels and
prove the first main result on the ISS of the closed-loop system by using the
generalized Lyapunov method. In Section 4, we design an observer and present
the second main result on the ISS of the state estimation error system. The
output feedback control and the third main result on the ISS of the closed-
loop system under output feedback control, as well as its proof, are presented
in Section 5. Numerical simulations are conducted in Section 6 to illustrate
the effectiveness of the proposed control scheme. Some conclusions are given
in Section 7.

Notation: Let ${\mathbb{N}_{0}}:=\\{0,1,2,...\\}$,
${\mathbb{N}}:=\mathbb{N}_{0}\setminus\\{0\\}$,
$\mathbb{R}:=(-\infty,+\infty)$, $\mathbb{R}_{\geq 0}:=[0,+\infty)$,
$\mathbb{R}_{>0}:=(0,+\infty)$, and $\mathbb{R}_{\leq 0}:=(-\infty,0]$.

For a domain $\Omega$ (either open or closed) in $\mathbb{R}^{1}$ or
$\mathbb{R}^{2}$, let
$C\left({\Omega}\right):=\\{v:{\Omega}\rightarrow\mathbb{R}|~{}v$ is
continuous on $\Omega\\}$. For $i\in\mathbb{N}$,
$C^{i}\left({\Omega}\right):=\\{v:{\Omega}\rightarrow\mathbb{R}|~{}v$ has
continuous derivatives up to order $i$ on ${\Omega}$}. Let
$C^{2,1}(\Omega)$$:=$$\\{v:$ $\Omega$$\rightarrow$$\mathbb{R}|~{}$$v$,
$v_{x}$, $v_{xx}$, $v_{t}$$\in$ $C(\Omega)\\}$.

For $T\in\mathbb{R}_{>0}$, let $Q_{T}:=(0,1)\times(0,T)$ and
$\overline{Q}_{T}$$:=$$[0,1]$$\times$$[0,T]$. Let
$Q_{\infty}$$:=(0,1)\times{\mathbb{R}_{>0}}$ and
$\overline{Q}_{\infty}:=[0,1]\times{\mathbb{R}_{\geq 0}}$. For
$v:Z\rightarrow\mathbb{R}$ with $Z\subset\overline{Q}_{\infty}$, the notation
$v[t]$ (or $v[{z}]$) denotes the profile at certain $t\in\mathbb{R}_{\geq 0}$
(or ${z}\in[0,1]$), i.e., $v[t]({z})=v({z},t)$ (or $v[{z}](t)=v({z},t)$).

The norms of a function $v$ in the Lebesgue spaces $L^{p}(0,1)$ with
$p\in[1,+\infty)$ and $L^{\infty}(0,1)$ are defined by
$\|v\|_{L^{p}(0,1)}:=\left(\int_{0}^{1}|v(x)|\text{d}x\right)^{\frac{1}{p}}$
and $\|v\|_{L^{\infty}(0,1)}:=\operatorname{ess\sup}_{x\in(0,1)}|v(x)|$,
respectively. For $T\in\mathbb{R}_{>0}$, the norm of a function $v$ in the
Lebesgue space $L^{\infty}((0,T);L^{\infty}(0,1))$ is defined by
$\|v\|_{L^{\infty}((0,T);L^{\infty}(0,1))}:=\operatorname{ess\sup}_{t\in(0,T)}\|v[t]\|_{L^{\infty}(0,1)}$.

Let $\mathcal{K}:=\\{\gamma:\mathbb{R}_{\geq 0}\rightarrow\mathbb{R}_{\geq
0}|$$\gamma(0)$$=$$0$}, $\gamma$ is continuous, strictly increasing},
$\mathcal{L}$$:=$$\\{\gamma:$$\mathbb{R}_{\geq
0}$$\rightarrow$$\mathbb{R}_{\geq 0}|$ $\gamma$ is continuous, strictly
decreasing, $\lim_{s\rightarrow\infty}\gamma(s)$$=$$0$},
$\mathcal{KL}:=\\{\beta:$$\mathbb{R}_{\geq 0}$$\times$$\mathbb{R}_{\geq 0}$
$\rightarrow$$\mathbb{R}_{\geq 0}|$${\beta}$ is continuous,
$\beta[t]\in\mathcal{K}$, $\forall t\in\mathbb{R}_{\geq 0}$;
$\beta[s]\in\mathcal{L},\forall s\in{\mathbb{R}_{>0}}\\}$.

##  2 Problem Statement

In this paper, we address the problem of stabilization of one-dimensional
linear parabolic equation:

$\displaystyle u_{t}(x,t)=$ $\displaystyle
u_{xx}(x,t)+\lambda(t)u(x,t)+f(x,t),(x,t)\in Q_{\infty},$ (1a) $\displaystyle
u_{x}(0,t)=$ $\displaystyle qu(0,t)+d_{0}(t),t\in\mathbb{R}_{>0},$ (1b)
$\displaystyle u(1,t)=$ $\displaystyle U(t)+d_{1}(t),t\in\mathbb{R}_{>0},$
(1c) $\displaystyle u(x,0)=$ $\displaystyle u_{0}(x),x\in(0,1),$ (1d)

where $\lambda(t)$ is the reaction coefficient, which is time-varying,
$f(x,t)$ denotes in-domain disturbance, $d_{0}(t)$ and $d_{1}(t)$ denote Robin
and Dirichlet boundary disturbance, respectively, $q$ is a constant,
$u_{0}(x)$ is the initial datum, and $U(t)$ is a boundary input that needs to
be designed.

The main objective of this work is to design an observer-based output feedback
control law $U(t)$ via time-invariant kernel functions such that both the
state estimation error system and the closed-loop system are ISS with respect
to (w.r.t.) external disturbances.

Throughout this paper, we always make the following assumptions:

$\displaystyle q\in$ $\displaystyle\mathbb{R}_{>0},\lambda\in
C(\mathbb{R}_{\geq 0}),\mathop{\text{sup}}\limits_{t\in\mathbb{R}_{\geq
0}}\lambda(t)<+\infty,$ (2a) $\displaystyle f\in$ $\displaystyle
C\left(\overline{Q}_{\infty}\right),d_{0},d_{1}\in C(\mathbb{R}_{\geq 0}),$
(2b) $\displaystyle u_{0}\in$ $\displaystyle{C^{2,1}(0,1)}\cap C([0,1]).$ (2c)

Moreover, the initial datum $u_{0}$ and boundary disturbances $d_{0},d_{1}$
satisfy certain compatibility conditions, which will be specified later in
each section.

##  3 State Feedback Control and ISS of the Closed-loop System

Before designing an observer for output feedback control, we show first that
there exists a state feedback control that can ensure the ISS in the
$L^{\infty}$-norm of system (1).

###  3.1 State Feedback Control Design

####  3.1.1 Selection of Kernel Functions

To adopt time-invariant kernel functions in backstepping control design,
following [3], we first select a kernel function, and then select a target
system with time-varying coefficient. More specifically, by the assumption
(2a), we take any positive constant $c_{0}$ such that

$\displaystyle c_{0}>\mathop{\text{sup}}\limits_{t\in\mathbb{R}_{\geq
0}}\lambda(t).$ (3)

Denote

$\displaystyle c(t):=c_{0}-\lambda(t)\text{ and
}\underline{c}:=\inf_{t\in{\mathbb{R}_{\geq 0}}}c(t)>0.$ (4)

Let $D{:=}\\{(x,{z})\in\mathbb{R}^{2}|~{}0\leq{z}\leq x\leq 1\\}$. We consider
the solution $k$ to the following equation defined over $D$:

$\displaystyle k_{xx}(x,{z})-k_{{zz}}(x,{z})=$ $\displaystyle c_{0}k(x,{z}),$
(5a) $\displaystyle\frac{\text{d}}{\text{d}x}(k(x,x))=$
$\displaystyle-\frac{1}{2}c_{0},$ (5b) $\displaystyle k_{{z}}(x,0)=$
$\displaystyle qk(x,0),$ (5c) $\displaystyle k(0,0)=$ $\displaystyle 0,$ (5d)

where
$\frac{\text{d}}{\text{d}x}\left(k(x,x)\right):=k_{x}(x,{z})|_{{z}=x}+k_{{z}}(x,{z})|_{{z}=x}$.
Indeed, the existence and regularity of the solution $k$ to equation (5) are
guaranteed by the following Lemma, whose proof can be found in [26].

######  Lemma 3.1.

Equation (5) has a unique solution $k\in C^{2}\left(D\right)$, which can be
expressed as

$\displaystyle k(x,{z})=$ $\displaystyle-
c_{0}x\frac{\mathcal{I}_{1}\left(\sqrt{c_{0}\left(x^{2}-{z}^{2}\right)}\right)}{\sqrt{c_{0}\left(x^{2}-{z}^{2}\right)}}$
$\displaystyle+\frac{qc_{0}}{\sqrt{c_{0}+q^{2}}}\int_{0}^{x-{z}}e^{\frac{-q\tau}{2}}\mathcal{I}_{0}\left(\sqrt{c_{0}(x+{z})(x-{z}-\tau)}\right)\sinh\left(\frac{\sqrt{c_{0}+q^{2}}}{2}\tau\right){\rm
d}\tau,$ (6)

where $\mathcal{I}_{0}$ and $\mathcal{I}_{1}$ are the modified Bessel
functions of order zero and order one, respectively (see, e.g., [2, pp.
174-175]).

####  3.1.2 Selection of Target System

Applying the Volterra transformation

$\displaystyle w(x,t):=u(x,t)-\int_{0}^{x}k(x,{z})u({z},t)\text{d}{z},$ (7)

we obtain the following target system involving a time-varying reaction
coefficient and also Dirichlet and Robin boundary disturbances:

$\displaystyle w_{t}(x,t)=$ $\displaystyle w_{xx}(x,t)-c(t)w(x,t)+\psi(x,t),$
$\displaystyle(x,t)\in Q_{\infty},$ (8a) $\displaystyle w_{x}(0,t)=$
$\displaystyle qw(0,t)+d_{0}(t),$ $\displaystyle t\in\mathbb{R}_{>0},$ (8b)
$\displaystyle w(1,t)=$ $\displaystyle d_{1}(t),$ $\displaystyle
t\in\mathbb{R}_{>0},$ (8c) $\displaystyle w(x,0)=$ $\displaystyle w_{0}(x),$
$\displaystyle x\in(0,1),$ (8d)

where

$\displaystyle\psi(x,t){:=}$ $\displaystyle
f(x,t)-\int_{0}^{x}k(x,{z})f({z},t)\text{d}{z}+k(x,0)d_{0}(t),$ (9)
$\displaystyle w_{0}(x){:=}$ $\displaystyle
u_{0}(x)-\int_{0}^{x}k(x,{z})u_{0}({z})\text{d}{z}.$

Note that the transformation (7) is invertible, and the equivalence between
system (1) and system (8) is guaranteed by the following lemma, whose proof
can be proceeded in a standard way as in, e.g., [27] and hence, is omitted.

######  Lemma 3.2.

Let $l$ be the unique solution of

$\displaystyle l_{xx}(x,{z})-l_{{zz}}(x,{z})=$ $\displaystyle-c_{0}l(x,{z}),$
(10a) $\displaystyle\frac{\text{d}}{\text{d}x}(l(x,x))=$
$\displaystyle-\frac{1}{2}c_{0},$ (10b) $\displaystyle l_{{z}}(x,0)=$
$\displaystyle ql(x,0),$ (10c) $\displaystyle l(0,0)=$ $\displaystyle 0.$
(10d)

Then, the inverse transformation of (7) is given by

$\displaystyle u(x,t):=w(x,t)+\int_{0}^{x}l(x,{z})w({z},t)\mathrm{d}{z}.$ (11)

Therefore, system (1) is equivalent to system (8).

####  3.1.3 State Feedback Controller

By using the function $k$ satisfying (5) and in view of the transformation
(7), we define the state feedback control law as

$\displaystyle U(t):=\int_{0}^{1}k(1,{z})u({z},t)\text{d}{z}.$ (12)

###  3.2 ISS Assessment of the Closed-loop System

In this section, we assess the ISS of the closed-loop system under the state
feedback control law (12).

####  3.2.1 ISS of the Target System

To ensure the existence of a classical solution to the original system (1) and
the target system (8), we assume further that the initial datum $u_{0}$ and
the boundary disturbances $d_{0},d_{1}$ satisfy the compatibility conditions:

$\displaystyle
u_{0x}(0)=qu_{0}(0)+d_{0}(0),u_{0}(1)=\int_{0}^{1}k(1,{z})u_{0}({z})\text{d}{z}+d_{1}(0).$

The classical PDE theory (see, e.g., [28, Chapter IV]) ensures that system (8)
admits a unique solution $w\in C^{2,1}(Q_{T})\cap
C\left(\overline{Q}_{T}\right)$ and hence, system (1) admits a unique solution
$u\in C^{2,1}(Q_{T})\cap C\left(\overline{Q}_{T}\right)$ for any
$T\in\mathbb{R}_{>0}$. In addition, for the target system (8), we can prove
the following result, which indicates the ISS w.r.t. external disturbances.

######  Proposition 3.1.

The target system (8) is ISS in the $L^{\infty}$-norm w.r.t. $f,d_{0},d_{1}$,
having the following estimate for all $T\in\mathbb{R}_{>0}$:

$\displaystyle\left\|{w}[T]\right\|_{L^{\infty}(0,1)}\leq
c_{1}\left(e^{-\sigma
T}\left\|{w}_{0}\right\|_{L^{\infty}(0,1)}+\left\|{d}_{0}\right\|_{L^{\infty}(0,T)}\right.\left.+\left\|{d}_{1}\right\|_{L^{\infty}(0,T)}+\left\|{f}\right\|_{L^{\infty}\left((0,T);L^{\infty}(0,1)\right)}\right),$

where $\sigma$ is an arbitrary constant satisfying
$\sigma\in(0,\underline{c})$, and $c_{1}$ is a positive constant depending
only on $q,\underline{c},\sigma$, and $\max_{(x,{z})\in D}|k(x,{z})|$.

The proof of Proposition 3.1 is given in Section 3.2.3.

####  3.2.2 ISS of the Closed-loop System

By virtue of Lemma 3.2 and Proposition 3.1, we state the first main result,
which indicates the ISS of the closed-loop system (1) under state feedback
control.

######  Theorem 3.2.

Under the state feedback control law (12), the closed-loop system (1) is ISS
in the $L^{\infty}$-norm w.r.t. $f,d_{0},d_{1}$, having the following estimate
for all $T\in\mathbb{R}_{>0}$:

$\displaystyle\left\|{u}[T]\right\|_{L^{\infty}(0,1)}\leq{c_{2}}\left(e^{-\sigma
T}\left\|{u}_{0}\right\|_{L^{\infty}(0,1)}+\left\|{d}_{0}\right\|_{L^{\infty}(0,T)}+\left\|{d}_{1}\right\|_{L^{\infty}(0,T)}+\left\|{f}\right\|_{L^{\infty}\left((0,T);L^{\infty}(0,1)\right)}\right),$

where $\sigma$ is an arbitrary constant satisfying
$\sigma\in(0,\underline{c})$, and $c_{2}$ is a positive constant depending
only on $q$, $\underline{c}$, $\sigma$, $\max_{(x,{z})\in D}|k(x,{z})|$, and
$\max_{(x,{z})\in D}|l(x,{z})|$.

The proof of Theorem 3.2 is standard and hence, is omitted.

####  3.2.3 Proof of Proposition 3.1

In this section, we apply the generalized Lyapunov method, which is proposed
in [16], to prove the ISS of the target system (8).

Proof of Proposition 3.1:  We proceed with the proof in four steps.

Step 1: define truncation functions. For an arbitrary constant $r>1$, define

$\displaystyle g(\theta):=\begin{cases}\theta^{r},&\theta\geq 0\\\
0,&\theta<0\end{cases}\ \text{and}\
G(\theta):=\int_{0}^{\theta}g(\tau)\mathrm{d}\tau,$ (13)

which satisfy

$\displaystyle g(\theta)\geq$ $\displaystyle 0,g^{\prime}(\theta)\geq
0,G(\theta)\geq 0,\forall\theta\in\mathbb{R},$ (14a) $\displaystyle
g(\theta)=$ $\displaystyle G(\theta)=0,\forall\theta\in\mathbb{R}_{\leq 0}.$
(14b)

Step 2: prove the upper bound of ${w}$. For any
${\sigma}\in(0,\underline{c})$, let

$\displaystyle{{v}(x,t):=}e^{{\sigma}t}{w}(x,t),{v}_{0}(x):={w}_{0}(x),{\check{d}_{0}(t):=}e^{{\sigma}t}{d}_{0}(t),\check{d}_{1}(t):=e^{{\sigma}t}{d}_{1}(t),\check{\psi}(x,t):=e^{{\sigma}t}{\psi}(x,t).$

For any $T\in\mathbb{R}_{>0}$, let
$\check{\mathcal{D}}:=\max\left\\{\left\|{v}_{0}\right\|_{L^{\infty}(0,1)},\frac{1}{{q}}\left\|\check{d}_{0}\right\|_{L^{\infty}(0,T)},\left\|\check{d}_{1}\right\|_{L^{\infty}(0,T)},\frac{1}{\underline{c}-{\sigma}}\left\|\check{\psi}\right\|_{L^{\infty}\left((0,T);L^{\infty}(0,1)\right)}\right\\}.$
Note that

$\displaystyle{v}(1,t)\leq\check{\mathcal{D}},\forall t\in(0,T).$ (15)

By integrating by parts and using the properties of $g$ and $G$, we get

$\displaystyle\frac{\text{d}}{\text{d}t}\int_{0}^{1}G\left({v}-\check{\mathcal{D}}\right)\text{d}x=$
$\displaystyle
g\left({v}(1,t)-\check{\mathcal{D}}\right){v}_{x}(1,t)-g\left({v}(0,t)-\check{\mathcal{D}}\right)\left({q}{v}(0,t)+\check{d}_{0}(t)\right)-\int_{0}^{1}g^{\prime}\left({v}-\check{\mathcal{D}}\right){v}_{x}^{2}\text{d}x$
$\displaystyle+\int_{0}^{1}g\left({v}-\check{\mathcal{D}}\right)\left(({\sigma}-c(t)){v}+\check{\psi}\right)\text{d}x$
$\displaystyle\leq$ $\displaystyle 0,\forall t\in(0,T).$ (16)

Therefore, it holds that

$\displaystyle\int_{0}^{1}G\left({v}(x,T)-\check{\mathcal{D}}\right)\text{d}x\leq\int_{0}^{1}G\left({v}(x,0)-\check{\mathcal{D}}\right)\text{d}x,$

which, along with the continuity of ${v}$, implies that

$\displaystyle{v}(x,T)\leq\check{\mathcal{D}},\forall x\in(0,1).$

Then, we have

$\displaystyle{w}(x,T)=e^{-{\sigma}T}{v}(x,T)\leq e^{-\sigma
T}\left\|{w}_{0}\right\|_{L^{\infty}(0,1)}+\frac{1}{{q}}\left\|{d}_{0}\right\|_{L^{\infty}(0,T)}+\left\|{d}_{1}\right\|_{L^{\infty}(0,T)}+\frac{1}{\underline{c}-{\sigma}}\left\|{\psi}\right\|_{L^{\infty}\left((0,T);L^{\infty}(0,1)\right)}.$
(17)

Step 3: prove the lower bound of ${w}$. For the same
${\sigma}\in(0,\underline{c})$ as in Step 2, let

$\displaystyle{\overline{v}(x,t):=}-e^{{\sigma}t}{w}(x,t),\overline{v}_{0}(x):=-{w}_{0}(x),{\overline{d}_{0}(t):=}-e^{{\sigma}t}{d}_{0}(t),\overline{d}_{1}(t):=-e^{{\sigma}t}{d}_{1}(t),\overline{\psi}(x,t):=-e^{{\sigma}t}{\psi}(x,t).$

Let $g$, $G$ be defined by (13), and
$\overline{\mathcal{D}}:=\max\left\\{\left\|\overline{v}_{0}\right\|_{L^{\infty}(0,1)},\frac{1}{{q}}\left\|\overline{d}_{0}\right\|_{L^{\infty}(0,T)},\left\|\overline{d}_{1}\right\|_{L^{\infty}(0,T)},\frac{1}{\underline{c}-{\sigma}}\left\|\overline{\psi}\right\|_{L^{\infty}((0,T);L^{\infty}(0,1))}\right\\}.$

Analogous to Step 2, for any $t\in(0,T)$, by deriving
$\frac{\text{d}}{\text{d}t}\int_{0}^{1}G\left(\overline{v}-\overline{\mathcal{D}}\right)\text{d}x\leq
0,$ we get:

$\displaystyle\int_{0}^{1}G\left(\overline{v}(x,T)-\overline{\mathcal{D}}\right)\text{d}x\leq\int_{0}^{1}G\left(\overline{v}(x,0)-\overline{\mathcal{D}}\right)\text{d}x.$

Therefore, it holds that

$\displaystyle\overline{v}(x,T)\leq\overline{\mathcal{D}},\forall x\in(0,1),$

which implies that

$\displaystyle{-{w}(x,T)}\leq
e^{-{\sigma}T}\left\|{w}_{0}\right\|_{L^{\infty}(0,1)}+\frac{1}{q}\left\|{d}_{0}\right\|_{L^{\infty}(0,T)}+\left\|{d}_{1}\right\|_{L^{\infty}(0,T)}+\frac{1}{\underline{c}-{\sigma}}\left\|{\psi}\right\|_{L^{\infty}((0,T);L^{\infty}(0,1))}.$
(18)

Step 4: conclusion. By the definition of $\psi$ (see (9)), we have

$\displaystyle{\left\|{\psi}\right\|_{L^{\infty}((0,T);L^{\infty}(0,1))}}\leq\overline{k}\left\|{d}_{0}\right\|_{L^{\infty}(0,T)}+\left(\overline{k}+1\right)\left\|{f}\right\|_{L^{\infty}((0,T);L^{\infty}(0,1))},$
(19)

where $\overline{k}:=\max_{(x,{z})\in D}|k(x,{z})|$.

By (17), (18), and (19), we deduce that the target system (8) is ISS in the
$L^{\infty}$-norm w.r.t. $f,d_{0}$, and $d_{1}$. $\blacksquare$

##  4 Observer Design and ISS of the State Estimation Error System

In this section, we design an observer, which ensures the ISS in the
$L^{\infty}$-norm of the state estimation error system.

###  4.1 Observer Design

We design an observer in the anti-collocated setup where the sensor and the
actuator are placed at the opposite ends. More specifically, for system (1)
with Dirichlet actuation, we assume that the only available measurement of the
system is located at $x=0$, the opposite end to the actuation located at
$x=1$. Thus, $u(0,t)$ is the system output. Note that in practice the output
may also be affected by external disturbances. Thus, denoting the measurement
disturbance by ${d_{m}(t)}$ and considering the output

$\displaystyle{y}(t):=u(0,t)+{d_{m}(t)},$ (20)

we propose an observer under the following form:

$\displaystyle\hat{u}_{t}(x,t)=$
$\displaystyle\hat{u}_{xx}(x,t)+\lambda(t)\hat{u}(x,t)+p(x)\left({y}(t)-\hat{u}(0,t)\right),(x,t)\in
Q_{\infty},$ (21a) $\displaystyle\hat{u}_{x}(0,t)=$ $\displaystyle
q{\hat{u}}(0,t)-p_{0}\left({y}(t)-\hat{u}(0,t)\right),t\in\mathbb{R}_{>0},$
(21b) $\displaystyle\hat{u}(1,t)=$ $\displaystyle U(t),t\in\mathbb{R}_{>0},$
(21c) $\displaystyle\hat{u}(x,0)=$ $\displaystyle\hat{u}_{0}(x),x\in(0,1),$
(21d)

where $\hat{u}_{0}(x)$ is the initial datum, and the function $p(x)$ and the
constant $p_{0}$ are observer gains to be determined.

To determine the observer gains $p(x)$ and $p_{0}$, we first consider the
following equation defined over $D$:

$\displaystyle m_{xx}(x,{z})-m_{{zz}}(x,{z})=$ $\displaystyle c_{0}m(x,{z}),$
(22a) $\displaystyle\frac{\text{d}}{\text{d}x}\left(m(x,x)\right)=$
$\displaystyle-\frac{1}{2}c_{0},$ (22b) $\displaystyle m(1,{z})=$
$\displaystyle 0,$ (22c)

where
$\frac{\text{d}}{\text{d}x}\left(m(x,x)\right):=m_{x}(x,{z})|_{{z}=x}+m_{{z}}(x,{z})|_{{z}=x}$.
The existence and regularity of a solution $m$ to (22) are guaranteed by the
following Lemma.

######  Lemma 4.1.

Equation (22) admits a unique solution $m\in C^{2}\left(D\right)$, which can
be expressed as

$\displaystyle
m(x,{z})=c_{0}(1-x)\frac{\mathcal{J}_{1}\left(\sqrt{c_{0}({z}^{2}-x^{2}+2(x-{z}))}\right)}{\sqrt{c_{0}\left({z}^{2}-x^{2}+2(x-{z})\right)}},$

where $\mathcal{J}_{1}$ is the Bessel function of order one (see, e.g., [2,
pp. 173-174]).

Proof:  Let us make a change of variables

$\displaystyle\overline{x}:=1-{z},\overline{{z}}:=1-x,\overline{m}(\overline{x},\overline{{z}}):=m(1-\overline{{z}},1-\overline{x}).$

Then, (22) becomes

$\displaystyle\overline{m}_{\overline{x}\overline{x}}(\overline{x},\overline{{z}})-\overline{m}_{\overline{{z}}\overline{{z}}}(\overline{x},\overline{{z}})=$
$\displaystyle-c_{0}\overline{m}(\overline{x},\overline{{z}}),$
$\displaystyle\frac{\text{d}}{\text{d}\overline{x}}\left(\overline{m}(\overline{x},\overline{x})\right)=$
$\displaystyle\frac{1}{2}c_{0},$ $\displaystyle\overline{m}(\overline{x},0)=$
$\displaystyle 0,$

whose solution is expressed as (see [2, (4.33), p. 35])

$\displaystyle\overline{m}(\overline{x},\overline{{z}})=c_{0}\overline{{z}}\frac{\mathcal{I}_{1}\left(\sqrt{-c_{0}(\overline{x}^{2}-\overline{{z}}^{2})}\right)}{\sqrt{-c_{0}(\overline{x}^{2}-\overline{{z}}^{2})}}=c_{0}\overline{{z}}\frac{\mathcal{I}_{1}\left(\text{i}\sqrt{c_{0}(\overline{x}^{2}-\overline{{z}}^{2})}\right)}{\text{i}\sqrt{c_{0}(\overline{x}^{2}-\overline{{z}}^{2})}},$

where $\mathcal{I}_{1}$ is the modified Bessel function of order one. Then,
using the fact that $\mathcal{I}_{1}(\text{i}x)=\text{i}\mathcal{J}_{1}(x)$
(see [2, (A.11), p. 174]), we obtain

$\displaystyle\overline{m}(\overline{x},\overline{{z}})=c_{0}\overline{{z}}\frac{\mathcal{J}_{1}\left(\sqrt{c_{0}(\overline{x}^{2}-\overline{{z}}^{2})}\right)}{\sqrt{c_{0}(\overline{x}^{2}-\overline{{z}}^{2})}}.$

$\blacksquare$

Now, based on the function $m$, we choose the observer gains $p_{0}$ and
$p(x)$ that satisfy

$\displaystyle{p_{0}}>$ $\displaystyle m(0,0)-q=\frac{c_{0}}{2}-q,$ (23)
$\displaystyle p(x)=$ $\displaystyle-
m_{{z}}(x,0)+{(q+p_{0})}m(x,0)+\int_{0}^{x}m(x,{z})p({z})\text{d}{z}.$ (24)

The existence and regularity of a solution $p$ to the integral equation (24)
are guaranteed by the following Lemma.

######  Lemma 4.2 ([29, Theorem 1.3.1, p. 33]).

Equation (24) admits a unique solution $p\in C(D)$. Furthermore, $p\in
C^{1}(D)$ due to the fact that $m\in C^{2}(D)$.

###  4.2 ISS of the State Estimation Error System

Let the estimation error of the observer given in (21) be defined by

$\displaystyle\widetilde{u}(x,t):=u(x,t)-\hat{u}(x,t).$ (25)

Thus, the state estimation error system will be given by

$\displaystyle\widetilde{u}_{t}(x,t)=$
$\displaystyle\widetilde{u}_{xx}(x,t)+\lambda(t)\widetilde{u}(x,t)+\widetilde{f}(x,t)-p(x)\widetilde{u}(0,t),(x,t)\in
Q_{\infty},$ (26a) $\displaystyle\widetilde{u}_{x}(0,t)=$
$\displaystyle\widetilde{d}_{0}(t)+{(q+p_{0})}\widetilde{u}(0,t),t\in\mathbb{R}_{>0},$
(26b) $\displaystyle\widetilde{u}(1,t)=$
$\displaystyle\widetilde{d}_{1}(t),t\in\mathbb{R}_{>0},$ (26c)
$\displaystyle\widetilde{u}(x,0)=$
$\displaystyle\widetilde{u}_{0}(x),x\in(0,1),$ (26d)

where
$\widetilde{f}(x,t):=f(x,t)-p(x){d_{m}(t)},\widetilde{d}_{0}(t):=d_{0}(t)+p_{0}{d_{m}(t)},\widetilde{d}_{1}(t):=d_{1}(t)$,
and $\widetilde{u}_{0}(x):=u_{0}(x)-\hat{u}_{0}(x)$.

By using the transformation

$\displaystyle\widetilde{w}(x,t):=\widetilde{u}(x,t)-\int_{0}^{x}m(x,{z})\widetilde{u}({z},t)\text{d}{z},$
(27)

we obtain the following target system involving also a time-varying reaction
coefficient and external disturbances:

$\displaystyle\widetilde{w}_{t}(x,t)=$
$\displaystyle\widetilde{w}_{xx}(x,t)-{c}(t)\widetilde{w}(x,t)+\widetilde{\psi}(x,t),(x,t)\in
Q_{\infty},$ (28a) $\displaystyle\widetilde{w}_{x}(0,t)=$ $\displaystyle
b\widetilde{w}(0,t)+\widetilde{d}_{0}(t),t\in\mathbb{R}_{>0},$ (28b)
$\displaystyle\widetilde{w}(1,t)=$
$\displaystyle\widetilde{d}_{1}(t),t\in\mathbb{R}_{>0},$ (28c)
$\displaystyle\widetilde{w}(x,0)=$
$\displaystyle\widetilde{w}_{0}(x),x\in(0,1),$ (28d)

where

$\displaystyle b:=$ $\displaystyle{q+p_{0}}-m(0,0)>0\ {(\text{by}\
\eqref{p10}),}$ (29) $\displaystyle\widetilde{\psi}(x,t):=$
$\displaystyle\widetilde{f}(x,t)-\int_{0}^{x}m(x,{z})\widetilde{f}({z},t)\text{d}{z}+m(x,0)\widetilde{d}_{0}(t),$
(30) $\displaystyle\widetilde{w}_{0}(x):=$
$\displaystyle\widetilde{u}_{0}(x)-\int_{0}^{x}m(x,{z})\widetilde{u}_{0}({z})\text{d}{z}.$

The equivalence between the state estimation error system (26) and the
corresponding target system (28) is guaranteed by the following lemma, whose
proof is standard and hence, is omitted, and the existence and regularity of a
solution $n$ to the following kernel function are guaranteed by [26].

######  Lemma 4.3.

Let $n$ be the unique solution of

$\displaystyle n_{xx}(x,{z})-n_{{zz}}(x,{z})=$ $\displaystyle-c_{0}n(x,{z}),$
(31a) $\displaystyle\frac{\text{d}}{\text{d}x}(n(x,x))=$
$\displaystyle-\frac{1}{2}c_{0},$ (31b) $\displaystyle n(1,{z})=$
$\displaystyle 0.$ (31c)

Then, the inverse transformation of (27) is given by

$\displaystyle\widetilde{u}(x,t):=\widetilde{w}(x,t)+\int_{0}^{x}n(x,{z})\widetilde{w}({z},t)\mathrm{d}{z}.$
(32)

Therefore, the state estimation error system (26) is equivalent to the
corresponding target system (28).

To ensure the existence of a solution $\widetilde{u}$ to the error system
(26), as well as a solution $(u,\hat{u})$ to the system coupled by (1) and
(21), we assume that the disturbance ${d_{m}}\in C(\mathbb{R}_{\geq 0})$ and
the initial datum $\hat{u}_{0}\in C^{2,1}(0,1)\cap C([0,1])$ satisfy the
following compatibility conditions:

$\displaystyle\hat{u}_{0x}(0)=$ $\displaystyle
q\hat{u}_{0}(0)-p_{0}\left(u_{0}(0)+{d_{m}}(0)-\hat{u}_{0}(0)\right),\hat{u}_{0}(1)=U(0).$

Analogous to the target system (8) of the $u$-system, for any
$T\in\mathbb{R}_{>0}$, the target system (28) admits a unique solution
$\widetilde{w}\in C^{2,1}(Q_{T})\cap C\left(\overline{Q}_{T}\right)$ and
hence, the system coupled by the original system (1) and the observer system
(21) admits a unique solution $(u,\hat{u})\in\left(C^{2,1}(Q_{T})\cap
C\left(\overline{Q}_{T}\right)\right)\times\left(C^{2,1}(Q_{T})\cap
C\left(\overline{Q}_{T}\right)\right)$.

Note that the target system (28) has the same form as the target system (8) of
the plant (1). By virtue of Lemma 4.3 and Proposition 3.1, we have the
following theorem, which is the second main result. The proof is omitted as it
is standard.

######  Theorem 4.1.

The state estimation error system (26) is ISS in the $L^{\infty}$-norm w.r.t.
$f,{d_{m}},d_{0},d_{1}$, having the following estimate for all
$T\in\mathbb{R}_{>0}$:

$\displaystyle\left\|\widetilde{u}[T]\right\|_{L^{\infty}(0,1)}\leq
c_{3}\left(e^{-\sigma
T}\left\|\widetilde{u}_{0}\right\|_{L^{\infty}(0,1)}+\left\|{d_{m}}\right\|_{L^{\infty}(0,T)}+\left\|{d}_{0}\right\|_{L^{\infty}(0,T)}+\left\|{d}_{1}\right\|_{L^{\infty}(0,T)}+\left\|{f}\right\|_{L^{\infty}\left((0,T);L^{\infty}(0,1)\right)}\right),$

where $\sigma$ is an arbitrary constant satisfying
$\sigma\in(0,\underline{c})$, and $c_{3}$ is a positive constant depending
only on $\underline{c},\sigma,p_{0},{q}$, $\max_{x\in[0,1]}|p(x)|$,
$\max_{(x,{z})\in D}|m(x,{z})|$, and $\max_{(x,{z})\in D}|n(x,{z})|$.

##  5 Output Feedback Control and ISS of the Closed-loop System

###  5.1 Observer-based Output Feedback Control

In this section, we present an observer-based output feedback control scheme,
which ensures the ISS in the $L^{\infty}$-norm of the closed-loop system.

Indeed, by using the function $k$ in closed form (3.1) and replacing $u$ with
$\hat{u}$ in (12), we obtain the observer-based output feedback control law:

$\displaystyle U(t):=\int_{0}^{1}k(1,{z})\hat{u}({z},t)\text{d}{z}.$ (33)

Note that by using the transformation

$\displaystyle\hat{w}(x,t):=\hat{u}(x,t)-\int_{0}^{x}k(x,{z})\hat{u}({z},t)\text{d}{z},$
(34)

we can transform the observer (21) into

$\displaystyle\hat{w}_{t}(x,t)=$
$\displaystyle\hat{w}_{xx}(x,t)-c(t)\hat{w}(x,t){+\breve{\psi}(x,t)}{+K_{p}(x)\widetilde{u}(0,t)},(x,t)\in
Q_{\infty},$ (35a) $\displaystyle\hat{w}_{x}(0,t)=$ $\displaystyle
q\hat{w}(0,t)-p_{0}{d_{m}(t)}{{-p_{0}}\widetilde{u}(0,t),t\in\mathbb{R}_{>0}},$
(35b) $\displaystyle{\hat{w}(1,t)=}$ $\displaystyle 0,t\in\mathbb{R}_{>0},$
(35c) $\displaystyle\hat{w}(x,0)=$ $\displaystyle\hat{w}_{0}(x),x\in(0,1),$
(35d)

where

$\displaystyle{\breve{\psi}(x,t){:=}}$ $\displaystyle
p(x){d_{m}(t)}-\int_{0}^{x}k(x,{z})p({z}){d_{m}(t)}\text{d}{z}-p_{0}k(x,0){d_{m}(t)},$
$\displaystyle K_{p}(x):=$ $\displaystyle
p(x){-p_{0}k(x,0)}-\int_{0}^{x}k(x,{z})p({z})\text{d}{z},$
$\displaystyle\hat{w}_{0}(x):=$
$\displaystyle\hat{u}_{0}(x)-\int_{0}^{x}k(x,{z})\hat{u}_{0}({z})\text{d}{z},.$

It is clear that system (35) has the same form as the target system (8) of the
$u$-system. Therefore, system (35) admits a solution $\hat{w}\in
C^{2,1}(Q_{T})\cap C\left(\overline{Q}_{T}\right)$ for any
$T\in\mathbb{R}_{>0}$. Moreover, by virtue of Lemma 3.2, the transformation
(34) is invertible. Thus, it can be proved the following Lemma, whose proof is
standard and hence, is omitted.

######  Lemma 5.1.

The inverse of the transformation (34) is given by

$\displaystyle\hat{u}{(x,t)}:=\hat{w}{(x,t)}+\int_{0}^{x}l(x,{z})\hat{w}({z},t)\mathrm{d}{z},$
(36)

where $l$ is determined by (3.2). Moreover, the observer system (21) is
equivalent to its target system (35).

In addition, for the target system (35), we can prove the following robust
stability property, which, along with Theorem 4.1, plays a vital role in
establishing the ISS of the closed-loop system under output feedback control.

######  Proposition 5.1.

The target system (35) admits the following estimate for all
$T\in\mathbb{R}_{>0}$:

$\displaystyle\left\|\hat{w}[T]\right\|_{L^{\infty}(0,1)}\leq$ $\displaystyle
c_{4}\left(e^{-{\sigma
T}}\left\|\hat{w}_{0}\right\|_{L^{\infty}(0,1)}+e^{-\sigma
T}\left\|\widetilde{u}_{0}\right\|_{L^{\infty}(0,1)}+\left\|f\right\|_{L^{\infty}((0,T);L^{\infty}(0,1))}+\left\|d_{0}\right\|_{L^{\infty}(0,T)}+\left\|d_{1}\right\|_{L^{\infty}(0,T)}\right.$
$\displaystyle\left.+{\left\|d_{m}\right\|_{L^{\infty}(0,T)}}\right),$

where $\sigma$ is an arbitray constant satisfying
$\sigma\in(0,\underline{c})$, and $c_{4}$ is a positive constant depending
only on $\underline{c}$, $\sigma$, $p_{0}$, $q$, $\max_{(x,{z})\in
D}|k(x,{z})|$, $\max_{x\in[0,1]}|p(x)|$, $\max_{(x,{z})\in D}|m(x,{z})|$, and
$\max_{(x,{z})\in D}|n(x,{z})|$.

The proof of Proposition 5.1 is given in Section 5.2.

Now, by virtue of Lemma 5.1, Proposition 5.1, and applying Theorem 4.1, we
obtain the following theorem, which is the third main result, indicating the
ISS of the closed-loop system under output feedback control.

######  Theorem 5.2.

Under the output feedback control law (33), the closed-loop
$(u,\widetilde{u})$-system is ISS in the $L^{\infty}$-norm, having the
following estimate for all $T\in\mathbb{R}_{>0}$:

$\displaystyle\left\|{u}[T]\right\|_{L^{\infty}(0,1)}+\left\|\widetilde{u}[T]\right\|_{L^{\infty}(0,1)}\leq$
$\displaystyle c_{5}\left(e^{-{\sigma
T}}\left\|u_{0}\right\|_{L^{\infty}(0,1)}+e^{-\sigma
T}\left\|\widetilde{u}_{0}\right\|_{L^{\infty}(0,1)}+\left\|f\right\|_{L^{\infty}((0,T);L^{\infty}(0,1))}+\left\|d_{0}\right\|_{L^{\infty}(0,T)}\right.$
$\displaystyle\left.+\left\|d_{1}\right\|_{L^{\infty}(0,T)}+{\left\|d_{m}\right\|_{L^{\infty}(0,T)}}\right),$

where $\sigma$ is an arbitrary constant satisfying
$\sigma\in(0,\underline{c})$, and $c_{5}$ is a positive constant depended only
on $q,\underline{c},\sigma,p_{0}$, $\max_{x\in[0,1]}|p(x)|$, $\max_{(x,{z})\in
D}|k(x,{z})|$, $\max_{(x,{z})\in D}|l(x,{z})|$, $\max_{(x,{z})\in
D}|m(x,{z})|$, and $\max_{(x,{z})\in D}|n(x,{z})|$.

The proof of Theorem 5.2 is standard and hence, is omitted.

###  5.2 ISS Assessment of the Closed-loop System

Proof of Proposition 5.1:  We proceed with the proof in four steps.

Step 1: define truncation functions. Let $g$ and $G$ be defined by (13). In
addition to (14), $G$ admits the following properties:

$\displaystyle G(\theta)=$
$\displaystyle\frac{1}{r+1}g(\theta)\theta,\forall\theta\in\mathbb{R},$ (37a)
$\displaystyle G(\theta+\rho)\leq$ $\displaystyle
2^{r}(G(\theta)+G(\rho)),\forall\theta,\rho\in\mathbb{R}.$ (37b)

For any ${\sigma}\in(0,\underline{c})$ and any $T\in\mathbb{R}_{>0}$, let

$\displaystyle\mathcal{D}:=\max$ $\displaystyle\left\\{\frac{e^{\sigma
T}p_{0}}{q}\left\|{d_{m}}\right\|_{L^{\infty}(0,T)},\frac{e^{\sigma
T}}{\underline{c}-{\sigma}}\left\|\breve{\psi}\right\|_{L^{\infty}((0,T);L^{\infty}(0,1))}\right\\}.$
(38)

Step 2: estimate $\int_{0}^{1}G\left(e^{\sigma
T}\hat{w}(x,T)-\mathcal{D}\right)\text{d}x$. Let

$\displaystyle\hat{v}(x,t):=e^{{\sigma}t}\hat{w}(x,t),\hat{v}_{0}(x):=\hat{w}_{0}(x),\acute{d}_{0}(t):=-p_{0}e^{{\sigma}t}{d_{m}(t)},{\acute{\psi}(x,t):=e^{{\sigma}t}\breve{\psi}(x,t)},$

where $\hat{w}$ is the observer target system (see (35)) .

By integrating by parts and applying the definitions of $g$ and $G$, we get

$\displaystyle\frac{\text{d}}{\text{d}t}\int_{0}^{1}G(\hat{v}-\mathcal{D})\text{d}x$
$\displaystyle=$ $\displaystyle
g({\hat{v}}(1,t)-\mathcal{D}){\hat{v}}_{x}(1,t)-g({\hat{v}}(0,t)-\mathcal{D})\left(q{\hat{v}}(0,t)+\acute{d}_{0}(t){-p_{0}e^{\sigma
t}}\widetilde{u}(0,t)\right)-\int_{0}^{1}g^{\prime}(\hat{v}-\mathcal{D}){\hat{v}}_{x}^{2}\text{d}x$
$\displaystyle+\int_{0}^{1}g({\hat{v}}-\mathcal{D})(\sigma-c(t))(\hat{v}-\mathcal{D})\text{d}x+\int_{0}^{1}g({\hat{v}}-\mathcal{D})\Big{(}(\sigma-c(t))\mathcal{D}+\acute{\psi}\Big{)}+\int_{0}^{1}g({\hat{v}}-\mathcal{D})e^{\sigma
t}K_{p}(x)\widetilde{u}(0,t)\text{d}x$ $\displaystyle=$ $\displaystyle
I_{1}+I_{2}+I_{3}+I_{4}+I_{5}+I_{6},\forall t\in(0,T),$ (39)

where

$\displaystyle{I_{1}}:=$ $\displaystyle
g\left({\hat{v}}(1,t)-\mathcal{D}\right){\hat{v}}_{x}(1,t),$
$\displaystyle{I_{2}}:=$
$\displaystyle-g({\hat{v}}(0,t)-\mathcal{D})(q\hat{v}(0,t)+\acute{d}_{0}(t){-p_{0}e^{\sigma
t}}\widetilde{u}(0,t)),$ $\displaystyle{I_{3}}:=$
$\displaystyle-\int_{0}^{1}g^{\prime}\left({\hat{v}}(x,t)-\mathcal{D}\right){\hat{v}}_{x}^{2}(x,t)\text{d}x,$
$\displaystyle{I_{4}}:=$
$\displaystyle\int_{0}^{1}g({\hat{v}}(x,t)-\mathcal{D})(\sigma-c(t))(\hat{v}(x,t)-\mathcal{D})\text{d}x,$
$\displaystyle{I_{5}}:=$
$\displaystyle\int_{0}^{1}g({\hat{v}}(x,t)-\mathcal{D})\Big{(}(\sigma-c(t))\mathcal{D}+\acute{\psi}(x,t)\Big{)}\text{d}x,$
$\displaystyle{I_{6}}:=$
$\displaystyle\int_{0}^{1}g({\hat{v}}(x,t)-\mathcal{D})K_{p}(x)e^{\sigma
t}\widetilde{u}(0,t)\text{d}x.$

It is easy to see that $I_{1}=0$, $I_{3}\leq 0$, and $I_{5}\leq 0$ for all
$t\in(0,T)$. In addition, by the definitions of $\mathcal{D}$ and
$\acute{d}_{0}$, the Young’s inequality with $\varepsilon_{1}\in(0,q)$, and
(37a), we deduce that

$\displaystyle{I_{2}}\leq$
$\displaystyle-q(r+1)G(\hat{v}(0,t)-\mathcal{D})+r\varepsilon_{1}G(\hat{v}(0,t)-\mathcal{D})+\varepsilon_{1}^{-r}G\left({|p_{0}|}e^{\sigma
t}|\widetilde{u}(0,t)|\right)$ $\displaystyle\leq$
$\displaystyle\frac{\varepsilon_{1}^{-r}}{r+1}{|p_{0}|}^{r+1}e^{(r+1)\sigma
t}|\widetilde{u}(0,t)|^{r+1},\forall t\in(0,T).$ (40)

For $I_{4}$, by $\underline{c}-{\sigma}>0$ and the properties of $g$ and $G$
(see (37a)), we obtain

$\displaystyle{I_{4}}\leq-\left(\underline{c}-{\sigma}\right)(r+1)\int_{0}^{1}G(\hat{v}-\mathcal{D})\text{d}x,\forall
t\in(0,T).$ (41)

For $I_{6}$, applying the Young’s inequality, (37a), and the definition of $g$
(see (13)), we get for all $t\in(0,T)$:

$\displaystyle{I_{6}}\leq$ $\displaystyle
r\varepsilon_{2}\int_{0}^{1}G(\hat{v}-\mathcal{D})\text{d}x+\frac{\varepsilon_{2}^{-r}}{r+1}\overline{K}_{p}^{r+1}e^{(r+1)\sigma
t}|\widetilde{u}(0,t)|^{r+1},$ (42)

where $\varepsilon_{2}$ is a positive constant that will be determined later
and $\overline{K}_{p}:=\max_{x\in[0,1]}|K_{p}(x)|$.

Let $\varepsilon_{2}\in(0,\underline{c}-{\sigma})$ and
$\varepsilon_{3}:=\min\\{\varepsilon_{1},\varepsilon_{2}\\}$. Since $I_{i}\leq
0~{}(i=1,3,5)$, by (5.2), (5.2), (41), and (42), we get for all $t\in(0,T)$:

$\displaystyle\frac{\text{d}}{\text{d}t}\int_{0}^{1}G(e^{\sigma
t}\hat{w}-\mathcal{D})\text{d}x=$
$\displaystyle\frac{\text{d}}{\text{d}t}\int_{0}^{1}G(\hat{v}-\mathcal{D})\text{d}x$
$\displaystyle\leq$ $\displaystyle-r_{0}\int_{0}^{1}G\left(e^{\sigma
t}\hat{w}-\mathcal{D}\right)\text{d}x+\frac{\varepsilon_{3}^{-r}K_{0}^{r+1}}{r+1}e^{(r+1)\sigma
t}|\widetilde{u}(0,t)|^{r+1},$

where
$r_{0}:=r_{0}(\varepsilon_{2},r):=\left(\underline{c}-{\sigma}\right)(r+1)-r\varepsilon_{2}>0$
and $K_{0}:={|p_{0}|}+\overline{K}_{p}$.

By using the Gronwall’s inequality, we deduce that

$\displaystyle\int_{0}^{1}G(e^{\sigma T}\hat{w}(x,T)-\mathcal{D})\text{d}x\leq
e^{-r_{0}T}\int_{0}^{1}G\left(\hat{w}_{0}(x)-\mathcal{D}\right)\text{d}x+\frac{\varepsilon_{3}^{-r}K_{0}^{r+1}}{r+1}\int_{0}^{T}e^{-r_{0}(T-s)+(r+1)\sigma
s}|\widetilde{u}(0,s)|^{r+1}\text{d}s.$

Note that the spatial $L^{\infty}$-estimate of $\widetilde{u}$ (see Theorem
4.1) ensures that

$\displaystyle|\widetilde{u}(0,T)|^{r+1}$ $\displaystyle\leq
2^{r}{C_{1}}^{r+1}\left(e^{-\sigma(r+1)T}\left\|\widetilde{u}_{0}\right\|_{L^{\infty}(0,1)}^{r+1}+B_{1}^{r+1}\right),$

where

$\displaystyle{B_{1}:=}$
$\displaystyle{\left\|{d_{m}}\right\|_{L^{\infty}(0,T)}+}\left\|{d}_{0}\right\|_{L^{\infty}(0,T)}+\left\|{d}_{1}\right\|_{L^{\infty}(0,T)}+\left\|{f}\right\|_{L^{\infty}((0,T);L^{\infty}(0,1))},$
(43)

and $C_{1}$ is a positive constant depending only on
$\underline{c},\sigma,p_{0},{q}$, $\max_{x\in[0,1]}|p(x)|$, $\max_{(x,{z})\in
D}|m(x,{z})|$, and $\max_{(x,{z})\in D}|n(x,{z})|$. Then, we have

$\displaystyle\int_{0}^{1}G\left(e^{\sigma
T}\hat{w}(x,T)-\mathcal{D}\right)\text{d}x\leq$ $\displaystyle
e^{-r_{0}T}\int_{0}^{1}G\left(|\hat{w}_{0}(x)|-\mathcal{D}\right)\text{d}x+\frac{{C_{2}}}{r_{0}(r+1)}\left\|\widetilde{u}_{0}\right\|_{L^{\infty}(0,1)}^{r+1}$
$\displaystyle+\frac{{C_{2}}e^{(r+1)\sigma
T}}{(r_{0}+(r+1)\sigma)(r+1)}B_{1}^{r+1},$ (44)

where
${C_{2}:=C_{2}(\varepsilon_{3},r):=2^{r}C_{1}^{r+1}}\varepsilon_{3}^{-r}K_{0}^{r+1}$.

Step 3: estimate $\int_{0}^{1}G(-e^{\sigma
T}\hat{w}(x,T)-\mathcal{D})\text{d}x$.

Analogous to the derivations in Step 2, by calculating
$\frac{\text{d}}{\text{d}t}\int_{0}^{1}G\left(-e^{\sigma
T}\hat{w}(x,T)-{{\mathcal{D}}}\right)\text{d}x$ and using the Gronwall’s
inequality, we can obtain

$\displaystyle\int_{0}^{1}G\left(-e^{\sigma
T}\hat{w}(x,T)-{{\mathcal{D}}}\right)\text{d}x\leq$ $\displaystyle
e^{-r_{0}T}\int_{0}^{1}G\left(|\hat{w}_{0}(x)|-\mathcal{D}\right)\text{d}x+{\frac{C_{2}}{r_{0}(r+1)}}\left\|\widetilde{u}_{0}\right\|_{L^{\infty}(0,1)}^{r+1}$
$\displaystyle+\frac{{C_{2}}e^{(r+1)\sigma
T}}{(r_{0}+(r+1)\sigma)(r+1)}B_{1}^{r+1}.$ (45)

Step 4: establish the estimate of $\|\hat{w}[T]\|_{L^{\infty}(0,1)}$. By the
properties of $g$ and $G$ (see (37)), (5.2), and (5.2), we get

$\displaystyle\frac{e^{(r+1)\sigma
T}}{r+1}\int_{0}^{1}|\hat{w}(x,T)|^{r+1}\text{d}x\leq$ $\displaystyle
2^{r}\left(\int_{0}^{1}G\left(e^{\sigma
T}|\hat{w}(x,T)|-{\mathcal{D}}\right)\text{d}x+\int_{0}^{1}G({\mathcal{D}})\text{d}x\right)$
$\displaystyle\leq$
$\displaystyle\frac{2^{r}}{r+1}e^{-r_{0}T}\int_{0}^{1}|\hat{w}_{0}(x)|^{r+1}\text{d}x+\frac{2^{r}{C_{2}}}{r_{0}(r+1)}\left\|\widetilde{u}_{0}\right\|_{L^{\infty}(0,1)}^{r+1}$
$\displaystyle+\frac{2^{r}e^{(r+1)\sigma
T}{C_{2}}}{(r_{0}+(r+1)\sigma)(r+1)}B_{1}^{r+1}+\frac{2^{r}}{r+1}{\mathcal{D}}^{r+1},$

which implies that

$\displaystyle\|\hat{w}[T]\|_{L^{r+1}(0,1)}\leq$ $\displaystyle
2^{\frac{r}{r+1}}e^{-\left(\frac{r_{0}}{r+1}+\sigma\right)T}\left\|\hat{w}_{0}\right\|_{L^{r+1}(0,1)}+\left(\frac{2^{r}{C_{2}}}{r_{0}}\right)^{\frac{1}{r+1}}e^{-\sigma
T}\left\|\widetilde{u}_{0}\right\|_{L^{\infty}(0,1)}$
$\displaystyle+\left(\frac{2^{r}{C_{2}}}{r_{0}+(r+1)\sigma}\right)^{\frac{1}{r+1}}B_{1}+2^{\frac{r}{r+1}}e^{-\sigma
T}{\mathcal{D}}.$ (46)

Letting $r\to+\infty$ in (46), we obtain

$\displaystyle\|\hat{w}[T]\|_{L^{\infty}(0,1)}\leq
2e^{-\left(\underline{c}-\varepsilon_{2}\right){T}}\left\|\hat{w}_{0}\right\|_{L^{\infty}(0,1)}+{C_{3}}e^{-\sigma
T}\left\|\widetilde{u}_{0}\right\|_{L^{\infty}(0,1)}+{C_{3}}B_{1}+2e^{-\sigma
T}{\mathcal{D}},$ (47)

where ${C_{3}}:=4{C_{1}}K_{0}\varepsilon_{3}^{-1}$ and
$\underline{c}-\varepsilon_{2}>\sigma>0$. Then, in view of the definitions of
$B_{1}$ (see (43)) and $\mathcal{D}$ (see (38)), we conclude that Proposition
5.1 holds true. $\blacksquare$

##  6 Numerical Results

In this section, we present some numerical results on the ISS in the
$L^{\infty}$-norm for the considered systems with different disturbances under
the state feedback control law (12), and the observer-based output feedback
control law (33), respectively.

In simulations, we always set

$\displaystyle q=$ $\displaystyle 1,$ $\displaystyle\lambda(t)=$
$\displaystyle\left\\{\begin{aligned}
&1.2\pi^{2}\left(\sin^{2}5t+1\right),0\leq t\leq 1,\\\
&1.2\pi^{2}\left(\sin^{2}5t+e^{-t}-e^{-1}+1\right),t>1.\end{aligned}\right.$

In (3) and (3.1), we set $c_{0}=\frac{13}{5}\pi^{2}$. Let
$p_{0}=\frac{6}{5}\pi^{2}$. The gain $p(x)$ is obtained by using the
successive approximations.

Moreover, we set

$\displaystyle{u_{0}}(x)=$
$\displaystyle-\frac{1}{6}\left(5x-\frac{1}{4}\right)(2-x)\left(3x^{2}-1\right),$
$\displaystyle{\hat{u}_{0}}(x)=$ $\displaystyle 0,$
$\displaystyle{\widetilde{u}_{0}}(x)=$ $\displaystyle
u_{0}(x)-\hat{u}_{0}(x),$ $\displaystyle f(x,t)=$
$\displaystyle\frac{A}{6}\sin\left(60t+x\right),$ $\displaystyle d_{0}(t)=$
$\displaystyle\frac{A_{0}}{5}\left(\sqrt{t}e^{-t}+2\sin(25t)\right),$
$\displaystyle d_{1}(t)=$ $\displaystyle\frac{2}{5}A_{1}\sin(25t),$
$\displaystyle{d_{m}(t)}=$ $\displaystyle\frac{A_{2}}{44}\sin(40t),$

where $A\in\\{0,2\\}$ and $A_{0},A_{1},A_{2}\in\\{0,1,3\\}$ are used to
describe the amplitude of disturbances.

Figure 1 shows that the disturbance-free $u$-system (1) is unstable in open
loop, while Fig. 2(a), (b), and (c) show that, under the state feedback
control law (12), the state of the disturbance-free system (1) tends to the
origin and the states of system (1) with different disturbances remain
bounded. Moreover, when amplitudes of disturbances decrease, the amplitudes of
states decrease. This reflects well the ISS property of the $u$-system under
the state feedback control law (12) as shown in Fig. 2(d), where the evolution
of $\|u[t]\|_{L^{\infty}(0,1)}$ w.r.t. different disturbances are presented.
Similarly, Fig. 3 reflects the ISS property of the $\widetilde{u}$-system
(26).

Under the output feedback control law (33), Fig. 4 indicates the robust
stability of the $u$-system (1), and Fig. 5(a), (b), and (c) show the robust
convergence of the observer $\hat{u}$-system (21), respectively. In
particular, the results shown in Fig. 5(d) reflect well the ISS property of
the closed-loop $(u,\widetilde{u})$-system. It can be seen that the numerical
results are consistent with the theoretical analysis, demonstrating well the
effectiveness of the proposed control scheme.

(a) Evolution of $u$ (b) Evolution of $\|u[t]\|_{L^{\infty}(0,1)}$

Figure 1: Open-loop response of the $u$-system (1) without disturbances

(a) Evolution of $u$ when $A=A_{0}=A_{1}=0$ (b) Evolution of $u$ when
$A=2,A_{0}=A_{1}=1$

(c) Evolution of $u$ when $A=2,A_{0}=A_{1}=3$ (d) Evolution of
$\|u[t]\|_{L^{\infty}(0,1)}$ for different disturbances

Figure 2: Closed-loop responses of the $u$-system (1) with different
disturbances under the state feedback control law (12)

(a) Evolution of $\widetilde{u}$ when $A=A_{0}=A_{1}=A_{2}=0$ (b) Evolution of
$\widetilde{u}$ when $A=2,A_{0}=A_{1}=A_{2}=1$

(c) Evolution of $\widetilde{u}$ when $A=2,A_{0}=A_{1}=A_{2}=3$ (d) Evolution
of $\|\widetilde{u}[t]\|_{L^{\infty}(0,1)}$ for different disturbances

Figure 3: Evolution of $\widetilde{u}$ and
$\|\widetilde{u}[t]\|_{L^{\infty}(0,1)}$ for the error $\widetilde{u}$-system
(26) with different disturbances

(a) Evolution of $u$ when $A=A_{0}=A_{1}=A_{2}=0$ (b) Evolution of $u$ when
$A=2,A_{0}=A_{1}=A_{2}=1$

(c) Evolution of $u$ when $A=2,A_{0}=A_{1}=A_{2}=3$

Figure 4: Closed-loop responses of the $u$-system (1) with different
disturbances under the output feedback control law (33)

(a) Evolution of $\hat{u}$ when $A=A_{0}=A_{1}=A_{2}=0$ (b) Evolution of
$\hat{u}$ when $A=2,A_{0}=A_{1}=A_{2}=1$

(c) Evolution of $\hat{u}$ when $A=2,A_{0}=A_{1}=A_{2}=3$ (d) Evolution of
$\|u[t]\|_{L^{\infty}(0,1)}+\|\widetilde{u}[t]\|_{L^{\infty}(0,1)}$ for
different disturbances

Figure 5: Evolution of $\hat{u}$ and
$\|u[t]\|_{L^{\infty}(0,1)}+\|\widetilde{u}[t]\|_{L^{\infty}(0,1)}$ for the
observer $\hat{u}$-system (21) and the $(u,\widetilde{u})$-system coupled via
the output feedback control law (33)

##  7 Conclusion

In this paper, we designed an observer-based output feedback stabilizing
controller for a class of time-varying parabolic PDEs with in-domain and mixed
boundary disturbances under the framework of ISS. Time-invariant kernel
functions were used in both controller and observer design and implementation,
which required much less computation efforts compared to those required in the
standard backstepping control of PDEs. The generalized Lyapunov method, which
allows dealing with Dirichlet boundary disturbances, was applied to establish
the ISS in the $L^{\infty}$-norm of the considered system in closed loop. To
validate the effectiveness of the proposed control scheme, numerical
simulations were conducted for systems with different disturbances. It is
worth mentioning that establishing the ISS in the $L^{p}$-norm with
$p\in[2,+\infty)$ for the considered system in closed loop under output
feedback control is much challenging due to the appearance of the boundary
term $\widetilde{u}(0,t)$, which is difficult to estimate by using the
$L^{p}$-norm of $\hat{w}$ or $\widetilde{u}$, in the target system (35) of the
observer. This will be considered in our future work.

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# Accelerating Boolean Constraint Propagation for Efficient SAT-Solving on
FPGAs

H. Govindasamy1, B. Esfandiari1, P. Garcia2

1{hari<EMAIL_ADDRESS>Dpt. Systems Computer Engineering, Carleton
University, Canada<EMAIL_ADDRESS>International School of Engineering,
Chulalongkorn University, Thailand

###### Abstract

We present a hardware-accelerated SAT solver targeting processor/Field
Programmable Gate Arrays (FPGA) SoCs. Our solution accelerates the most
expensive subroutine of the Davis-Putnam-Logemann-Loveland (DPLL) algorithm,
Boolean Constraint Propagation (BCP) through fine-grained FPGA parallelism.
Unlike prior state-of-the-art solutions, our solver eliminates costly clause
look-up operations by assigning clauses directly to clause processors on the
FPGA and dividing large formulas into smaller partitions manageable by FPGA.
Partitions are hot-swapped during runtime as required and the supported
formula size is limited only by available external memory, not on-chip FPGA
memory.

We evaluate our solver on a Xilinx Zynq platform with results showing quicker
execution time across various formula sizes, subject to formula partitioning
strategy. Compared to prior state-of-the-art, we achieve 1.7x and 1.1x speed
up on BCP for 2 representative benchmarks and up to 6x total speedup over
software-only implementation.

###### Index Terms:

FPGA, SAT, acceleration, embedded, boolean, satisfiability

††publicationid: pubid: 0000–0000/00$00.00 © 2021 IEEE

##  I Introduction

The Boolean Satisfiability problem (SAT) is a fundamental problem in computer
science, the first NP-Complete problem [1]. SAT solvers have become the
backbone of several engineering domains, as any NP-Complete problem can be
encoded as instance of SAT [1, 2]. SAT solvers determine whether a given
boolean formula is satisfiable by identifying an assignment to the formulas’
free variables that evaluate the formula to true. The formula is unsatisfiable
otherwise. Most SAT solvers target CNF-SAT, a subset of SAT that determines
the satisfiability of formulas encoded in _Conjunctive Normal Form_ (CNF) .
Formulas in CNF are conjunctions of clauses, where each clause is a
disjunction of one or more literals (a variable or its negation).

With the advent of modern Systems-on-Chip (SoC) comprised of both hard
embedded processors and configurable FPGA fabric offering myriad
implementation opportunities [3], deployed from the embedded to the high
performance computing domain [4], accelerating SAT-solving through hardware is
an attractive approach. We present a novel architecture for hardware-
accelerated SAT-solving that outperforms state of the art solutions, released
in open-source form for the Xilinx Zynq platform. Specifically, this article
offers the following contributions:

  * •

We describe a methodology to map and runtime-manage clauses across a processor
and connected FPGA, making efficient use of FPGA resources and avoiding
recurring performance pitfalls.

  * •

We describe the implementation of an open-source prototype system, deployed on
a Xilinx Zynq chip, identifying how the hardware architecture effects the
aforementioned strategy.

  * •

We evaluate our design against the state of the art using two representative
benchmarks, showing speed-ups of 1.7x and 1.1x, respective, and overall a 6x
improvement over vanilla software execution.

Section II describes necessary background knowledge on a particular SAT-
solving algorithm required to understand the remainder of this paper. Section
III presents an overview of historical solutions and state of the art,
directly compared against in this paper. Section IV presents our contribution,
evaluated in Section V, with concluding remarks and suggestions for future
work described in Section VI.

Figure 1: Interface between processor and FPGA-based BCP Coprocessor. DPLL’s
BCP is accelerated through fine-grained parallelization across Clause
Processors.

##  II Background: DPLL and BCP

SAT solvers are categorized into complete and incomplete solvers[5]. Complete
solvers evaluate every possible variable assignment, ending on the first
satisfying assignment or after exhausting the search space. A formula is
unsatisfiable if the complete solver concludes without finding a satisfying
assignment.

Most incomplete solvers use Stochastic Local Search (SLS) to greedily search
for a satisfying assignment in the formula’s variable assignment search
space[6]. While typically quicker than complete solvers, incomplete solvers do
not guarantee results as they tend to get stuck in local maxima or skip
satisfying assignments. Since they don’t explore the solution space
exhaustively they can never conclude that a formula is unsatisfiable.

Davis-Putnam-Logemann-Loveland (DPLL) and DPLL-based algorithms are the most
predominant complete solvers [6]. DPLL performs two primary operations: 1)
_decision_ and 2) _Boolean Constraint Propagation_ (BCP). DPLL-based
algorithms follow DPLL’s core structure, and propose improved decision
heuristics, learning and BCP mechanisms.

During decision, DPLL heuristically picks and assigns truth values to free
variables. BCP subsequently propagates the effect of the decision using the
_unit implication rule_[7]. The unit implication rule identifies unit clauses
where all but one of its literals are false. Unit clauses can only be
satisfied by assigning the variable to true if the literal is positive or to
false on negative literals. The resulting assignment is known as an
implication. BCP repeatedly applies this rule until all clauses are satisfied
(formula is therefore satisfiable) or at least one clause evaluates false
(conflict). On conflicts, DPLL backtracks by retracting and/or inverting
assignments from earlier decisions.

BCP is expensive, accounting for 80-90% of DPLL’s CPU time, rendering it a
prime candidate for hardware acceleration [7, 8]. BCP coprocessors accelerate
DPLL by implementing specialized BCP processing engines on FPGA. These run
alongside a General Purpose Processor (GPP) that performs the remaining DPLL
operations: decision heuristics and backtracking. Using this architecture, the
BCP coprocessor is first configured with the clauses, and then waits to
evaluate decisions from the GPP. Any DPLL-based software solver can integrate
with a BCP-coprocessor by replacing software BCP with the hardware accelerated
BCP-coprocessor [8, 9].

FPGA-based BCP coprocessors are either instance-specific or application-
specific. Instance-specific solver are built to solve a single SAT instance
and designed by translating an input formula into its equivalent logical
circuit. However, to solve new instances, the FPGA requires a complete rebuild
(synthesis and FPGA programming may take up to several hours). Although these
solvers can be significantly quicker than their software counterparts, their
performance becomes notably slower when build times are included. For
instance, Ivan et al’s best result against the hole7 benchmark achieves a
6.66x speedup against MiniSAT[10, 11]; however, when build times are included,
compilation alone takes 50 seconds, whereas MiniSAT finishes in under 0.064
seconds [12].

Application-specific solvers eliminate the need to rebuild the FPGA by
instantiating general-purpose processing units capable of tackling any SAT
instance (given that it fits in hardware). The BCP coprocessor is configured
with the target problem by simply overwriting FPGA memory.

##  III State of the art

Algorithmic techniques for efficient SAT solving have been extensively
researched, and the literature contains several surveys that describe the
history and state of the art of the problem ([13], [14]). Techniques aimed at
accelerating the execution of a particular SAT solving algorithm include
software parallelization [15], deployment on specialized GPUs [16], and even
acceleration through machine-learning approaches [17].

Our approach sits within FPGA-based acceleration, which began roughly 3
decades ago [18], with a few prominent results at the turn of the century
([19], [20]). However, it was not until significant advances in FPGA
performance occurred in the last decade, and the rise of SoC platforms
combining FPGA fabric with hard processors, that FPGA-based SAT acceleration
matured. The most notable architectures were proposed by Davis et al [7] and
Thong et al [8, 9]: both exploring the use of FPGA to implement BCP
coprocessors, keeping the remainder of DPLL in software, described in detail
below:

###  III-A Davis et al.

Davis et al calculate implications in parallel by using several inference
engines (IE), each assigned a list of clauses (partitions) [7]. For every
decision/implication, the clause containing the assignment variable is first
retrieved before calculating implications. Implications are forwarded to a
conflict detector that ensures that two or more IEs have not implied opposing
values for the same variable. Implications are then sent to the processor and
queued up for propagation.

To keep clause retrieval time low, a variable only occurs once in each IEs
partition (i.e clauses within the same IE share no common variables). This
limits the number of clauses affected by a decision to one, thereby also
limiting implications per IE to one, constraining the effected performance.
While some strategies to increase this limit have been proposed [21], they
remain unexplored.

###  III-B Thong et al.

Thong et al. propose a concurrent BCP coprocessor comprising multiple
sequential processing engines (PE) [9]. Identifying that Davis et al.’s clause
lookup is slower than direct access [8], they develop a clause storage and
encoding scheme that efficiently links clauses with shared variables. The
processor sends decisions to the FPGA and starts BCP execution at a single PE.
Using the linked list, the PE traverses every clause containing the decision
variable and calculates implications, which are then added to a local queue
and propagated. The running PE triggers BCP execution in another PE when it
arrives at a link to a clause that is located elsewhere. The coprocessor
supports multithreaded software execution, hiding communication and software
latency by keeping the coprocessor busy while software threads make decisions
when possible.

Davis et al. and Thong et al. have laid a strong foundation in developing
application-specific FPGA-based BCP coprocessors; we extend their work and
propose a solution that processes clauses in parallel without the need for
clause lookup.

##  IV The SAT solver Architecture

We present a BCP coprocessor that works alongside vanilla DPLL (and should, in
theory, work seamlessly with any DPLL-based solver). Like Thong et al., we
forgo clause lookup and allow clauses to share variables within the same
partition. However, we still achieve Davis et al.’s high degree of parallelism
by placing clauses directly in clause processors (explained in Section IV-A).

SAT instances larger than the available number of Clause Processors (CPs) are
partitioned, stored in external memory (i.e., software) and hot-swapped into
the BCP coprocessors as required during runtime. Solvable instance size is
limited only by the GPP’s RAM, not on-chip FPGA memory. We deploy our solution
on the Zynq chip, and available
here111https://github.com/harigovind1998/FPGA_BCP_acceleration for use. To our
knowledge, this is the first open-source hardware-accelerated SAT solver.

###  IV-A The BCP accelerator architecture

Figure 1 illustrates our approach, comprising a GPP and an FPGA accelerated
BCP coprocessor. The GPP executes DPLL’s remaining elements (decisions,
backtrack, etc.), partitions large SAT instances (explained in Section IV-B)
and swaps partitions into hardware as required. Figure 2 depicts the
coprocessor’s behavior. Its default state is idle, awaiting instructions to
execute. Once a decision is received, the systems loops until all unit clauses
are exhausted.

The BCP coprocessor, depicted in Figure 1, comprises a control unit (1), an
array of clause processors (2) and an implication selector (3). The central
control unit communicates directly with the GPP and each CP. Based on the
received GPP command, it loads clauses into CPs, broadcasts decisions, or
clears assignments during backtrack.

At its core, the BCP coprocessor consists of an array of CPs that calculate
decision and implication results in parallel. CPs store clauses as an array of
literals maintain a local copy of each literal’s respective variable
assignment. Partitions are hot-swapped into FPGA by overwriting a CPs array of
literals with the literals of the new clause. Variable assignments are updated
during decisions and BCP, and cleared during backtrack.

Finally, the implication selector chooses a single implication to propagate
when multiple implications arise as a result of BCP. Rather than using an
explicit implication conflict detector, as done by Davis et al [7], we
propagate the chosen implication, and identify conflicts during evaluation.

Figure 2: BCP Engine - Flowchart

###  IV-B Formulae partitioning

SAT instances contain an arbitrary number of variables and clauses. The
problem size solvable on FPGA is limited by its available Configurable Logic
Block (CLB) and memory, and requires large problems be partitioned into
smaller manageable sizes. Partitions are stored in the GPP, and swapped into
FPGA during run time by overwriting CPs clauses. BCP is performed individually
on each partition, and implications are relayed back to the GPP. Implications
are subsequently propagated to other partitions.

Algorithm 1 describes our simple greedy approach to partitioning large SAT
instances. We aim to make partitions as large as possible, limited by the
coprocessor’s clause and variable threshold. The partitioning algorithm begins
with a single empty partition and loops through each clause, adding them to
the last partition in a list of partitions (line 7). New partitions are
created if adding a clause results the last partition exceeding the systems
clause or variable threshold (line 4).

1 Partition $(f,C,V)$;

Input : CNF formula $f$, number of clauses supported by FPGA $C$, number of
variables supported by FPGA $V$

Output : partitions of clauses from $f$

2 partitions $\leftarrow$ list with 1 empty partition;

3 foreach  _clause in f_ do

4 if  _adding clause to last partition results in it exceeding C or V_ then

5 create and add new partition to partitions;

6

7 end if

8 add clause to last partition in partitions;

9

10 end foreach

11return partitions;

Algorithm 1 A greedy approach to partitioning large SAT instances

Consider Equation 1, composed of four clauses, and an instance of our
coprocessor that supports two clauses and three variables. Equation 2 and 3
outline the two possible ways to partition Equation 1 using Algorithm 1.
Equation 2 describes a scenario where the partitions reach the clause limit,
while the Equation 3 reaches the variable limit.

$f=(\neg a\lor b\lor\neg c)\wedge(a\lor\neg b\lor\neg c)\wedge(\neg d\lor
e\lor f)\wedge(d\lor e\lor f)$ (1)

$variation\\_1=\\{\\{(\neg a\lor b\lor\neg c)\wedge(a\lor\neg b\lor\neg
c)\\},\\\ \\{(\neg d\lor e\lor f)\wedge(d\lor e\lor f)\\}\\}$ (2)
$variation\\_2=\\{\\{(\neg a\lor b\lor\neg c)\\},\\{(a\lor\neg b\lor\neg
c)\\},\\\ \\{(\neg d\lor e\lor f)\\},\\{(d\lor e\lor f)\\}\\}$ (3)

Figure 3: (a) Davis et al. store the formula directly on FPGA. Clauses within
partitions contain no shared variables, and partitions are mapped directly to
Implication Engines. (b) Thong et al. store formula directly on FPGA. Clauses
are linked to other clauses with shared variables and are processed
sequentially. (c) Formula stored in external memory (”software” view). Clauses
in partitions mapped directly to Clause Processors, and hot-swapped as
required.

Results (refer to Section V) indicate that partitioning is a bottleneck in our
approach. Performance improvement is dictated by the amount of required
partition swapping and the number of unused CPs (occurs when the number of
clauses in a partition is less than the available number of CPs). Thus,
performance improvement is observed with certain partition assignments, while
others lead to performance degradation. System performance can be improved by
developing a more effective partitioning algorithm, but beyond the scope of
this paper and reserved for future work.

###  IV-C Execution

Each clause processor is only associated with a single clause; thus, no clause
look-up or traversal is required to retrieve the affected clause for
processing. All clauses on the FPGA are processed in parallel as soon as a
decision is received. Consider Equation 1’s mapping of partitions to hardware
as presented in Figure 3. Figure 4 summarizes the execution stages for Davis
et al.’s, Thong et al.’s and our approach for the theoretical execution for a
decision of variable $a$.

In our approach, clauses 1 and 2 are processed by Clause Processor 1 and 2 in
parallel once the decision is received. Since clauses 3 and 4 do not contain
variable $a$, Partition 2 remains in external memory and is not processed.
Though Davis et al. also process clause 1 and 2 in parallel, each Implication
Engine first performs a clause look-up to retrieve the affected clause.
Results of the decision on the affected clause are then calculated.

Thong et al.’s approach starts BCP on Processing Engine 1. After clause 1 is
processed Processing Engine 1 traverses to clause 2. In the manner, clauses in
a partition are processed sequentially. Execution concludes after computing
Partitions 1’s final element, clause 2. Processing Engine 2 remains idle for
the entire duration as clauses in partition 2 do not contain variable $a$

Figure 4: Execution steps of each described approach.

###  IV-D Processor-FPGA interface

The BCP coprocessor implements the Advanced eXtensible Interface 4-Lite (AXI-
Lite) IP interface, acting as a subordinate to a processor (AXI master). Using
AXI, the processor writes directly to the coprocessor’s registers to send
instructions and data, and continues polling for status updates and new
implication until the coprocessor completes.

Status changes dictate DPLL’s flow, either allowing the search to continue
assigning additional variables, or triggers backtracking on conflicts. A copy
of all the implications are saved on the processor to avoid re-assigning
implied variables, and further propagated to the remaining partitions.

##  V Experiments and Results

On a Xilinx Zynq chip with total capacity of 14400 LUTs and 28800 FF, our
solution supports 224 parallel Clause Processors and 63 variables. We achieve
a clock frequency of 106.66 MHz, utilizing 647 LUTRAM of on-chip memory, 13151
LUTs, and 11059 FFs.

| Millions of BCP/s
---|---
SAT Instance | Davis et al [7] |  Thong et al [9] | Our Design
bmc-galileo-8 | 40 | 102 | 175
bmc-ibm-12 | 33 | 150 | 169
TABLE I: Comparison of BCP engine throughput (BCPs/s) with related work.
Results reflect maximum theoretical throughput, achieved only data is fully
available to BCP engines. Figure 5: Breakdown of the total execution time
across constituent components.

Related work calculates throughput (in BPCs performed per second), assuming
full data availability: i.e., not taking into account software execution and
communication/data transfer latency. Whilst this is a useful metric to assess
hardware performance in isolation (and we report equivalent results in Table
I), it does not accurately depict system performance; to do so, we break down
the full execution in Figure 5 and evaluate speedup over vanilla software
implementation, evaluating combinations of clause and variable sizes, with
speedup depicted in Table II for meaningful combinations. For each
combination, we also depict real throughput, in the form of BCPs/s averaged
over total execution time (63 variables and 224 clauses is the theoretical
upper bound, without the need for hot swapping). To evaluate the different
effects of clause/variable sizes on execution, we fix one and vary the other,
measuring total execution time: results are depicted in Figures 6 and 7.

|  | Variables
---|---|---
|  | 63 | 126 | 252 | 630
Clauses | 224 |  362M BCP/s 2.2x |  17K BCP/s 0.17x | NA | NA
448 |  702K BCP/s 1.6x |  21K BCP/s 0.21x |  13K BCP/s 0.08x | NA
2240 |  441K BCP/s 1.91x |  22K BCP/s 1.26x |  16K BCP/s 0.61x |  12K BCP/s 0.10x
22400 |  313K BCP/s 6.32x |  20K BCP/s 5.04x |  16K BCP/s 4.86x |  14K BCP/s 3.31x
TABLE II: Varied clause/variable sizes and their impact on the relative
speedup of hardware/software and the effective throughput of BCP engines.
Figure 6: Effect of increasing clauses size on total execution time, for 63
variables. Figure 7: Effect of increasing variables size on total execution
time, for 22400 clauses.

##  VI Conclusions

We described a SAT-solver hardware-accelerated architecture that outperforms
state of the art by hot-swapping clause assignment at runtime, making
efficient use of FPGA resources. Our solution prototype, on a Xilinx Zynq
chip, is available in open-source. Practitioners may use the presented
solution in their designs, whenever a problem is encoded in SAT form and
performance is critical.

An important open question remains: our performance is constrained by how
clauses are partitioned. A partitioning scheme that minimizes the distribution
of variables among clauses will minimize runtime swapping, resulting in
improved execution. However, how to best partition a formula to achieve this
is not yet known. Future work must formulate this challenge as an optimization
problem, and methods for its efficient solution must be devised. Once that is
achieved, they can be applied (offline) prior to deployment on our
architecture.

## Acknowledgments

We acknowledge the support of the Natural Sciences and Engineering Research
Council of Canada (NSERC)

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