# Exponential stability for a nonlinear porous-elastic system with delay

M.J. Dos Santos  C.A. Raposo  L.G.R. Miranda and B. Feng  Manoel Jeremias Dos
Santos – Faculty of Exact Sciences and Technology, Federal University of Pará,
Abaetetuba, Pará, 68440-000, Brazil<EMAIL_ADDRESS>Carlos Alberto Raposo
da Cunha – Department of Mathematics, Federal University of Bahia, Salvador,
Bahia, 40.170-110, Brazil<EMAIL_ADDRESS>Luiz Gutemberg Rosário
Miranda – Faculty of Mathematics, Federal University of Pará, Salinópolis,
Pará, 68721-000, Brazil<EMAIL_ADDRESS>Baowei Feng – Department of
Economic Mathematics, Southwestern University of Finance and Economics,
Chengdu, 611130, People’s Republic of China<EMAIL_ADDRESS>
###### Abstract

In this work, we consider the existence of a global solution and the
exponential decay of a nonlinear porous elastic system with time delay. The
nonlinear term, as well as the delay acting in the equation of the volume
fraction. In order to obtain the existence and uniqueness of a global
solution, we will use the semigroup theory of linear operators and under a
certain relation involving the coefficients of the system together with a
Lyapunov functional, we will establish the exponential decay of the energy
associated to the system.

###### keywords:

Global solution, exponential decay, nonlinear porous elastic system, time
delay

35A02, 35B35, 35B40 31 2023 1 https://doi.org/10.46298/cm.10439

##  1 Introduction

This paper is concerned with a nonlinear porous elastic system with a time
delay given by

$\displaystyle\begin{array}[]{rcl}\rho u_{tt}(x,t)\\!-\\!\mu
u_{xx}(x,t)\\!-\\!b\phi_{x}(x,t)\\!=\\!0,\\\
\\!\\!\\!J\phi_{tt}(x,t)\\!-\\!\delta\phi_{xx}(x,t)\\!+\\!bu_{x}(x,t)\\!+\\!\xi\phi(x,t)\\!+\\!\mu_{1}\phi_{t}(x,t)\\!+\\!\mu_{2}\phi_{t}(x,t-\tau)\\!+\\!f(\phi(x,t))\\!=\\!0,\end{array}$
(3)

with $(x,t)\in(0,1)\times(0,+\infty)$, were $\rho$, $\mu$, $J$, $\delta$,
$\xi$, $\mu_{1}$, $\mu_{2}$ are positive constants and $b$ is a non-zero
constant satisfying $b^{2}\leq\mu\xi$. We consider Dirichlet boundary
conditions

$\displaystyle u(0,t)=u(1,t)=\phi(0,t)=\phi(1,t)=0,\quad t>0$ (4)

and initial conditions

$\displaystyle\begin{array}[]{rl}u(x,0)=u_{0}(x),\,\,u_{t}(x,0)=u_{1}(x),&x\in(0,1),\\\
\phi(x,0)=\phi_{0}(x),\,\,\phi_{t}(x,0)=\phi_{0}(x),&x\in(0,1).\end{array}$
(7)

Since there exists a delay term in (3) we must provide information for
$\phi_{t}$ in time interval $(0,\tau)$ i.e., in addition to the initial
conditions given earlier, we still have

$\phi_{t}(x,t-\tau)=f_{0}(x,t-\tau),\,(x,t)\in(0,1)\times(0,\tau).$

where $(u_{0},u_{1},\phi_{0},\phi_{1},f_{0})$ will be taken in an appropriate
space. The importance of porous elastic materials lies in the fact that they
are present in various fields of human activity such as the petroleum
industry, material science, soil mechanics, foundation engineering, powder
technology, biology and others (cf. [IESANBOOK], [STRAUGHAN]). Among the
various theories dealing with a porous material, we can find a linear theory
proposed by Cowin and Nunziato [COWIN-NUNZIATO], [NUNZIATO-COWIN] which is a
generalization of the elastic theory for materials with voids, considering
besides the material elasticity property, the volume fraction of the voids in
the material. In this theory, the bulk density $\rho=\rho(x,t)$ is given by
the product of matrix density of the material $\gamma=\gamma(x,t)$ and the
volume fraction $\nu=\nu(x,t)$

$\rho(x,t)=\gamma(x,t)\nu(x,t).$

They also consider a reference configuration (generally considered as an
initial configuration)

$\rho_{0}(x)=\gamma_{0}(x)\nu_{0}(x).$

Let $u_{i}=u_{i}(x,t)$ denote the components of the displacement vector field
and so the components of the infinitesimal strain field are given by

$e_{ij}=\frac{1}{2}(u_{i,j}+u_{j,i}),$

where the comma after the letter indicates the partial derivative with respect
to the indicated coordinate. In addition, $\phi=\phi(x,t)$ represents the
change in volume fraction with respect to the reference configuration. In a
framework of evolution equations and in a setting of three-dimensional theory,
the porous-elastic theory is described as

$\displaystyle\rho\ddot{u}_{i}$ $\displaystyle=$ $\displaystyle T_{ij,j}+\rho
f_{i},$ $\displaystyle\rho k\ddot{\phi}$ $\displaystyle=$ $\displaystyle
h_{i,i}+g+\rho\ell,$

which are called the balance of linear momentum and the balance of
equilibrated force equations, respectively. In the above system, $T_{ji}$ are
the components of the stress tensor, $f_{i}$ is the body force vector, $h_{i}$
are the components of the equilibrated stress vector, $k$ is the equilibrated
inertia, $g$ is the intrinsic equilibrated body force and $\ell$ is the
extrinsic equilibrated body force. The constitutive equations for homogeneous
and isotropic elastic bodies are (cf. [COWIN-NUNZIATO]):

$\displaystyle T_{ij}$ $\displaystyle=$
$\displaystyle\lambda\delta_{ij}e_{rr}+2\mu e_{ij}+\beta\phi\delta_{ij},$
$\displaystyle h_{i}$ $\displaystyle=$ $\displaystyle\alpha\phi_{,i},$
$\displaystyle g$ $\displaystyle=$
$\displaystyle-\omega\dot{\phi}-\xi\phi-\beta e_{rr},$

where $\lambda$, $\mu$, $\beta$, $\alpha$, $\omega$, $\xi$ and $\omega$, are
constitutive constants that depend on the reference state $\nu_{0}$ and
$\delta_{ij}$ is Kronecker’s delta. The necessary and sufficient conditions
for the internal energy density to be positive definite quadratic form are
(cf.[COWIN-NUNZIATO])

$\mu>0,\ \alpha>0,\ \xi>0,\ \kappa>0,\ \omega>0,\ 3\lambda+2\mu>0\ \mbox{and}\
3b^{2}\leq(3\lambda+2\mu)\xi.$

###  1.1 The stabilization scenario

The stabilization study constitutes an important issue in research areas into
mathematics and engineering and has been aiming for investigation for years.
Naturally, significant mathematical properties are extracted from undamped
partial differential equations, and we pointed out the important mathematical
properties from wave equations, plate equations, shell equations and beam
(plane and curved) beams [GRAFFBOOK], [LAGNESE-LIONS-1]. However, dissipative
mechanisms make more realistic the phenomenons translated in terms of partial
differential equations. See for example the classical books due to Komornik
[KOMORNIK] and Lions and Lagnese [LAGNESEBOOK], [LIONSBOOK] on boundary
stabilization. In this direction, we will say that the energy of the solutions
of dissipative partial differential equations decay exponentially if they are
controlled by an exponential negative. On the contrary, we will say that the
energy of the solution slowly decays. Thus, in both cases is important to
determine conditions to achieve some type of decay. To the best knowledge, the
analysis of the temporal decay in one-dimensional porous-elasticity theory was
first studied by Quintanilla [QUINTANILLA]. He showed, by taking into account
a particular set of solutions and by using the Routh-Hurwitz theorem, that the
porous viscosity acting on the second equation of the following porous elastic
system

$\displaystyle\rho_{0}u_{tt}-\mu u_{xx}-\beta\varphi_{x}$ $\displaystyle=$
$\displaystyle 0,$
$\displaystyle\rho_{0}\kappa\varphi_{tt}-\alpha\varphi_{xx}+\beta
u_{x}+\xi\varphi+\tau\varphi_{t}$ $\displaystyle=$ $\displaystyle 0,$

was not powerful enough to obtain the exponential decay of solutions when
$\alpha\neq\mu k$. For this problem was used the denomination of slow decay
was to characterize the decay obtained. Damping effects like temperature,
elastic viscosity, porous viscosity and micro-temperature acting on porous-
elasticity equations were considered in the past years and obtained several
contributions to porous-elasticity systems with an emphasis on lack of
exponential decay, the exponential decay, the polynomial decay, the optimality
of the polynomial decay and numerical computations results, see [Casas_PS],
[CASAS-QUINTANILLA-2005-2], [MAGANA-QUINTANILLA-2006], [MAGANA-
QUINTANILLA-2007], [PAMPLONA-RIVERA-QUINTANILLA-2009], [PAMPLONA-RIVERA-
QUINTANILLA-2011], [SANTOS-ALMEIDAJR-2016], [SANTOS-ALMEIDAJR-2017], [SANTOS-
CAMPELO-ALMEIDAJR-2016], [SANTOS-CAMPELO-ALMEIDAJR-2017]. A porous
thermoelastic system was investigated in [FAREH-MESSAOUDI-2017] where the heat
conduction is given by Cattaneo’s law and was proved an exponential decay
result. Analyticity of the semigroup associated with the transmission problem
in porous elasticity and delay was studied in [RAPOSO-APALARA-RIBEIRO]. It is
interesting to note a particular case of the equations (3), when
$k:=\mu=\xi=b$ and $\rho_{0}:=\rho$ e $\rho_{1}:=\rho k$, we obtain the
following nonlinear Timoshenko system that modeling a thin elastic beam with
delay

$\displaystyle\begin{array}[]{rcl}\rho_{0}u_{tt}-k(u_{x}+\phi)_{x}&=&0,\\\
\rho_{1}\phi_{tt}-\delta\phi_{xx}+k(u_{x}+\phi)+\mu_{1}\phi_{t}+\mu_{2}\phi_{t}(x,t-\tau)+f(\phi)&=&0.\end{array}$
(10)

The exponential decay for the system (10) was studied in [FENG-PELICER] which
extended the result previously obtained in [HOUARI-LASKRI]. In both cases,
linear and non-linear, exponential decay was obtained through equality

$\displaystyle\frac{\rho_{0}}{k}=\frac{\rho_{1}}{\delta}.$ (11)

This paper intends to answer the following question: Is it possible to add
terms of delay, friction and forcing in the porous elastic system similar to
that found in (10) and also to obtain the exponential decay? The answer is
yes, and in this situation, a relation between the coefficients will appear
similar to (11), in the porous elastic case (3) the equality

$\displaystyle\frac{\rho}{\mu}=\frac{J}{\delta}$ (12)

will play an important role. This article is divided as follows: in section 2
we will establish the notations, as well as put the system (3)-(7) in the form
of an abstract Cauchy problem. In section 3 we will establish the existence of
a global solution for the system (3)-(3) and in section 4 we will deal with
the main result of this work, which is the exponential decay of the energy of
the solution for the problem (3)-(7).

##  2 Preliminaries and main results

In this work, we will use the standard notations $L^{p}(0,1)$ with $1\leq
p\leq\infty$ for the Lebesgue spaces and $H_{0}^{1}(0,1)$ for Sobolev space.
The norm in $L^{p}(0,1)$ will be represented by $\|\cdot\|_{p}$ and we will
write $\|\cdot\|$ instead of $\|\cdot\|_{2}$ to norm in $L^{2}(0,1)$. For the
forcing term $f(\phi)$, we assume $f:\mathbb{R}\to\mathbb{R}$ and that there
are constants $k_{0}>0$ and $\theta>0$ satisfying

$\displaystyle|f(x)-f(y)|\leq
k_{0}(|x|^{\theta}+|y|^{\theta})|x-y|\quad\forall x,y\in\mathbb{R}.$ (13)

In addition, we will also assume that

$\displaystyle 0\leq\hat{f}(x)\leq f(x)x\quad\forall x\in\mathbb{R},$ (14)

with $\hat{f}(x)=\displaystyle\int_{0}^{x}f(s)ds$. Using an argument found in
[FENG-PELICER], [HOUARI-LASKRI], [NICAISE-PIGNOTTI-2006], [NICAISE-
PIGNOTTI-2008], in order to deal with the delay term, we define the following
variable:

$\displaystyle z(x,y,t)=\phi_{t}(x,t-\tau y),\quad x\in(0,1),\ y\in(0,1),\
t>0.$ (15)

Then it is easy to verify

$\displaystyle\tau
z_{t}(x,y,t)+z_{y}(x,y,t)=0,\quad\forall(x,y,t)\in(0,1)\times(0,1)\times(0,\infty).$
(16)

Thus, the system (3)-(7) takes the following form

$\displaystyle\rho u_{tt}(x,t)-\mu u_{xx}(x,t)-b\phi_{x}(x,t)\\!=\\!0,$ (17)
$\displaystyle\\!\\!\\!\\!J\phi_{tt}(x,t)-\delta\phi_{xx}(x,t)+bu_{x}(x,t)+\xi\phi(x,t)+\mu_{1}\phi_{t}(x,t)+\mu_{2}z(x,1,t)+f(\phi(x,t))\\!=\\!0,$
(18) $\displaystyle\tau z_{t}(x,y,t)+z_{y}(x,y,t)\\!=\\!0,$ (19)

with $x\in(0,1)$, $y\in(0,1)$ and $t>0$. In addition, boundary conditions are
given by

$\displaystyle\begin{array}[]{rl}u(0,t)=u(1,t)=\phi(0,t)=\phi(1,t)=0,&t>0,\\\
z(x,0,t)=\phi_{t}(x,t),&(x,t)\in(0,1)\times(0,+\infty)\end{array}$ (22)

and initial conditions

$\displaystyle\begin{array}[]{rl}u(x,0)=u_{0}(x),\,u_{t}(x,0)=u_{1}(x),\,\phi(x,0)=\phi_{0}(x),\,\phi_{t}(x,0)=\phi_{0}(x),x\in(0,1),\\\
z(x,1,0)=f_{0}(x,t-\tau),(x,t)\in(0,1)\times(0,\tau).\end{array}$ (25)

Considering $U(t)=(u(t),u_{t}(t),\phi(t),\phi_{t}(t),z(t)$ and
$U_{0}=(u_{0},u_{1},\phi_{0},\phi_{1},f_{0})$, the system or (17)-(25) is
reduced to the following abstract first order evolution problem:

$\left\\{\begin{array}[]{rcl}\dot{U}(t)&=&\pazocal{A}U(t)+\pazocal{F}(U(t)),\quad
t>0,\\\ U(0)&=&U_{0},\end{array}\right.$ (26)

where $\dot{}=\frac{d}{dt}$, and

$\pazocal{A}U=\left(\begin{array}[]{c}v\\\
\frac{\mu}{\rho}u_{xx}+\frac{b}{\rho}\phi_{x}\\\ \psi\\\
\frac{\delta}{J}\phi_{xx}-\frac{b}{J}u_{x}-\frac{\xi}{J}\phi-\frac{\mu_{1}}{J}\psi-\frac{\mu_{2}}{J}z(\cdot,1)\\\
-\frac{1}{\tau}z_{y}\end{array}\right),\quad\pazocal{F}(U)=\left(\begin{array}[]{c}0\\\
0\\\ 0\\\ -\frac{1}{J}f(\phi)\\\ 0\end{array}\right)$ (27)

with the domain

$\displaystyle D(\pazocal{A})=\\{(u,v,\phi,\psi,z)^{T}\in H;\ \psi(x)=z(x,0),\
x\in(0,1)\\},$ (28)

where

$\displaystyle H=(H^{2}(0,1)\cap
H^{1}_{0}(0,1))\\!\times\\!H_{0}^{1}(0,1)\\!\times\\!(H^{2}(0,1)\cap
H_{0}^{1}(0,1))\\!\times\\!H_{0}^{1}(0,1)\\!\times\\!L^{2}(0,1;H^{1}(0,1)).$
(29)

We define the energy space $\mathcal{H}$ by

$\displaystyle\mathcal{H}:=H^{1}_{0}(0,1)\times L^{2}(0,1)\times
H_{0}^{1}(0,1)\times L^{2}(0,1)\times L^{2}(0,1;L^{2}(0,1)).$ (30)

For $U=(u,v,\phi,\psi,z)^{T}$,
$V=(\tilde{u},\tilde{v},\tilde{\phi},\tilde{\psi},\tilde{z})^{T}$ in
$\mathcal{H}$ and for $\eta$ a positive constant satisfying

$\displaystyle\tau\mu_{2}\leq\eta\leq\tau(2\mu_{1}-\mu_{2}),$ (31)

we equip $\mathcal{H}$ with the following inner product:

$\displaystyle\langle U,V\rangle_{\mathcal{H}}=$
$\displaystyle\int_{0}^{1}[\rho
v\tilde{v}+J\psi\tilde{\psi}+\delta\phi_{x}\tilde{\phi}_{x}+\mu
u_{x}\tilde{u}_{x}+b(u_{x}\tilde{\phi}+\tilde{u}_{x}\phi)+\xi\phi\tilde{\phi}]dx$
$\displaystyle\,\,+\eta\int_{0}^{1}\int_{0}^{1}z(x,y)\tilde{z}(x,y)dydx$ (32)

and so

$\displaystyle\|U\|^{2}_{\mathcal{H}}=\int_{0}^{1}[\rho
v^{2}+J\psi^{2}+\delta\phi_{x}^{2}+\mu
u_{x}^{2}+2bu_{x}\phi+\xi\phi^{2}]dx+\eta\int_{0}^{1}\int_{0}^{1}z^{2}(x,y)dydx.$
(33)

######  Remark 2.1.

Since $b^{2}\leq\mu\xi$ and

$\displaystyle\mu
u_{x}^{2}+2bu_{x}\phi+\xi\phi^{2}=\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}^{2}+\bigg{(}\mu-\frac{b^{2}}{\xi}\bigg{)}u_{x}^{2},$
(34)

it is easy to see that $\|U\|^{2}_{\mathcal{H}}\geq 0$ for all
$U\in\mathcal{H}$.

######  Definition 2.2.

Let $S(t)$ be a $C_{0}$-semigroup of contractions on $\mathcal{H}$. A
continuous function $U\,:\,(0,T]\rightarrow\mathcal{H}$ is a mild solution to
the problem (26) if it satisfies

$\displaystyle U(t)=S(t)U_{0}+\int_{0}^{t}S(t-s)f(U(s))ds,\,\,\,0<t\leq T,$
(35)

where $f$ is locally Lipschitz on $\mathcal{H}$.

The first main result regarding the existence and uniqueness of the global
solution to the system (26) and which will play an important role in the
result of the exponential decay is

######  Theorem 2.3.

Assume that (13)-(14) and $\mu_{2}\leq\mu_{1}$ hold, then we have the
following results.

(i)

If $U_{0}\in\mathcal{H}$, then problem (26) has a unique mild solution $U\in
C([0,\infty),\mathcal{H})$ with $U(0)=U_{0}$;

(ii)

If $U_{1}$ and $U_{2}$ are two mild solutions of problem (26), then there
exists a positive constant $C_{0}=C(U_{1}(0),U_{2}(0))$ such that

$\displaystyle\|U_{1}(t)-U_{2}(t)\|_{\mathcal{H}}\leq
e^{C_{0}T}\|U_{1}(0)-U_{2}(0)\|_{\mathcal{H}},\quad\forall 0\leq t\leq T.$
(36)

(iii)

If $U_{0}\in D(\pazocal{A})$, then the above mild solution is a strong
solution.

With the solution $U(t)=(u(t),u_{t}(t),\phi(t),\phi_{t}(t),z(t))$ of (17)-(25)
obtained by Theorem 2.3, we can define the energy associated with this
solution as follows:

$E(t)=\frac{1}{2}\int_{0}^{1}[\rho
u_{t}^{2}+J\phi_{t}^{2}+\delta\phi_{x}^{2}+\mu
u_{x}^{2}+2bu_{x}\phi+\xi\phi^{2}+2\hat{f}(\phi)]dx\\\
+\frac{\eta}{2}\int_{0}^{1}\int_{0}^{1}z^{2}(x,y,t)dydx.$ (37)

The second main result of this work, tells us about the energy behavior of the
solution over time and its statement is

######  Theorem 2.4.

Assume that (13)-(14) and $\mu_{2}<\mu_{1}$ hold. Assume that (12) also holds.
Then, with respect to mild solution, there exist $C>0$ and $\gamma>0$ such
that

$\displaystyle E(t)\leq Ce^{-\gamma t},\quad t\geq 0.$ (38)

##  3 Well-posedness of the problem

######  Lemma 3.1.

The operator $\pazocal{A}$ defined in (27) is the infinitesimal generator of a
$C_{0}$-semigroup of contractions on $\mathcal{H}$.

######  Proof 3.2.

Firstly it is not difficult to see that $D(\pazocal{A})$ is dense in
$\mathcal{H}$. In addition, it follows of definition of $\pazocal{A}$, (32)
and by integration by parts that for any $U=(u,v,\phi,\psi,z)\in
D(\pazocal{A})$

$\displaystyle\langle\pazocal{A}U,U\rangle_{\mathcal{H}}=\bigg{(}-\mu_{1}+\frac{\eta}{2\tau}\bigg{)}\int_{0}^{1}\psi^{2}dx-\mu_{2}\int_{0}^{1}z(x,1)\psi
dx-\frac{\eta}{2\tau}\int_{0}^{1}z^{2}(x,1)dx,$ (39)

and applying Hölder’s inequality we have

$\displaystyle\langle\pazocal{A}U,U\rangle_{\mathcal{H}}\leq\bigg{(}-\mu_{1}+\frac{\eta}{2\tau}+\frac{\mu_{2}}{2}\bigg{)}\int_{0}^{1}\psi^{2}dx+\bigg{(}-\frac{\eta}{2\tau}+\frac{\mu_{2}}{2}\bigg{)}\int_{0}^{1}z^{2}(x,1)dx.$
(40)

Keeping in mind the condition (31), we observe that

$\displaystyle-\mu_{1}+\frac{\eta}{2\tau}+\frac{\mu_{2}}{2}\leq
0\quad\text{and}\quad-\frac{\eta}{2\tau}+\frac{\mu_{2}}{2}\leq 0,$ (41)

with implies that the operator $\pazocal{A}$ is dissipative. We will now prove
that $0\in\varrho(\pazocal{A})$, where $\varrho(\pazocal{A})$ is resolvent set
of $\pazocal{A}$, that is, given
$V=(f_{1},f_{2},f_{3},f_{4},f_{5})\in\mathcal{H}$, we must find
$U=(u,v,\phi,\psi,z)\in D(\pazocal{A})$ solution of

$\displaystyle-\pazocal{A}U=V,\quad\text{with}\quad\|U\|_{\mathcal{H}}\leq
C\|V\|_{\mathcal{H}},$ (42)

where $C$ is a constant that is independent of $U$ and $V$. The previous
equality results in the following system:

$\displaystyle-v$ $\displaystyle=$ $\displaystyle f_{1},$ (43)
$\displaystyle-\mu u_{xx}-b\phi_{x}$ $\displaystyle=$ $\displaystyle\rho
f_{2},$ (44) $\displaystyle-\psi$ $\displaystyle=$ $\displaystyle f_{3},$ (45)
$\displaystyle-\delta\phi_{xx}+bu_{x}+\xi\phi+\mu_{1}\psi+\mu_{2}z(\cdot,1)$
$\displaystyle=$ $\displaystyle Jf_{4},$ (46) $\displaystyle z_{y}$
$\displaystyle=$ $\displaystyle-\tau f_{5}.$ (47)

By (43) and (45) we have, $v\in H_{0}^{1}(0,1)$ and $\psi\in H_{0}^{1}(0,1)$
and by (47)

$\displaystyle z(\cdot,y)$ $\displaystyle=$
$\displaystyle-\tau\int_{0}^{y}f_{5}(\cdot,s)ds-f_{3}(\cdot)\in
L^{2}(0,1;H^{1}(0,1)).$ (48)

Substituting (45) and (47) into (46) we obtain the system

$\displaystyle\begin{array}[]{rcl}-\mu u_{xx}-b\phi_{x}&=&F_{1}\in
L^{2}(0,1),\\\ -\delta\phi_{xx}+bu_{x}+\xi\phi&=&F_{2}\in
L^{2}(0,1),\end{array}$ (51)

where

$\displaystyle F_{1}$ $\displaystyle=$ $\displaystyle\rho f_{2},$
$\displaystyle F_{2}$ $\displaystyle=$
$\displaystyle(\mu_{1}+1)f_{3}+\mu_{2}\tau\int_{0}^{1}f_{5}(\cdot,s)ds+Jf_{4},$

with boundary conditions

$\displaystyle u(0)=u(1)=\phi(0)=\phi(1)=0.$ (52)

Solving system (51)-(52) is equivalent to finding $(u,\phi)\in
H_{0}^{1}(0,1)\times H_{0}^{1}(0,1)$ such that

$\displaystyle\int_{0}^{1}[\mu
u_{x}\bar{u}_{x}+b(u_{x}\bar{\phi}+\bar{u}_{x}\phi)+\xi\phi\bar{\phi}+\delta\phi_{x}\bar{\phi}_{x}]dx=\int_{0}^{1}F_{1}\bar{u}dx+\int_{0}^{1}F_{2}\bar{\phi}dx,$
(53)

for all $(\bar{u},\bar{\phi})\in H_{0}^{1}(0,1)\times H_{0}^{1}(0,1)$. Problem
(53) is equivalent to

$\displaystyle a((u,\phi),(\bar{u},\bar{\phi}))=h(\bar{u},\bar{\phi})$ (54)

where the bilinear form $a:(H_{0}^{1}(0,1)\times
H_{0}^{1}(0,1))^{2}\to\mathbb{R}$ and the linear form

$h:H_{0}^{1}(0,1)\times H_{0}^{1}(0,1)\to\mathbb{R}$

are given by

$\displaystyle a((u,\phi),(\bar{u},\bar{\phi}))=\int_{0}^{1}[\mu
u_{x}\bar{u}_{x}+b(u_{x}\bar{\phi}+\bar{u}_{x}\phi)+\xi\phi\bar{\phi}+\delta\phi_{x}\bar{\phi}_{x}]dx$
(55)

and

$\displaystyle
h(\bar{u},\bar{\phi})=\int_{0}^{1}F_{1}\bar{u}dx+\int_{0}^{1}F_{2}\bar{\phi}dx.$
(56)

The bilinear form $a$ is coercive because thanks to Young’s inequality we have

$\begin{array}[]{lll}a((u,\phi),(u,\phi))&=&\int_{0}^{1}[\mu
u_{x}^{2}+2bu_{x}\phi+\xi\phi^{2}+\delta\phi_{x}^{2}]dx\\\
&\geq&\mu\int_{0}^{1}u_{x}^{2}dx-\frac{b^{2}}{\xi}\int_{0}^{1}u_{x}^{2}dx\\\
&&-\xi\int_{0}^{1}\phi^{2}dx+\xi\int_{0}^{1}\phi^{2}dx+\delta\int_{0}^{1}\phi_{x}^{2}dx\\\
&\geq&\bigg{(}\frac{\mu\xi-b^{2}}{\xi}\bigg{)}\int_{0}^{1}u_{x}^{2}dx+\delta\int_{0}^{1}\phi_{x}^{2}dx\\\
&\geq&C_{1}(\|u_{x}\|^{2}+\|\phi_{x}\|^{2}),\end{array}$

where $C_{1}=\min\\{(\mu\xi-b^{2})/\xi,\delta\\}$. It is not difficult to
verify that $a$ and $h$ are continuous. So applying the Lax-Milgram’s Theorem,
we deduce that problem (51) admits a unique solution $(u,\phi)\in
H_{0}^{1}(0,1)\times H_{0}^{1}(0,1)$. Thanks to the classical elliptic
regularity (c.f. [NIRENBERG]) we obtain $(u,\phi)\in H^{2}(0,1)\times
H^{2}(0,1)$ and $C>0$. Therefore, $0\in\varrho(\pazocal{A})$. Consequently,
the result of the lemma follows from Theorem 1.2.4 in [LIU-ZHANG].

######  Lemma 3.3.

The operator $\pazocal{F}$ defined in (26) is locally Lipschitz in
$\mathcal{H}$.

######  Proof 3.4.

Let $U=(u,v,\phi,\psi,z)$ and
$V=(\tilde{u},\tilde{v},\tilde{\phi},\tilde{\psi},\tilde{z})$ be in
$\mathcal{H}$, with

$\displaystyle\|U\|_{\mathcal{H}},\|V\|_{\mathcal{H}}\leq M,\quad M>0.$ (57)

By using (13), Hölder’s inequality and the embedding
$H^{1}(0,1)\hookrightarrow L^{2(\theta+1)}(0,1)$ we obtain

$\displaystyle\|F(U)-F(V)\|^{2}_{\mathcal{H}}$
$\displaystyle=J^{-1}\int_{0}^{1}|f(\phi)-f(\tilde{\phi})|^{2}$
$\displaystyle\leq\
J^{-1}k_{0}^{2}\int_{0}^{1}(|\phi|^{\theta}+|\tilde{\phi}|^{\theta})^{2}|\phi-\tilde{\phi}|^{2}dx$
$\displaystyle\leq
C_{0}\bigg{(}\int_{0}^{1}(|\phi|^{\theta}+|\tilde{\phi}|^{\theta})^{\frac{2(\theta+1)}{\theta}}\bigg{)}^{\frac{\theta}{\theta+1}}\bigg{(}\int_{0}^{1}|\phi-\tilde{\phi}|^{2(\theta+1)}dx\bigg{)}^{\frac{1}{\theta+1}}$
$\displaystyle\leq
C_{0}(\|\phi\|_{2(\theta+1)}^{2\theta}+\|\tilde{\phi}\|_{2(\theta+1)}^{2\theta})\|\phi-\tilde{\phi}\|_{2(\theta+1)}^{2}$
$\displaystyle\leq
C_{0}(\|\phi\|^{2\theta}+\|\tilde{\phi}\|^{2\theta})\|\phi-\tilde{\phi}\|^{2}$
$\displaystyle\leq C_{0}(M)\|U-V\|_{\pazocal{H}}^{2},$

where $C_{0}$ is a positive constant depending on $M$.

Based on the energy defined in (37), we have the following result

######  Lemma 3.5.

The functional energy $E(t)$ defined in (37) of a solution

$U(t)=(u(t),u_{t}(t),\phi(t),\phi_{t}(t),z(t))$

of (17)-(25) is non-increasing function. More precisely, there exists a
constant $C_{E}>0$ such that for all $t\geq 0$

$\displaystyle\frac{dE}{dt}(t)\leq-
C_{E}\bigg{(}\int_{0}^{1}\phi_{t}^{2}dx+\int_{0}^{1}z^{2}(x,1,t)dx\bigg{)}\leq
0.$ (58)

######  Proof 3.6.

Multiplying (17) by $u_{t}$, (18) by $\phi_{t}$, integrating the result over
$[0,1]$ with respect to $x$ and using Young’s inequality, we obtain

$\displaystyle\begin{split}\frac{1}{2}\frac{d}{dt}\bigg{(}\int_{0}^{1}[\rho
u_{t}^{2}+J\phi_{t}^{2}+\delta\phi_{x}^{2}&+\mu
u_{x}^{2}+2bu_{x}\phi+\xi\phi^{2}+2\hat{f}(\phi)]dx\bigg{)}\\\
&=-\mu_{1}\int_{0}^{1}\phi_{t}^{2}dx-\mu_{2}\int_{0}^{1}z(x,1,t)\phi_{t}dx\\\
&\leq-\bigg{(}\mu_{1}-\frac{\mu_{2}}{2}\bigg{)}\int_{0}^{1}\phi_{t}^{2}dx+\frac{\mu_{2}}{2}\int_{0}^{1}z^{2}(x,1,t)dx.\end{split}$
(59)

Multiplying (19) by $\dfrac{\eta}{\tau}z$ and integrating the result over
$[0,1]\times[0,1]$ with respect $y$ and $x$, respectively we have

$\displaystyle\begin{split}\frac{\eta}{2}\frac{d}{dt}\int_{0}^{1}\int_{0}^{1}z^{2}(x,y,t)dydx&=-\frac{\eta}{2\tau}\int_{0}^{1}\int_{0}^{1}\frac{\partial}{\partial
y}z^{2}(x,y,t)dydx\\\
&=\frac{\eta}{2\tau}\int_{0}^{1}(z^{2}(x,0,t)-z^{2}(x,1,t))dx\\\
&=\frac{\eta}{2\tau}\int_{0}^{1}(\phi_{t}^{2}-z^{2}(x,1,t))dx,\end{split}$
(60)

adding (59) and (60) we obtain

$\displaystyle\frac{dE}{dt}(t)\leq-\bigg{(}\mu_{1}-\frac{\mu_{2}}{2}-\frac{\eta}{2\tau}\bigg{)}\int_{0}^{1}\phi_{t}^{2}dx-\bigg{(}\frac{\eta}{2\tau}-\frac{\mu_{2}}{2}\bigg{)}\int_{0}^{1}z^{2}(x,1,t)dx.$
(61)

Considering

$\displaystyle
C_{E}=\min\bigg{\\{}\mu_{1}-\frac{\mu_{2}}{2}-\frac{\eta}{2\tau},\frac{\eta}{2\tau}-\frac{\mu_{2}}{2}\bigg{\\}},$
(62)

we obtain (58).

######  Remark 3.7.

It follows from Theorem 3.5 that

$\displaystyle E(t)\leq E(0),\quad\forall t\geq 0.$ (63)

Furthermore, if $U(t)=(u(t),u_{t}(t),\phi(t),\phi_{t}(t),z(t))$ is solution of
(17)-(25), then it follows from (14), (33) and (37)

$\displaystyle E(t)$ $\displaystyle\geq\frac{1}{2}\bigg{\\{}\int_{0}^{1}[\rho
u_{t}^{2}+J\phi_{t}^{2}+\delta\phi_{x}^{2}+\mu
u_{x}^{2}+2bu_{x}\phi+\xi\phi^{2}dx+\eta\int_{0}^{1}\int_{0}^{1}z^{2}(x,y,t)dydx\bigg{\\}}$
(64) $\displaystyle\geq\frac{1}{2}\|U(t)\|_{\mathcal{H}}^{2}.$

######  Remark 3.8.

By Using Young’s and Poincaré’s inequalities in

$\displaystyle\frac{b^{2}}{\xi}\int_{0}^{1}u^{2}_{x}dx=\int_{0}^{1}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}^{2}dx-\int_{0}^{1}bu_{x}\phi
dx-\sqrt{\xi}\int_{0}^{1}\phi\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}dx$

we obtain,

$\displaystyle\int_{0}^{1}u^{2}_{x}dx\leq\frac{3\xi}{b^{2}}\int_{0}^{1}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}^{2}dx+\frac{2\xi^{2}c_{p}}{b^{2}}\int_{0}^{1}\phi_{x}^{2}dx.$
(65)

We are now able to prove the main result of this section which is:

######  Proof 3.9 (Proof of Theorem 2.3).

i) Since $\pazocal{A}$ generates a $C_{0}$-semigroup of contractions (Lemma
3.1) and $\pazocal{F}$ is locally Lipschitz (Lemma 3.3) follows from Pazy
[PAZY], Theorem 6.1.4 that (26) has a unique mild solution

$\displaystyle
U(t)=e^{\pazocal{A}t}U_{0}+\int_{0}^{t}e^{\pazocal{A}(t-s)}\pazocal{F}(U(s))ds,$
(66)

defined in $[0,t_{max})$, where $t_{max}$ depend on $U_{0}$. Moreover, if
$t_{max}<\infty$, then

$\displaystyle\lim_{t\to t_{\text{max}}}\|U(t)\|_{\mathcal{H}}=+\infty.$ (67)

However, from (63) and (64) we have

$\displaystyle\|U(t)\|_{\mathcal{H}}^{2}\leq 2E(0)\quad\text{for all}\quad
t\geq 0.$ (68)

Therefore, $t_{max}=+\infty$. ii) It is not difficult to obtain (36), just
consider a standard procedure found in Pazy [PAZY], Theorem 6.1.2, p. 184,
which consists in using the locally Lipschitz condition of $\pazocal{F}$ and
the Grönwall’s Inequality. iii) By using Theorem 6.1.5 in Pazy [PAZY], we know
that any mild solutions with initial data in $D(\pazocal{A})$ are strong
solution.

##  4 Exponential stability

In this section, we will prove the second main result of this work, Theorem
2.4, which will be divided into the following lemmas.

######  Lemma 4.1.

Let $U(t)=(u(t),u_{t}(t),\phi(t),\phi_{t}(t),z(t))$ be the solution to
(17)-(25). Then there is $\delta_{0}>0$ (depending only on the coefficients of
(17)-(19)) such that

$\begin{split}E(t)\\!\leq\\!\delta_{0}\bigg{\\{}\\!\\!\int_{0}^{1}\\!\\!\bigg{[}u_{t}^{2}+\phi_{t}^{2}+\phi_{x}^{2}+\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}^{2}\\!\\!+\hat{f}(\phi)\bigg{]}dx+\\!\int_{0}^{1}\\!\\!\int_{0}^{1}\\!\\!\\!z^{2}(x,y,t)dydx\bigg{\\}}.\end{split}$
(69)

######  Proof 4.2.

It is enough to consider (37), (34) and (65).

Let us now consider a solution $U(t)=(u(t),u_{t}(t),\phi(t),\phi_{t}(t),z(t))$
to (17)-(25) and for this solution we define the following functionals:

$\displaystyle I_{1}(t)$ $\displaystyle:=$ $\displaystyle-\int_{0}^{1}(\rho
uu_{t}+J\phi\phi_{t})dx-\frac{\mu_{1}}{2}\int_{0}^{1}\phi^{2}dx,$ (70)
$\displaystyle I_{2}(t)$ $\displaystyle:=$
$\displaystyle\int_{0}^{1}(J\phi_{t}\phi+\rho
u_{t}w)dx+\frac{\mu_{1}}{2}\int_{0}^{1}\phi^{2}dx,$ (71)

where $w$ is the solution of

$\displaystyle-w_{x}=\frac{b}{\mu}\phi\quad w(0)=w(1)=0,$ (72) $\displaystyle
I_{3}(t)$ $\displaystyle:=$ $\displaystyle
J\int_{0}^{1}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}\phi_{t}dx+\frac{bJ}{\sqrt{\xi}}\int_{0}^{1}\phi_{x}u_{t}dx,$
(73) $\displaystyle I_{4}(t)$ $\displaystyle:=$
$\displaystyle\int_{0}^{1}\int_{0}^{1}e^{-2\tau y}z^{2}(x,y,t)dydx.$ (74)

Thereby, we have the following results.

######  Lemma 4.3.

For any $\varepsilon>0$ the functional $I_{1}$ satisfies,

$\displaystyle\frac{d}{dt}I_{1}(t)\leq{}$ $\displaystyle{}-\int_{0}^{1}(\rho
u_{t}^{2}+J\phi_{t}^{2})dx+\bigg{[}\delta+\varepsilon+c_{1}+\frac{2\xi^{2}c_{p}}{b^{2}}\bigg{(}\mu-\frac{b^{2}}{\xi}\bigg{)}\bigg{]}\int_{0}^{1}\phi_{x}^{2}dx$
(75)
$\displaystyle{}+\bigg{[}3\bigg{(}\frac{\mu\xi}{b^{2}}-1\bigg{)}+1\bigg{]}\int_{0}^{1}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}^{2}dx+\frac{\mu_{2}^{2}c_{p}}{4\varepsilon}\int_{0}^{1}z^{2}(x,1,t)dx,$

where $c_{1}>0$ is a constant depending on Poincaré’s constant $c_{p}$.

######  Proof 4.4.

Taking the derivative of $I_{1}$, we have

$\displaystyle\frac{d}{dt}I_{1}(t)=-\int_{0}^{1}(\rho
u^{2}_{t}+J\phi_{t}^{2})dx-\int_{0}^{1}(\rho
u_{tt}u+J\phi_{tt}\phi)dx-\mu_{1}\int_{0}^{1}\phi\phi_{t}dx.$

Using (17), (18) and (34) we arrive at

$\begin{split}\frac{d}{dt}I_{1}(t)=-\int_{0}^{1}(\rho
u^{2}_{t}+J\phi_{t}^{2})dx+\delta\int_{0}^{1}\phi_{x}^{2}dx+\int_{0}^{1}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}^{2}dx\\\
+\bigg{(}\mu-\frac{b^{2}}{\xi}\bigg{)}\int_{0}^{1}u_{x}^{2}dx+\mu_{2}\int_{0}^{1}z(x,1,t)\phi
dx+\int_{0}^{1}f(\phi)\phi dx.\end{split}$ (76)

It follows from Young’s and Poincaré’s inequalities that for any
$\varepsilon>0$,

$\displaystyle\mu_{2}\int_{0}^{1}z(x,1,t)\phi dx{}$
$\displaystyle{}\leq\frac{\varepsilon}{c_{p}}\int_{0}^{1}\phi^{2}dx+\frac{\mu_{2}^{2}c_{p}}{4\varepsilon}\int_{0}^{1}z^{2}(x,1,t)dx$
(77)
$\displaystyle{}\leq\varepsilon\int_{0}^{1}\phi_{x}^{2}dx+\frac{\mu_{2}^{2}c_{p}}{4\varepsilon}\int_{0}^{1}z^{2}(x,1,t)dx,$
$\displaystyle\int_{0}^{1}f(\phi)\phi dx\leq
k_{0}\int_{0}^{1}|\phi|^{\theta}|\phi||\phi|dx{}$
$\displaystyle{}\leq\|\phi\|_{2(\theta+1)}^{\theta}\|\phi\|_{2(\theta+1)}\|\phi\|_{2}\leq
c_{1}\int_{0}^{1}\phi_{x}^{2}dx.$ (78)

Combining (76), (77), (78) and (65) we obtain (75).

######  Lemma 4.5.

For any $\lambda_{2},\tilde{\lambda}_{2}>0$, the functional $I_{2}$ satisfies,

$\begin{split}\frac{d}{dt}I_{2}(t)\leq(\mu_{2}\lambda_{2}-\delta)\int_{0}^{1}\phi_{x}^{2}dx+\bigg{(}J+\frac{\rho
b^{2}}{4\mu^{2}\tilde{\lambda}_{2}}\bigg{)}\int_{0}^{1}\phi_{t}^{2}dx+\rho\tilde{\lambda}_{2}c_{p}\int_{0}^{1}u_{t}^{2}dx\\\
+\frac{\mu_{2}^{2}c_{p}}{4\lambda_{2}}\int_{0}^{1}z^{2}(x,1,t)dx-\int_{0}^{1}\hat{f}(\phi)dx.\end{split}$
(79)

######  Proof 4.6.

Taking derivative of $I_{2}$, considering (17), (18) and integrating by parts,
we obtain

$\begin{split}\frac{d}{dt}I_{2}(t)=-\delta\int_{0}^{1}\phi_{x}^{2}dx+J\int_{0}^{1}\phi_{t}^{2}dx-\xi\int_{0}^{1}\phi^{2}dx+\rho\int_{0}^{1}u_{t}w_{t}dx\\\
+\mu\int_{0}^{1}w_{x}^{2}dx-\mu_{2}\int_{0}^{1}z(x,1,t)\phi
dx-\int_{0}^{1}f(\phi)\phi dx.\end{split}$ (80)

By (72) we can get

$\displaystyle\mu\int_{0}^{1}w_{x}^{2}dx\leq\frac{b^{2}}{\mu}\int_{0}^{1}\phi^{2}dx\leq\xi
c_{p}\int_{0}^{1}\phi_{x}^{2}dx,$ (81)
$\displaystyle\int_{0}^{1}w_{t}^{2}dx\leq
c_{p}\int_{0}^{1}w_{xt}^{2}dx\leq\frac{c_{p}b^{2}}{\mu^{2}}\int_{0}^{1}\phi_{t}^{2}dx.$
(82)

By using Young’s and Poincaré’s inequalities and (82), we have

$\displaystyle\mu_{2}\int_{0}^{1}z(x,1,t)\phi dx$
$\displaystyle\leq\frac{\mu_{2}\lambda_{2}}{c_{p}}\int_{0}^{1}\phi^{2}dx+\frac{\mu_{2}c_{p}}{4\lambda_{2}}\int_{0}^{1}z^{2}(x,1,t)dx$
(83)
$\displaystyle\leq\mu_{2}\lambda_{2}\int_{0}^{1}\phi_{x}^{2}dx+\frac{\mu_{2}c_{p}}{4\lambda_{2}}\int_{0}^{1}z^{2}(x,1,t)dx,$
$\displaystyle\rho\int_{0}^{1}u_{t}w_{t}dx$
$\displaystyle\leq\rho\tilde{\lambda}_{2}c_{p}\int_{0}^{1}u_{t}^{2}dx+\frac{\rho}{4\tilde{\lambda}_{2}c_{p}}\int_{0}^{1}w_{t}^{2}dx$
(84)
$\displaystyle\leq\rho\tilde{\lambda}_{2}c_{p}\int_{0}^{1}u_{t}^{2}dx+\frac{\rho
b^{2}}{4\mu^{2}\tilde{\lambda}_{2}}\int_{0}^{1}\phi_{t}^{2}dx.$

From (80)-(84) we obtain (79).

######  Lemma 4.7.

Assume that (12) holds. Then the functional $I_{3}$ satisfies

$\frac{d}{dt}I_{3}(t)\leq\delta\big{[}\phi_{x}u_{x}\big{]}_{x=0}^{x=1}-\frac{3\sqrt{\xi}}{8}\int_{0}^{1}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}^{2}dx+\bigg{(}J\sqrt{\xi}+\frac{\mu_{1}^{2}}{\sqrt{\xi}}\bigg{)}\int_{0}^{1}\phi_{t}^{2}dx\\\
+\bigg{(}\frac{\xi^{2}c_{p}}{12\sqrt{\xi}}+\frac{6k_{0}^{2}}{\sqrt{\xi}}\bigg{)}\int_{0}^{1}\phi_{x}^{2}dx+\frac{\mu_{2}^{2}}{\sqrt{\xi}}\int_{0}^{1}z^{2}(x,1,t)dx-\sqrt{\xi}\int_{0}^{1}\hat{f}(\phi)dx.$
(85)

######  Proof 4.8.

Notice that

$\displaystyle\frac{d}{dt}I_{3}(t)=$ $\displaystyle
J\int_{0}^{1}\\!\\!\\!\phi_{tt}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}dx+J\int_{0}^{1}\\!\\!\\!\phi_{t}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}_{t}dx$
$\displaystyle\,\,\,+\frac{bJ}{\sqrt{\xi}}\int_{0}^{1}\\!\\!\\!\phi_{xt}u_{t}dx+\frac{bJ}{\sqrt{\xi}}\int_{0}^{1}\\!\\!\\!\phi_{x}u_{tt}dx.$
(86)

By using (17) and (18) and integration by parts, we have

$\frac{d}{dt}I_{3}(t)=\delta[\phi_{x}u_{x}]_{x=0}^{x=1}-\sqrt{\xi}\int_{0}^{1}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}^{2}dx-\\\
\mu_{1}\int_{0}^{1}\phi_{t}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}dx-\mu_{2}\int_{0}^{1}z(x,1,t)\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}dx\\\
-\frac{b}{\sqrt{\xi}}\int_{0}^{1}f(\phi)u_{x}dx-\sqrt{\xi}\int_{0}^{1}f(\phi)\phi
dx+J\sqrt{\xi}\int_{0}^{1}\phi_{t}^{2}dx.$ (87)

By using Young’s and Poincaré’s inequality, we have

$\begin{array}[]{c}-\mu_{1}\int_{0}^{1}\phi_{t}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}dx\\\
\leq\frac{\sqrt{\xi}}{4}\int_{0}^{1}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}^{2}dx+\frac{\mu_{1}^{2}}{\sqrt{\xi}}\int_{0}^{1}\phi_{t}^{2}dx,\end{array}$
(88)
$\begin{array}[]{c}-\mu_{2}\int_{0}^{1}z(x,1,t)\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}dx\\\
\leq\frac{\sqrt{\xi}}{4}\int_{0}^{1}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}^{2}dx+\frac{\mu_{2}^{2}}{\sqrt{\xi}}\int_{0}^{1}z^{2}(x,1,t)dx,\end{array}$
(89)

and by using (65), for any $\varepsilon>0$

$\begin{array}[]{c}-\frac{b}{\sqrt{\xi}}\int_{0}^{1}f(\phi)u_{x}dx\leq\frac{k_{0}|b|}{\sqrt{\xi}}\|u_{x}\|_{2}\|\phi\|_{2(\theta+1)}^{\theta}\|\phi\|_{2(\theta+1)}\\\
\leq\frac{k_{0}b^{2}\varepsilon}{\sqrt{\xi}}\int_{0}^{1}u_{x}^{2}dx+\frac{k_{0}}{4\sqrt{\xi}\varepsilon}\int_{0}^{1}\phi_{x}^{2}dx\\\
\leq 3\varepsilon
k_{0}\sqrt{\xi}\int_{0}^{1}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}^{2}dx+\bigg{(}\frac{2k_{0}\xi^{2}c_{p}\varepsilon}{\sqrt{\xi}}+\frac{k_{0}}{4\varepsilon\sqrt{\xi}}\bigg{)}\int_{0}^{1}\phi_{x}^{2}dx.\end{array}$

Taking $\varepsilon=1/24k_{0}$ in the previous inequality, we obtain

$\displaystyle-\frac{b}{\sqrt{\xi}}\int_{0}^{1}f(\phi)u_{x}dx\leq\frac{\sqrt{\xi}}{8}\int_{0}^{1}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}^{2}dx+\bigg{(}\frac{\xi^{2}c_{p}}{12\sqrt{\xi}}+\frac{6k_{0}^{2}}{\sqrt{\xi}}\bigg{)}\int_{0}^{1}\phi_{x}^{2}dx.$
(90)

Combining (87), (88), (89) and (90) we obtain (85).

Next we deal with the boundary term in (85). We define the function

$\displaystyle q(x)=2-4x,\quad x\in[0,1],$ (91)

so we have the following result.

######  Lemma 4.9.

For any $\varepsilon>0$, we have

$\delta[u_{x}\phi_{x}]_{x=0}^{x=1}\leq-\frac{J\delta}{4\varepsilon}\frac{d}{dt}\int_{0}^{1}q\phi_{t}\phi_{x}dx-\frac{\rho\varepsilon}{\mu}\frac{d}{dt}\int_{0}^{1}qu_{t}u_{x}dx+\frac{2\rho\varepsilon}{\mu}\int_{0}^{1}u_{t}^{2}dx\\\
+\bigg{(}\frac{\delta^{2}}{2\varepsilon}+\frac{\delta^{2}}{4\varepsilon^{3}}+\frac{\delta^{2}}{2}+\frac{\varepsilon}{\mu}+\frac{2\xi^{2}c_{p}\varepsilon}{\mu}+\frac{4\xi^{2}c_{p}\varepsilon}{b^{2}}\bigg{)}\int_{0}^{1}\phi_{x}^{2}dx+\bigg{(}\frac{J\delta}{2\varepsilon}+\frac{\mu_{1}^{2}}{4\varepsilon^{2}}\bigg{)}\int_{0}^{1}\phi_{t}^{2}dx\\\
+\bigg{(}\frac{\xi\varepsilon}{4}+\frac{3\xi\varepsilon}{\mu}+\frac{6\xi\varepsilon}{b^{2}}\bigg{)}\int_{0}^{1}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}^{2}dx+\frac{\mu_{2}^{2}}{4\varepsilon^{2}}\int_{0}^{1}z^{2}(x,1,t)dx.$
(92)

######  Proof 4.10.

By using Young’s inequality we have

$\displaystyle\delta[u_{x}\phi_{x}]_{x=0}^{x=1}\leq\varepsilon[u_{x}^{2}(1)+u_{x}^{2}(0)]+\frac{\delta^{2}}{4\varepsilon}[\phi_{x}^{2}(1)+\phi_{x}^{2}(0)].$
(93)

Now, by using (18) and integrating by parts, we have

$\frac{d}{dt}\int_{0}^{1}J\delta q\phi_{t}\phi_{x}dx=\\\
-\delta^{2}[\phi_{x}^{2}(1)+\phi_{x}^{2}(0)]+2\delta^{2}\int_{0}^{1}\phi_{x}^{2}dx-\delta\sqrt{\xi}\int_{0}^{1}q\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}\phi_{x}dx\\\
-\delta\mu_{1}\int_{0}^{1}q\phi_{t}\phi_{x}dx-\delta\mu_{2}\int_{0}^{1}qz(x,1,t)\phi_{x}dx-J\delta\int_{0}^{1}qf(\phi)\phi_{x}dx+2J\delta\int_{0}^{1}\phi_{t}^{2}dx.$
(94)

Using Young’s inequality

$\displaystyle\delta\sqrt{\xi}\int_{0}^{1}q\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}\phi_{x}dx\leq\xi\varepsilon^{2}\int_{0}^{1}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}^{2}dx+\frac{\delta^{2}}{\varepsilon^{2}}\int_{0}^{1}\phi_{x}^{2}dx,$
(95)
$\displaystyle\delta\mu_{1}\int_{0}^{1}q\phi_{t}\phi_{x}dx\leq\frac{\mu_{1}^{2}}{\varepsilon}\int_{0}^{1}\phi_{t}^{2}dx+\varepsilon\delta^{2}\int_{0}^{1}\phi_{x}^{2}dx,$
(96)
$\displaystyle\delta\mu_{2}\int_{0}^{1}qz(x,1,t)\phi_{x}dx\leq\frac{\mu_{2}^{2}}{\varepsilon}\int_{0}^{1}z^{2}(x,1,t)dx+\varepsilon\delta^{2}\int_{0}^{1}\phi_{x}^{2}dx.$
(97)

Combining (94), (95), (96), (97) and (14) we have

$\frac{d}{dt}\int_{0}^{1}J\delta
q\phi_{t}\phi_{x}dx\leq-\delta^{2}[\phi_{x}^{2}(1)+\phi_{x}^{2}(0)]+\bigg{(}2\delta^{2}+\frac{\delta^{2}}{\varepsilon^{2}}+2\varepsilon\delta^{2}\bigg{)}\int_{0}^{1}\phi_{x}^{2}dx\\\
+\xi\varepsilon^{2}\int_{0}^{1}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}^{2}dx+\bigg{(}2J\delta+\frac{\mu_{1}^{2}}{\varepsilon}\bigg{)}\int_{0}^{1}\phi_{t}^{2}dx+\frac{\mu_{2}^{2}}{\varepsilon}\int_{0}^{1}z^{2}(x,1,t)dx.$
(98)

On the other hand, by using (17) and integration by parts, we have

$\displaystyle\frac{d}{dt}\int_{0}^{1}\rho
qu_{t}u_{x}dx=-\mu[u_{x}^{2}(1)+u_{x}^{2}(0)]+2\mu\int_{0}^{1}u_{x}^{2}dx+b\int_{0}^{1}qu_{x}\phi_{x}dx+2\rho\int_{0}^{1}u_{t}^{2}dx$
(99)

and applying Young’s inequality and (65) in the previous equality

$\frac{d}{dt}\int_{0}^{1}\rho
qu_{t}u_{x}dx\leq-\mu[u_{x}^{2}(1)+u_{x}^{2}(0)]+\bigg{(}3\xi+\frac{6\mu\xi}{b^{2}}\bigg{)}\int_{0}^{1}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}^{2}dx\\\
+\bigg{(}1+2\xi^{2}c_{p}+\frac{4\mu\xi^{2}c_{p}}{b^{2}}\bigg{)}\int_{0}^{1}\phi_{x}^{2}dx+2\rho\int_{0}^{1}u_{t}^{2}dx.$
(100)

Combining (98) and (100) we obtain (92).

Following the same line of argument given found in [HOUARI-LASKRI] it is not
difficult to obtain

######  Lemma 4.11.

The functional $I_{4}$ satisfies

$\displaystyle\frac{d}{dt}I_{4}(t)=-2I_{4}(t)-\frac{e^{2\tau}}{\tau}\int_{0}^{1}z^{2}(x,1,t)dx+\frac{1}{\tau}\int_{0}^{1}\phi_{t}^{2}dx.$
(101)

Now we define the following Lyapunov functional $\mathcal{L}(t)$

$\mathcal{L}(t):=ME(t)+\frac{\sqrt{\xi}}{8}\beta
I_{1}(t)+NI_{2}(t)+I_{3}(t)+\frac{J\delta}{4\varepsilon}\int_{0}^{1}q\phi_{t}\phi_{x}dx+\frac{\rho\varepsilon}{\mu}\int_{0}^{1}qu_{t}u_{x}dx+I_{4}(t)$
(102)

where

$\displaystyle\beta:=\bigg{[}3\bigg{(}\frac{\mu\xi}{b^{2}}-1\bigg{)}+1\bigg{]}^{-1}.$
(103)

######  Lemma 4.12.

Let $U(t)=(u(t),u_{t}(t),\phi(t),\phi_{t}(t),z(t))$ be the solution of problem
(17)-(25). For $M$ large enough, there exist two positives $\gamma_{1}$ and
$\gamma_{2}$ depending on $M$, $N$ and $\varepsilon$ such that for any $t\geq
0$,

$\displaystyle\gamma_{1}E(t)\leq\mathcal{L}(t)\leq\gamma_{2}E(t).$ (104)

######  Proof 4.13.

We have

$\displaystyle|\mathcal{L}(t)-ME(t)|\leq{}$
$\displaystyle{}\frac{\sqrt{\xi}}{8}\beta\bigg{|}-\int_{0}^{1}(\rho
uu_{t}+J\phi\phi_{t})dx-\frac{\mu_{1}}{2}\int_{0}^{1}\phi^{2}dx\bigg{|}$ (105)
$\displaystyle{}+N\bigg{|}\int_{0}^{1}(J\phi_{t}\phi+\rho
u_{t}w)dx+\frac{\mu_{1}}{2}\int_{0}^{1}\phi^{2}dx\bigg{|}$
$\displaystyle{}+\bigg{|}J\int_{0}^{1}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}\phi_{t}dx+\frac{bJ}{\sqrt{\xi}}\int_{0}^{1}\phi_{x}u_{t}dx\bigg{|}$
$\displaystyle{}+\frac{J\delta}{4\varepsilon}\int_{0}^{1}|q\phi_{t}\phi_{x}|dx+\frac{\rho\varepsilon}{\mu}\int_{0}^{1}|qu_{t}u_{x}|dx$
$\displaystyle{}+\bigg{|}\int_{0}^{1}\int_{0}^{1}e^{-2\tau
y}z^{2}(x,y,t)dydx\bigg{|}.$

By using Young’s and Poincaré’s inequalities, (65) and (81) we obtain:
$\displaystyle\frac{\sqrt{\xi}}{8}\beta\bigg{|}-\int_{0}^{1}(\rho
uu_{t}+J\phi\phi_{t})dx-\frac{\mu_{1}}{2}\int_{0}^{1}\phi^{2}dx\bigg{|}\leq{}$
$\displaystyle{}\frac{\sqrt{\xi}\beta\rho}{16}\int_{0}^{1}u_{t}^{2}dx+\frac{\sqrt{\xi}\beta
J}{16}\int_{0}^{1}\phi_{t}^{2}dx$ (106)
$\displaystyle+\bigg{[}\frac{\sqrt{\xi}\beta
c_{p}}{16}(J+\mu)+\frac{\xi^{5/2}\beta\rho
c_{p}^{2}}{8b^{2}}\bigg{]}\int_{0}^{1}\phi_{x}^{2}dx$
$\displaystyle+\frac{3\xi^{3/2}\beta\rho
c_{p}}{16b^{2}}\int_{0}^{1}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}^{2}dx,$
$\displaystyle N\bigg{|}\int_{0}^{1}(J\phi_{t}\phi+\rho
u_{t}w)dx+\frac{\mu_{1}}{2}\int_{0}^{1}\phi^{2}dx\bigg{|}\leq{}$
$\displaystyle{}\frac{N\rho}{2}\int_{0}^{1}u_{t}^{2}dx+\frac{NJ}{2}\int_{0}^{1}\phi_{t}^{2}dx$
(107) $\displaystyle+\frac{Nc_{p}}{2}\bigg{(}J+\mu_{1}+\frac{\rho\xi
c_{p}}{\mu}\bigg{)}\int_{0}^{1}\phi_{x}^{2}dx,$
$\displaystyle\bigg{|}J\int_{0}^{1}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}\phi_{t}dx+\frac{bJ}{\sqrt{\xi}}\int_{0}^{1}\phi_{x}u_{t}dx\bigg{|}\leq{}$
$\displaystyle{}\frac{Jb^{2}}{2}\int_{0}^{1}u_{t}^{2}dx+\frac{J}{2}\int_{0}^{1}\phi_{t}^{2}dx$
(108)
$\displaystyle+\frac{J}{2}\int_{0}^{1}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}^{2}dx$
$\displaystyle+\frac{J}{2\xi}\int_{0}^{1}\phi_{x}^{2}dx,$
$\displaystyle\frac{J\delta}{4\varepsilon}\int_{0}^{1}|q\phi_{t}\phi_{x}|dx\leq{}$
$\displaystyle{}\frac{J\delta}{4\varepsilon}\int_{0}^{1}\phi_{t}^{2}dx+\frac{J\delta}{4\varepsilon}\int_{0}^{1}\phi_{x}^{2}dx,$
(109)
$\displaystyle\frac{\rho\varepsilon}{\mu}\int_{0}^{1}|qu_{t}u_{x}|dx\leq{}$
$\displaystyle{}\frac{\rho\varepsilon}{\mu}\int_{0}^{1}u_{t}^{2}dx+\frac{2\rho\xi^{2}c_{p}\varepsilon}{\mu
b^{2}}\int_{0}^{1}\phi_{x}^{2}dx$ (110)
$\displaystyle+\frac{3\rho\xi\varepsilon}{\mu
b^{2}}\int_{0}^{1}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}^{2}dx.$
From (105)-(110), we have

$|\mathcal{L}(t)-ME(t)|\leq\alpha_{1}\frac{\rho}{2}\int_{0}^{1}u_{t}^{2}dx+\alpha_{2}\frac{J}{2}\int_{0}^{1}\phi_{t}^{2}dx+\alpha_{3}\frac{\delta}{2}\int_{0}^{1}\phi_{x}^{2}dx\\\
+\alpha_{4}\frac{1}{2}\int_{0}^{1}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}^{2}dx+\alpha_{5}\frac{\eta}{2}\int_{0}^{1}\int_{0}^{1}z^{2}(x,y,t)dydx$
(111)

where

$\displaystyle\alpha_{1}$
$\displaystyle=\frac{\sqrt{\xi}\beta}{8}+N+\frac{Jb^{2}}{\rho}+\frac{2\varepsilon}{\mu},$
(112) $\displaystyle\alpha_{2}$
$\displaystyle=\frac{\sqrt{\xi}\beta}{8}+N+1+\frac{\delta}{2\varepsilon},$
(113) $\displaystyle\alpha_{3}$ $\displaystyle=\frac{\sqrt{\xi}\beta
c_{p}(J+\mu)}{8\delta}+\frac{\xi^{5/2}\beta\rho c_{p}^{2}}{4\delta
b^{2}}+\frac{Nc_{p}}{\delta}\bigg{(}J+\mu_{1}+\frac{\rho\xi
c_{p}}{\mu}\bigg{)}+\frac{J}{\delta}+\frac{J}{2\varepsilon}+\frac{4\rho\xi^{2}c_{p}\varepsilon}{\mu\delta
b^{2}},$ (114) $\displaystyle\alpha_{4}$
$\displaystyle=\frac{3\xi^{3/2}\beta\rho
c_{p}}{8b^{2}}+J+\frac{6\rho\xi\varepsilon}{\mu b^{2}},$ (115)
$\displaystyle\alpha_{5}$ $\displaystyle=\frac{2}{\eta}.$ (116)

By considering

$\displaystyle\tilde{C}=\max\\{\alpha_{i};\ i=1,\ldots,5\\}$ (117)

we arrive

$\displaystyle|\mathcal{L}(t)-ME(t)|\leq\tilde{C}E(t).$ (118)

So, we can choose $M$ large enough so that $\gamma_{1}=M-\tilde{C}>0$ and
$\gamma_{2}=M+\tilde{C}>0$ such that (104) holds.

We are ready to prove the main result of this paper.

######  Proof 4.14 (Proof of Theorem 2.4).

It follows from (58), (75), (79), (85), (92), and (101)

$\displaystyle\frac{d}{dt}\mathcal{L}(t)$
$\displaystyle\leq\bigg{[}-\frac{\rho\sqrt{\xi}\beta}{8}+N\rho\tilde{\lambda}_{2}c_{p}+\frac{2\rho\varepsilon}{\mu}\bigg{]}\int_{0}^{1}u_{t}^{2}dx$
$\displaystyle+\bigg{[}\\!-\\!MC_{E}\\!-\\!\frac{\sqrt{\xi}\beta
J}{8}\\!+\\!N\bigg{(}J+\frac{\rho
b^{2}}{4\mu^{2}\tilde{\lambda}_{2}}\bigg{)}\\!+\\!\bigg{(}J\sqrt{\xi}\\!+\\!\frac{\mu_{1}^{2}}{\sqrt{\xi}}\bigg{)}\\!+\\!\bigg{(}\frac{J\delta}{2\varepsilon}\\!+\\!\frac{\mu_{1}^{2}}{4\varepsilon^{2}}\bigg{)}\\!+\\!\frac{1}{\tau}\bigg{]}\int_{0}^{1}\\!\\!\phi_{t}^{2}dx$
$\displaystyle+\frac{\sqrt{\xi}\beta}{8}\bigg{[}\delta+\varepsilon+c_{1}+\frac{2\xi^{2}c_{p}}{b^{2}}\bigg{(}\mu-\frac{b^{2}}{\xi}\bigg{)}\bigg{]}\int_{0}^{1}\phi_{x}^{2}dx$
$\displaystyle+\bigg{[}N(\mu_{2}\lambda_{2}-\delta)+\frac{\xi^{2}c_{p}}{12\sqrt{\xi}}+\frac{6k_{0}}{\sqrt{\xi}}+\frac{\delta^{2}}{2\varepsilon}+\frac{\delta^{2}}{4\varepsilon^{2}}+\frac{\delta^{2}}{2}+\frac{\xi}{\mu}+\frac{2\xi^{2}c_{p}\varepsilon}{\mu}+\frac{4\xi^{2}c_{p}\varepsilon}{b^{2}}\bigg{]}\int_{0}^{1}\phi_{x}^{2}dx$
$\displaystyle+\bigg{[}-\frac{\sqrt{\xi}}{4}+\frac{\xi\varepsilon}{4}+\frac{3\xi\varepsilon}{\mu}+\frac{6\xi\varepsilon}{b^{2}}\bigg{]}\int_{0}^{1}\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}^{2}dx$
$\displaystyle+\bigg{[}-MC_{E}+\frac{\mu_{2}^{2}c_{p}}{32\varepsilon}+\frac{N\mu_{2}^{2}c_{p}}{4\lambda_{2}}+\frac{\mu_{2}^{2}}{\sqrt{\xi}}+\frac{\mu_{2}^{2}}{4\varepsilon^{2}}-\frac{e^{2\tau}}{\tau}\bigg{]}\int_{0}^{1}z^{2}(x,1,t)dx$
$\displaystyle-(N+\sqrt{\xi})\int_{0}^{1}\hat{f}(\phi)dx-e^{-2\tau}\int_{0}^{1}\int_{0}^{1}z^{2}(x,y,t)dydx.$
(120)

At this point, we have to choose our constants very carefully. First, let us
choose $\lambda_{2}$ small enough such that

$\displaystyle\lambda_{2}<\frac{\delta}{\mu_{2}}.$ (121)

Then, take $\varepsilon$ small enough such that

$\displaystyle\varepsilon\leq\min\bigg{\\{}\frac{\sqrt{\xi}}{8}\bigg{/}\bigg{(}\frac{\xi}{4}+\frac{3\xi}{\mu}+\frac{6\xi}{b^{2}}\bigg{)},\frac{\sqrt{\xi}\mu\beta}{64}\bigg{\\}}.$
(122)

Then, we select $N$ large enough so that

$\displaystyle N(\delta-\mu_{2}\lambda_{2})>$
$\displaystyle\frac{\sqrt{\xi}\beta}{8}\bigg{[}\delta+\varepsilon+c_{1}+\frac{2\xi^{2}c_{p}}{b^{2}}\bigg{(}\mu-\frac{b^{2}}{\xi}\bigg{)}\bigg{]}$
$\displaystyle\,\,+\frac{\xi^{2}c_{p}}{12\sqrt{\xi}}+\frac{6k_{0}}{\sqrt{\xi}}+\frac{\delta^{2}}{2\varepsilon}+\frac{\delta^{2}}{4\varepsilon^{2}}+\frac{\delta^{2}}{2}+\frac{\xi}{\mu}+\frac{2\xi^{2}c_{p}\varepsilon}{\mu}+\frac{4\xi^{2}c_{p}\varepsilon}{b^{2}}.$

Now, we pick $\tilde{\lambda}_{2}$ so small that

$\displaystyle\tilde{\lambda}_{2}<\frac{\sqrt{\xi}\beta}{32Nc_{p}}.$

Finally, we choose $M$ large enough so that, there exists a positive constant
$\beta_{1}>0$, such that (120) becomes

$\displaystyle\frac{d}{dt}\mathcal{L}(t)\leq$
$\displaystyle-\beta_{1}\int_{0}^{1}\bigg{[}u_{t}^{2}+\phi_{t}^{2}+\phi_{x}^{2}+\bigg{(}\frac{b}{\sqrt{\xi}}u_{x}+\sqrt{\xi}\phi\bigg{)}^{2}+z^{2}(x,1,t)+\hat{f}(\phi)\bigg{]}dx$
$\displaystyle-\beta_{1}\int_{0}^{1}\int_{0}^{1}z^{2}(x,y,t)dydx$ (123)

which implies that there exists also $\beta_{2}>0$, such that

$\displaystyle\frac{d}{dt}\mathcal{L}(t)\leq-\beta_{2}E(t),\quad\forall t\geq
0,$ (124)

and from (104)

$\displaystyle\frac{d}{dt}\mathcal{L}(t)\leq-\frac{\beta_{2}}{\gamma_{1}}\mathcal{L}(t),\quad\forall
t\geq 0.$ (125)

Integrating (126) and using again (104) we arrive

$\displaystyle
E(t)\leq-\frac{\gamma_{2}}{\gamma_{1}}E(0)e^{-(\beta_{2}/\gamma_{2})t},\quad\forall
t\geq 0,$ (126)

which gives us that the exponential stability holds for any $U_{0}\in D(A)$.
Noting that $D(A)$ is dense $\mathcal{H}$, we can extend the energy
inequalities to phase space $\mathcal{H}$. Thus we complete the proof of the
theorem.

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June 07,2019April 13, 2022Durvudkhan Suragan


# Framework for Certification of AI-Based Systems

Maxime Gariel  Xwing, Inc.
San Francisco, CA USA
<EMAIL_ADDRESS>Brian Shimanuki  Xwing, Inc.
San Francisco, CA USA
<EMAIL_ADDRESS>Rob Timpe  Xwing, Inc.
San Francisco, CA USA
<EMAIL_ADDRESS>Evan Wilson  Xwing, Inc.
San Francisco, CA USA
<EMAIL_ADDRESS>
###### Abstract

The current certification process for aerospace software is not adapted to
“AI-based” algorithms such as deep neural networks. Unlike traditional
aerospace software, the precise parameters optimized during neural network
training are as important as (or more than) the code processing the network
and they are not directly mathematically understandable. Despite their lack of
explainability such algorithms are appealing because for some applications
they can exhibit high performance unattainable with any traditional explicit
line-by-line software methods.

This paper proposes a framework and principles that could be used to establish
certification methods for neural network models for which the current
certification processes such as DO-178 cannot be applied. While it is not a
magic recipe, it is a set of common sense steps that will allow the applicant
and the regulator increase their confidence in the developed software, by
demonstrating the capabilities to bring together, trace, and track the
requirements, data, software, training process, and test results.

##  I Introduction

Recent technologies including deep neural networks and graphics cards have the
potential to enable the automation of functions previously impossible due to
algorithmic and computational resource constraints. A common neural networking
application is image processing, where cameras have the potential to
drastically reduce the size, weight and power of some sensor suites while
providing additional functionalities. These benefits arise because a single
sensor can be used for multiple purposes (e.g. using cameras for airborne
intruder detection or lane tracking while taxiing).

Before use on a certified aircraft, all software must undergo a certification
effort. The goal of the software certification process is to ensure that the
software follows a defined set of requirements and that it does not fail /
crash / return error during its execution. Traditionally, this certification
process involves linking individual requirements to specific lines of code to
check that all requirements are fulfilled and extraneous code is not present.
The performance characteristics of the software are evaluated separately.

For deep neural network (DNN), the precise parameters resulting from training
are as important as (or more important than) the code that drives it.
Additionally, requirements cannot be tied to specific lines of code because
the neural networks functionality is encoded in the large parameter set. We
therefore need to provide tools that can track the trained network and the
validation data along with the algorithm. Our framework will provide
capabilities to bring together, trace, and track the requirements, data,
software, training process, and test results.

While AI will have many applications in aerospace, we use the example of
“vision-based detect-and-avoid” to illustrate the proposed framework in this
paper. Specifically, we focus on training a DNN for object detection using
airplanes. The same reasoning can be applied to other applications like
detection of runways or lane tracking for taxiing.

To motivate the need for a new approach, compare a traditional sensor like a
radar to a camera for detecting other flying aircraft. Radars have been used
for decades and certified for both airborne and ground based applications.

As shown in Figure 1, a radar is an active sensor (emits energy) and its
physical properties are understood so that every step of the detection can be
modeled. Well understood mathematical algorithms including the Fourier
transform and the constant false alarm rate algorithm can then be applied to
processing radar return data. Additionally, radar data includes time of flight
information that is used to determine range which allows for easier
discernment of targets from the background.

On the other hand, camera images are more drastically affected by a large
range of physical phenomenon like lighting and weather conditions. The large
variations in sample data and the challenge of separating targets from
background data without range information makes the use of traditional
algorithms challenging. Instead, the “learned description” captured by a
trained DNN can often achieve drastically better results than traditional
approaches for image processing tasks.

In the case of the radar, every element is either known by the manufacturer of
the radar or from physics. In the case of vision-based systems, there are a
lot of unknowns that cannot be practically modeled, and the software needs to
be robust to changes in any of these parameters.

Figure 1: Process for tracking an object using a radar.

In other words, a radar-based approach could be analyzed using a bottom-up
approach, based on a physical model of the world. A vision based approach, on
the other hand, must be analyzed in a top-down manner, where a complex model
such as a DNN is trained to perform a task (such as object detection) and its
performance is then statistically analyzed to ensure it is adequate.

In particular, we need to ensure that the trained model is generalizable and
stable.

Generalizability

AI systems are trained with a subset of possible input scenarios but will be
used in scenarios across the operational domain. They should generalize to
scenarios not encountered during training and still maintain high accuracy.

Stability

AI systems should behave consistently in similar situations and be robust to
small variations. The nonlinear nature of neural networks that allows for
great model flexibility also allows for sudden changes in model outputs on
images that are nearly identical. The output of the network should be stable
and not change suddenly or unexpectedly.

A machine learning model is first trained and then can be used for inference.
If the code running the model is deterministic and there is no online
learning, then the model is a function from input data to labels/annotations
which is determined by the weight parameters that are optimized during
training. Thus in two phases, the model is optimized using annotated training
data, and then kept fixed and can be run on live data. In the remainder of
this paper and framework, we assume that the model is fixed during inference.

Figure 2: Example of a DNN.

This two step process of training and then inference is crucial because it
allows a thorough analysis to be performed on the trained model with the
confidence (at least, in a statistical sense) that the model will perform
predictably while in flight.

Once the model is trained, it can be viewed as a black box mapping inputs to
outputs. However, because we cannot completely characterize how the weights
deciding which additions and multiplications are applied cause certain
inferences to arise, it is essential to understand the generalization of the
model and where its limitations lie.

When evaluating neural network systems, the input space is too large to test
completely. Probability and statistics can be used to show that a system meets
performance requirements on a sample set. Thus it is important to ensure that
data used to train and test the model spans the operational domain of where
the model is to be used. One of the goals of our framework is to enable this
type of analysis to be performed.

The rest of this paper is organized as follows. In Section II we describe
relevant contributions from the fields of machine learning and certification.
In Section III we describe the requirements and high-level processes of our
proposed framework. In Section IV we describe our work in implementing the
core of this framework - a set of tools and processes for tracking aspects of
the development process, including datasets, codebases, and trained models. In
Section V we discuss the results of training and evaluting a DNN for detecting
airplanes using this framework. Finally, we give our conclusions in Section VI
and discuss directions for future work.

##  II Related Work

###  II-A Certification Processes

The intent of aircraft certification is to reduce and prevent aircraft
accidents. Each set of standards builds to prevent aircraft failures. The
widely adopted processes for traditional aerospace system safety and
determination of certification levels are defined in ARP-4754 and ARP-4761
[21, 20]. Those processes, as illustrated in Figure 3 guide the manufacturer
to determine the level of certification required for hardware and software.
DO-254 [18] is the widely adopted hardware certification framework and DO-178
[19] is the equivalent for software. DO-178 uses the principle that one must
trace every line of code to a requirement and vice-versa. That ensures that
every line of code is included for a reason and that all requirements are
addressed.

Figure 3: ARP-4754 Process [21].

For neural networks, the DO-178 process breaks down when requirements cannot
be simply flowed down to explicit lines of code.

EASA released a report [3] of guidelines for learning assurance of machine
learning models resulting from a collaboration between EASA and Daedalean in
the context of a runway detector project. They recommend verification to be
split between data, the learning process, and the inference model. Data should
be traceable to its origins and should cover the ConOps space. Learning should
be repeatable and convergent. Inference should be verified both in terms of
operational guarantees and performance.

###  II-B Machine Learning

Within the world of machine learning research, there are a number of areas
that are relevant to the certification process. Much recent work has been done
on the subject of so-called ”adversarial attacks”. This reflects a concerning
property of neural networks, namely that small input perturbations can have a
large effect on the output of the network. In adversarial attacks, adversarial
examples can be generated which “trick” the model into making the wrong
classification. These are constructed by making systematic changes to what
Ilya et al. [9] have called non-robust features. Models can be trained under
adversarial training to produce models that are robust to adversarial attacks
[14].

While most adversarial attacks have been done directly on the image space,
some have looked at more physical realizations. Eykholt et al. [4] have
successfully performed attacks by physically perturbing a stop sign before an
image is taken. However, for the use case of this paper, at the level of
airspace, it is impossible for an adversary to systematically affect large
portions of the image formation.

The existence of adversarial inputs has led to an interest in providing formal
guarantees that a model is robust to these types of attacks. For example, Katz
et al. [12] made some progress in formally proving adversarial robustness of
DNNs used to approximate the ACAS Xu collision avoidance algorithm. The
authors distinguish between local robustness (i.e. the DNN output does not
change under adversarial perturbations of an individually specified input) and
global robustness (i.e that the DNN output does not change under adversarial
perturbations of any possible input). Their focus is on proving local
robustness, while a certifiable system would want to show global robustness,
which remains an area of ongoing research [7]. Gehr et al. [5] took a
different approach, which was better able to scale to the kinds of large DNNs
typically used in computer vision. More recent work has focused on proving
robustness to adversarial attacks beyond the traditional $L_{p}$-norm balls
[15].

Of course, robustness to adversarial inputs is not the only property needed
for certification. The field of formal verification has made advances in
verifying other sorts of properties of DNNs as well. For example, Reluplex
[11] has been used to formally verify certain properties of DNNs used to
represent the ACAS Xu collision avoidance algorithm (such as the property that
certain areas of the input space all produce the same output). Yang et al.
[28] were able to fully verify a neural network in the context of computer
vision. Howevever, their approach requires a full model of the world and the
image formation process, which is impractical for our use case (see the
example in the Introduction). Unfortunately, current approaches have two major
limitations that make practical application in the certification context
difficult. First, these approaches do not scale well to the large networks
used in state of the art computer vision (although some recent work has shown
promise in this regard [5] [23]). And secondly, many of the properties we
might want to verify (i.e. the network performs well, regardless of weather
conditions) are difficult/impossible to specify formally. Nevertheless, this
is an area of ongoing research and future advances may allow these approaches
to be incorporated into our framework.

Another branch of related work is in the area of DNN testing. These methods
extrapolate traditional software engineering test methods to apply to DNNs.
They automatically generate test cases (i.e. synthetic images) in a way that
attempts to find edge cases where the model behaves unexpectedly [24, 16, 29,
27]. These test cases can help identify areas where more training data is
needed in a much more systematic and thorough manner than manual testing.

This is also a large body of work related to making DNNs more interpretable
and explainable [22, 13, 26]. However, these techniques do not provide the
level of interpretability present in traditional software, which leads to the
need for the type of framework proposed here.

##  III Framework

The objective of the proposed framework is to fill in the gaps where the
traditional processes break. We leverage industry standards such as ARP-4754
and ARP-4761 [21, 20], which describe the process of requirements,
implementation, and verification. Our framework is not a replacement for this
process but rather extends it to provide visibility for neural network
systems.

Figure 4: Certification processes for a system with an AI component.

As shown in Figure 4, a component such as a neural network based object
detector needs to be certified alongside more traditional software and
hardware components. This framework will serve to demonstrate what needs to be
addressed when verifying the neural network component.

Figure 5: V-model for systems development lifecycle [21].

Figure 5 shows a prototype for the V-model lifecycle from ARP-4574 [21].
Requirements are written, for which software and hardware are designed to
satisfy, and then are tested and verified. In our framework we will expand on
the bottom of the V to capture objectives important for verification for
neural networks.

With a neural network, software is written for accelerated hardware (eg. GPUs,
TPUs, ASICs, FPGAs) to perform the computations, but the millions to billions
of parameters used in the computation are optimized separately and usually
stored separately from the committed code base.

A model is a mathematical function performing some objective. In our use case,
this is object detection but could be one of a multitude of tasks. The types
of models we are considering under our framework are those structured as
neural networks.

When a neural network performs a computation, the weights and parameters are
just as essential if not more essential than the code running the network
computations. The ingredients that affect a model’s behavior are:

  * •

The code used to train the model

  * •

The initial weights of the model prior to training

  * •

The data used to train the model

  * •

The code used at inference time

  * •

The input image at inference time

If the processes for training and inference are deterministic, then the
inference model is a function of only these five components, the first four of
which do not change at run time.

Our framework assumes a deterministic neural network is used at inference.
Online learning is not feasible in our framework as it entails that the model
changes while in use and thus inferences are made with an unverified model.

Our verification of the model is threefold: we set requirements and processes
to verify the data being utilized to build and test the model, the overall
model performance on a test dataset, and the reliability of operation at
runtime. The dataset verification ensures that this performance spans the
types of conditions in the operational domain. The performance metrics on the
test set ensure that the model performs well. The runtime verification ensures
that the model is executable on the target hardware and environment. We
enumerate a series of steps for this in Figure 6 and how this fits into a
V-model with additional dependencies in Figure 7.

  1. 1.

Requirements development:

    1. (a)

Define the operational domain

    2. (b)

Define model requirements

    3. (c)

Define runtime requirements

  2. 2.

Data Management:

    1. (a)

Data collection

    2. (b)

Dataset verification

  3. 3.

Implementation:

    1. (a)

Model training

    2. (b)

Inference model implementation

    3. (c)

Guardrails/model uncertainty and monitoring

  4. 4.

Testing:

    1. (a)

Model evaluation

      1. i.

Evaluation (i.e. performance metrics)

      2. ii.

Sensitivity testing

    2. (b)

Runtime verification

    3. (c)

Guardrails and Monitoring Evaluation

Figure 6: Steps in certification framework.

Figure 7: V-model for an AI component, annotated with corresponding framework
steps.

###  III-A Requirements Development

Before a model can be evaluated with respect to any use cases, there is a need
to identify the operational domain, that is, the conditions in which the
system will operate. For our use case, the operational domain should be
defined in terms of parameters such as location, environmental/weather
conditions, time of day, level of traffic, intruder aircraft attributes, and
camera characteristics amongst others. The applicant should make an exhaustive
list of all the external conditions that may be encountered, and the
proportions in which they are expected to be encountered. This needs to be in
place so that analyses of the data can be measured with respect to the
expected set of conditions.

A system is expected to perform a function well and without faults. System
analysis is made much more tractable by splitting the requirements into
runtime requirements and performance requirements.

The runtime requirements can be verified and validated using traditional
methods. For a neural network, these requirements should be about how the
system processing the neural network computations functions and cover aspects
such as the system’s determinism, throughput, latency, and possibility for
execution errors.

The performance metrics need to be determined so that the model can be
evaluated. In a multiclass detection setting, the mAP (mean average precision)
is a commonly used high-level measurement of the model’s performance. So a
basic requirement might be that the model achieves a certain mAP on the test
set, determined by the allowable false negative and false positive rates.

However, the mAP (or any other single statistic, for that matter) is not
enough to give a complete picture of the model’s performance on its own.
Additional, more specific requirements might include:

  * •

Maximum overall false positive rate

  * •

Maximum overall false negative rate

  * •

Accuracy of detections with respect to true target position, typically
measured using the intersection over union (IOU) of the detected and ground
truth bounding boxes

  * •

Maximum false positive/negative rates in specific situations (i.e. weather
conditions, lighting conditions, intruder distance, etc.). It may allowable
for the model to have degraded performance in rare, less dangerous situations.
But it should be possible to show that model performs well, e.g. when the
intruder is very close, or when visibility is good. It is also expected that
during the development process, requirements for additional situations will be
added as they are encountered.

###  III-B Dataset Collection

Data must be gathered which spans all of the values of parameters determined
in the operational domain, ideally distributed in similar proportions to the
distribution of real conditions. This data should be compiled for two types of
datasets: the development datasets and the certification dataset. The
development datasets, often further split into training and validation sets,
comprise the data the applicant uses to construct and train the model. The
certification dataset must be kept separate and is used to test the final
model.

It would be best if the certification dataset were collected and constructed
by an independent party to minimize data dependence, but minimally
certification data should be collected on separate flights from data in the
development datasets and it should be possible to verify that no images from
the certification dataset were used in training.

The data collection process is ongoing throughout the development process. As
weaknesses are found in the trained models, new data will need to be collected
and added to the training and test sets. Additionally, developers may want to
create datasets focusing on specific scenarios, remove images from datasets
that are no longer relevant (for example, if different cameras are being used
and images taken by older cameras are no longer representative), further
refine labels on existing datasets, and otherwise modify datasets.

This creates a need for tools to track changes to a dataset over time.
Datasets should be versioned so that it is possible to fully reconstruct any
dataset used during the development process. Within each dataset, images
should be tracked, along with any relevant metadata, information about the
source of the image, and information about the labels in the image. Our
approach to maintaining this information is described in Subsection IV-B.

###  III-C Dataset Verification

In order for the performance metrics of the model to be useful for running
live inference, the dataset used to test the model needs to be representative
of real world conditions, that is, the operational domain. Formally, the
following generalization should hold true.

Goal

Acceptable performance on the certification dataset implies performance in the
operational domain.

To achieve this, certain properties need to hold for the certification
dataset.

  1. 1.

The examples in the certification dataset should cover the variations in
operational parameters of the operational domain.

  2. 2.

The certification dataset needs to contain examples from throughout the
operational domain.

Partitioning by operating parameters ensures that the test conditions vary
across the input space. Edge cases should be included to test the boundaries
of the model undergoing certification.

The development datasets shall be disjoint from the certification dataset. The
trained model needs to be developed on data that is independent from all data
in the certification dataset. This ensures that performance on the
certification dataset stems from generalization and not directly optimizing
for the tested conditions. This means that the data used for the certification
dataset should be independently collected from data used for development.

###  III-D Model Management

A typical development process for a deep learning project will involve
repeatedly training models. Each training run may involve different
hyperparameters (learning rate, regularization parameters, etc.), model
architectures, training code, training data, etc. The output of the training
process is a model file, consisting of the neural network architecture and the
parameters of the network. These model files should be tracked with the
following parameters:

  * •

The version of code used to generate and train the model, including all
dependencies

  * •

Any random seeds and initial weights

  * •

Datasets used during training/evaluation

  * •

Metrics computed during model evaluation (eg. accuracy, precision/recall, etc)

Given this information, we can easily re-train a model using the same
architecture, hyperparameters, etc. on the same training set to reproduce the
same model byte for byte.

###  III-E Inference Model

The inference model implementation encompasses turning the model into an
executable model to be used on the target hardware and environment. If
techniques such as batch normalization, dropout, pruning, or quantization are
used, the inference model differs from the model in the training process. The
metrics above should be computed on the model used for inference rather than
the one for training in these cases. Otherwise, it is necessary to verify that
the difference between the functions described by the two models are within
acceptable bounds.

The system running the inference model also needs to undergo traditional
software and hardware verification components such as memory usage and timing.
This is easier for models where computations are independent of the data (ie.
no branching). There should also be verification that the executable model
behaves identically to the model tested during model verification, if there
are any differences between the execution environment during evaluation and
runtime.

Note that these traditional certification methods do not apply to code used in
development (and in particular, code used to train a model) as they
deterministically produce model parameters prior to flight. Thus they do not
need any runtime guarantees. This code will be tracked and its outputs
verified using the processes outlined in our framework.

###  III-F Guardrails and Monitoring

It is beneficial to detect when a model is not performing as expected and
prevent any dangerous behavior that may result. This can be useful during the
development process, when unexpected behavior can be addressed by improving
the model. However, it is most useful after deployment of a model.

An example of a guardrail to prevent unsafe behavior might be limits on how
the model can influence the control systems (i.e. to prevent dangerous
maneuvers). On the monitoring side, it may be useful to detect when a given
input is dissimilar from the training data [8]. Assertions about the outputs
of the model may also be useful, for example to detect flickering (i.e. a
detected airplane disappearing/reappearing between frames of a video) [10].
However, we leave a full description of this topic out of the scope of this
document.

###  III-G Other Extensions to the Framework

The framework proposed here is largely focused on traceability of datasets and
trained models, with the goal of enabling the types of statistical analysis
that will be necessary for certification. And while various types of
statistical analysis will certainly be necessary for certification, other
techniques may also be useful.

For example, formal verification has the potential to offer much stronger
guarantees than any statistical analysis can provide. Challenges remain that
prevent formal verification from fully replacing statistical verification in
the certification process (see the “Related Work” section). However, as
progress is made in this area, formal verification techniques can be
increasingly useful. One area of particular focus has been in proving
robustness to adversarial inputs. Our framework could be extended by using
techniques similar to [5] in order to show local robustness. Although this
technique can’t prove robustness to any possible adversarial attack, it can be
used to increase confidence in the model by showing robustness to adversarial
examples in the locality of a selection of specific images. Similarly,
DeepSafe [7] has been used to show more general forms of robustness on small
networks for low-dimensional input spaces. Future work in this direction may
allow for stronger formal guarantees of robustness.

Other sorts of properties that might be formally verified have been less
studied and are subject to future research. For example, future work may allow
for an approach similar to that in [28] to be used for verifying the model on
a subset of the input space.

Our implementation of this framework provides methods for tracking and
analyzing data in the training/test sets, but we do not expect manual dataset
creation and analysis to be sufficient for fully covering the input space.
Automated testing frameworks directed at DNNs can help in this regard by
automatically generating images to test various edge cases of the network [24,
16, 29, 27]. As problematic areas of the input space are discovered and images
are added to the training/test sets they will be tracked and analyzed using
the process described below.

Automated testing can help find edge cases where the trained model fails, but
it may also be useful to understand why a model fails on a given input. This
is an area where fields like explainable AI (XAI) may be of use [13]. For
example, a technique such as LIME [17] may be useful during the development
process to understand why a model fails on a given input image, and what kind
of additional training data may be needed to improve the performance. During
the certification process it may be helpful to explain the failures of a model
on the certification dataset (since perfect performance is unlikely) to better
understand the model’s failure modes.

##  IV Methodology

We have used these concepts in building an object detection model for
detecting intruder aircraft in images.

###  IV-A Automatic Data Collection and Labelling

We collect images with potential intruders automatically during flights and
place initial detection bounding boxes using ADS-B and radar data. The quality
of these boxes vary as the ADS-B given by other aircraft is not always
completely accurate and depending on how well the cameras are calibrated.

These annotations can be refined manually. We track when and how these changes
are made.

###  IV-B Version Controlled Datasets

Data is a first-class citizen in our framework. We track data like we track
code, by making commits in git.

Data is abstracted into images, annotations, and datasets. Images are the
underlying visual data. Annotations are the labels, bounding boxes, and
metadata associated with images, either collected automatically or added
manually. Datasets are collections of versioned annotations and their
associated images. Images are immutable. Annotations and datasets can evolve
and update independently. The integrity of all components is guaranteed by
tracking the hash. Multiple datasets can coexist, even while pointing at
different versions of data items. We use DVC [2] to ensure data integrity of
images/videos while keeping the size of the git repository small.

Figure 8: Dataset, annotation, and image traceability.

In our use case, we refined bounding boxes in annotations using CVAT [1], an
open source image and video annotation tool. Updates were committed to our
dataset repository, where history is tracked.

###  IV-C Training Process

We train our network with Tensorflow with GPU deterministic operations
enabled. All of our batching and image preprocessing is deterministic. When a
training job starts, we produce a trace of the versions of all of the code
repositories, data repositories, libraries, GPUs, and drivers being used. This
way, the training is fully deterministic and repeatable.

###  IV-D Data Attributes

Images extracted from flight data are automatically labeled as described
above. In addition to bounding boxes, data is also labeled with additional
attributes that are useful in analyzing the resulting dataset. These
attributes include the callsign of any intruders (which can be used to look up
the type of aircraft), time of day, distance to intruder, and position and
orientation of the ownship (can be used along with time of day to determine
the position of the sun/other landmarks relative to cameras). Additional
information, such as weather/lighting conditions, background type, as well as
any other qualitative descriptions of the image, can be added manually.
Together, this information can be used to determine whether a given dataset
has an appropriate diversity of images.

Since images are extracted from video, they are also labeled with sequence
information. This makes it possible to analyze the performance of the model
from frame to frame, which is important in ensuring stability.

##  V Results

In order to test and refine our framework, we trained and evaluated a DNN for
object detection of airplanes.

###  V-A Datasets

Images from our flight tests were collected, labeled, and managed using the
process described in Subsection IV-A. The labeled data was then split into a
training set of 724 images and a test set of 379 images. A typical image with
predicted labels drawn is shown in Figure 10. The image has been cropped to a
region surrounding the airplane.

###  V-B Performance

We trained an SSD FPN with ResNet-50 on the training dataset. The model was
pretrained on the COCO dataset, and was trained for 50 epochs using the Adam
optimizer. Evaluation on the test dataset achieved an average precision of
79%. For the purposes of this paper we only trained on a single class, the
airplane class, and so we use the average precision instead of the more
typical mean average precision.

###  V-C Sensitivity by Partitions

During the certification process it will be important to determine how
sensitive a given model is to changes in various properties of the input. For
example, a model that loses its accuracy when an intruder aircraft is
approaching from a certain angle would need additional training before
certification.

To address this, we built a framework for analyzing the sensitivity of a
trained model to the types of attributes that we record with our data (see
“Data Attributes” section above). To test this approach we performed a basic
analysis of the impact of the distance to intruder aircraft on the performance
of a trained model. We gathered 379 images from a series of encounters to
perform the evaluation. The data was divided into 4 bins based on range -
0-375m, 375-750m, 750-1115m and 1115-1500m. The average precision was then
calculated for each bin. As expected, the model performed better at closer
ranges and worse at longer ranges.

Figure 9: Graph showing the average precision for each range bin.

Figure 10: Example aircraft intruder detection.

##  VI Conclusion

The certification of complex machine learning algorithms presents many new
challenges. This project lays out a framework for recommended practices in
building and verifying a certified neural network system. In addition, we
present some of our work to implement some of that framework by managing the
data, training, and verification processes on our use case of detecting
aircraft intruders in flight. We illustrate these processes by training a
neural network while tracking the origins of how the model was constructed and
all data used to train it. This model is not meant to be certified itself, but
rather to demonstrate how the types of analysis needed for certification could
be performed.

There are many challenges and risks in using machine learning systems in
safety-critical applications. By bringing together and tracking the
requirements, data, training, and testing, we can evaluate and make
statistical guarantees on the domain characterized by the data collected.

The view presented here is a heavily data focused approach for the use of a
deterministic vision based system. The ideas of online learning and state-
dependent systems (eg recurrent networks) have not been investigated due to
additional complexity with little to no additional benefit for our use case.
Moreover, we leave other parts of implementing the framework, such as the
development of monitoring systems and additional guardrails as realtime checks
for future work. Other possible extensions to the framework, such as formal
verification, automated testing/image generation, and methods for model
explainability/interpretability, are subject to change as these fields advance
and are also left for future work.

Our framework focuses on providing methods to ensure that a trained model is
generalizable across macroscopic variations in input changes (such as changes
in weather, range to target, etc.). We do not directly address the need to
demonstrate stability. Techniques such as training on adversarial examples can
be easily integrated into our framework. Analyses of model stability can be
integrated during the certification phase, for example by generating datasets
of perturbed images (adversarial or otherwise) and storing/tracing them using
our framework.

## Acknowledgements

This work was sponsored by the NASA SBIR/STTR program.

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