# Nonlinear inverse problem for identifying
a coefficient of the lowest term in hyperbolic equation with nonlocal
conditions

G. Yu. Mehdiyeva  Baku State University
Department of Computational Mathematics
23 Z.Khalilov Str.
Az1148
Baku
Azerbaijan<EMAIL_ADDRESS>,  Y. T. Mehraliyev  Baku State University
Department of Differential and Integral Equations
23 Z.Khalilov Str.
Az1148
Baku
Azerbaijan<EMAIL_ADDRESS>and  E. I. Azizbayov  The Academy of Public
Administration under the President of the Republic of Azerbaijan
Department of Control for Intelligent Systems
74, Lermontov Str.
Az 1001
Baku
Azerbaijan The Academy of Public Administration under the President of the
Republic of Azerbaijan
Department of Control for Intelligent Systems
74, Lermontov Str.
Az 1001
Baku
Azerbaijan<EMAIL_ADDRESS>
###### Abstract.

In this paper, a nonlinear inverse boundary value problem for the second-order
hyperbolic equation with nonlocal conditions is studied. To investigate the
solvability of the original problem, we first consider an auxiliary inverse
boundary value problem and prove its equivalence (in a certain sense) to the
original problem. Then using the Fourier method and contraction mappings
principle, the existence and uniqueness theorem for auxiliary problem is
proved. Further, on the basis of the equivalency of these problems the
existence and uniqueness theorem for the classical solution of the considered
inverse coefficient problem is proved for the smaller value of time.

###### Key words and phrases:

inverse problem, hyperbolic equation, overdetermination condition, classical
solution, existence, uniqueness

###### 1991 Mathematics Subject Classification:

35R30; 35L10; 35A01; 35A02; 35A09

##  1\. Introduction and problem statement

Let $0<T<+\infty$ be some fixed number and $D_{T}$ be a rectangular region in
the $xt$-plane defined by $0\leq x\leq 1,\ 0\leq t\leq T$. Consider the
problem of determining the unknown functions $u=u(x,t)$ and $a=a(t)$ such that
the pair $\\{u,a\\}$ satisfies the following hyperbolic equation of second
order

(1.1) $u_{tt}(x,t)-u_{xx}(x,t)=a(t)u(x,t)+f(x,t)\ \ (x,t)\in D_{T},$

with the nonlocal initial conditions

(1.2) $u(x,0)+\delta_{1}u(x,T)=\varphi(x),\ \
u_{t}(x,0)+\delta_{2}u_{t}(x,T)=\psi(x),\ 0\leq x\leq 1,$

the boundary conditions

(1.3) $u(0,t)=\beta u(1,t),\ 0\leq t\leq T,$ (1.4)
$\int\limits_{0}^{1}{u(x,t)dx}=0,\ 0\leq t\leq T,$

and overdetermination condition

(1.5) $u\left({\frac{1}{2},t}\right)=h(t),\ 0\leq t\leq T,$

in which $\delta_{1},\delta_{2}\geq 0$ and $\beta\neq\pm 1$ are given numbers,
$f(x,t),\varphi(x)$, and $\psi(x)$ are known functions of $x\in[0,1]$ and
$t\in[0,T]$.

In the present work we investigate a nonlinear inverse problem for identifying
a coefficient of the lowest term in hyperbolic equation from the
overdetermination data. Such problems are called inverse problems in
mathematical physics. The applied importance of inverse problems is so great
(seismology, mineral exploration, biology, medicine, desalination of seawater,
movement of liquid in a porous medium, acoustics, electromagnetics, fluid
dynamics, calculating the density of the Earth from measurements of its
gravity field, for example) which puts them a series of the most actual
problems of modern mathematics. Strictly speaking, the inverse problems for
hyperbolic/wave equations are of prime interest in seismology. Besides,
vibrations of structures (as buildings and beams) are modeled by hyperbolic
differential equations. However, it should be noted that an inverse problem is
called linear if the recovery function enters the given equation linearly, and
nonlinear otherwise.

Nowadays, inverse problems for hyperbolic equations have been well studied by
many authors using different methods (see, e.g., [2, 5, 6, 7, 8, 13, 18, 19,
20, 22], and the references given therein). Moreover, in [14, 15, 16, 17] the
authors present a regularity result for solutions of partial differential
equations in the framework of mixed Morrey spaces.

A distinctive feature of presented article is the investigation of an inverse
hyperbolic problem with both spatial and time nonlocal conditions.

This article is based on ideas close to those used in [1, 4, 10, 21].

######  Definition 1.

A pair of functions $\\{u(x,t),a(t)\\}$ is said to be a classical solution of
problem (1.1)–(1.5) if all three of the following conditions are satisfied:

  * a.

The function $u(x,t)$ with the derivatives $u_{xx}(x,t)$ and $u_{tt}(x,t)$ are
continuous in the domain $D_{T}$.

  * b.

The function $a(t)$ is continuous on the interval $[0,T]$.

  * c.

The Eq. (1.1) and conditions (1.2)–(1.5) are satisfied in the classical
(usual) sense.

Now, to study problem (1.1)–(1.5), we consider the following auxiliary inverse
boundary value problem: it is required to find a pair of functions $u(x,t)\in
C^{2}(D_{T}),\\\ \ a(t)\in C[0,T]$ from (1.1)–(1.3) and

(1.6) $u_{x}(0,t)=u_{x}(1,t),\ 0\leq t\leq T,$ (1.7)
$h^{\prime\prime}(t)-u_{xx}\left({\frac{1}{2},t}\right)=a(t)h(t)+f\left({\frac{1}{2},t}\right),\
0\leq t\leq T.$

Similarly (see [9], Theorem 2.2, p.4) it can be proved that

######  Theorem 1.

Suppose that $\varphi(x),\psi(x)\in C[0,1],\ h(t)\in C^{2}[0,T],\ h(t)\neq 0$,
$f(x,t)\in C(D_{T}),\ \int\limits_{0}^{1}{f(x,t)dx}=0,\ 0\leq t\leq T$ and the
compatibility conditions

$\int\limits_{0}^{1}{\varphi(x)dx}=0,\ \ \int\limits_{0}^{1}{\psi(x)dx}=0,$
$h(0)+\delta_{1}h(T)=\varphi\left({\frac{1}{2}}\right),\ \
h^{\prime}(0)+\delta_{2}h^{\prime}(T)=\psi\left({\frac{1}{2}}\right),$

hold. Then the following statements are true:

  * (_i_)

Each classical solution $\\{u(x,t),a(t)\\}$ of problem (1.1)–(1.5) is the
solution of problem (1.1)–(1.3), (1.6), (1.7), as well

  * (_ii_)

each solution $\\{u(x,t),a(t)\\}$ of problem (1.1)–(1.3), (1.6), (1.7) is a
classical solution to the problem (1.1)–(1.5), if

$\frac{{(1+2\delta_{1}+3\delta_{2}+\delta_{1}\delta_{2})T^{2}}}{{2(1+\delta_{1})(1+\delta_{2})}}\left\|{a(t)}\right\|_{C[0,T]}<1.$

##  2\. Auxiliary facts and denotations

It is known that sequences of functions [11]

(2.1) $X_{0}(x)=px+q,\ \ X_{2k-1}(x)=(px+q)\cos\lambda_{k}x,\ \
X_{2k}(x)=\sin\lambda_{k}x,\ \ k=1,2,\ldots,$ (2.2) $Y_{0}(x)=2,\ \
Y_{2k-1}(x)=4\sin\lambda_{k}x,\ \ Y_{2k}(x)=q(1-q-px)\cos\lambda_{k}x,\ \
k=1,2,\ldots,$

form a biorthogonal system and system (2.1) forms a Riesz basis in
$L_{2}(0,1)$ for $\lambda_{k}=2k\pi$ $(k=1,2,\ldots)$. Here $p$ and $q$
denotes, in turn, the numbers

$p=\frac{{1-\beta}}{{1+\beta}}\neq 0,\ \ q=\frac{\beta}{{1+\beta}}.$

We state the following lemmas without proof.

######  Lemma 1.

(see [3, 12]) For any function $v(x)$ with the properties:

$v(x)\in C^{2i-1}[0,1],\ \ v^{(2i)}(x)\in L_{2}(0,1),$ $v^{(2s)}(0)=\beta
v^{(2s)}(1),\ \ v^{(2s+1)}(0)=v^{(2s+1)}(1)\ \ (i\geq 1;\ \
s=\overline{0,i}),$

the estimates are valid

$\left({\sum\limits_{k=0}^{\infty}{(\lambda_{k}^{2i}v_{2k-1})^{2}}}\right)^{\frac{1}{2}}\leq
2\sqrt{2}\left\|{v^{(2i)}(x)}\right\|_{L_{2}(0,1)},$ (2.3)
$\left({\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{2i}v_{2k})^{2}}}\right)^{\frac{1}{2}}\leq
2\sqrt{2}\left\|{v^{(2i)}(x)(1-q-px)-2ipv^{(2i-1)}(x)}\right\|_{L_{2}(0,1)},$

where

$v_{k}=\int\limits_{0}^{1}{v(x)Y_{k}(x)dx},\ k=0,1,....$

######  Lemma 2.

(see [3, 12]) Under the assumptions:

$v(x)\in C^{2i}[0,1],\ \ v^{(2i+1)}(x)\in L_{2}(0,1),$ $v^{(2s)}(0)=\beta
v^{(2s)}(1),\ \ v^{(2s+1)}(0)=v^{(2s+1)}(1)\ \ (i\geq 1;\ \
s=\overline{0,i}),$

we establish the validity of the estimates

$\left({\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{2i+1}v_{2k-1})^{2}}}\right)^{\frac{1}{2}}\leq
2\sqrt{2}\left\|{v^{(2i+1)}(x)}\right\|_{L_{2}(0,1)},$
$\left({\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{2i+1}v_{2k-1})^{2}}}\right)^{\frac{1}{2}}$
(2.4) $\leq
2\sqrt{2}\left\|{v^{(2i+1)}(x)(1-q-px)-(2i+1)pv^{(2i)}(x)}\right\|_{L_{2}(0,1)}.$

We now look at the following functional spaces:

$B_{2,T}^{3}$ [12] denotes a set of all functions of the form

$u(x,t)=\sum\limits_{k=1}^{\infty}{u_{k}(t)X_{k}(x)},$

considered in $D_{T}$, where $u_{k}(t)\in C[0,T]$ and

$J(u)\equiv\left\|{u_{0}(t)}\right\|_{C[0,T]}$
$+\left\\{{\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{3}\left\|{u_{2k-1}(t)}\right\|_{C[0,T]})^{2}}}\right\\}^{\frac{1}{2}}+\left\\{{\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{3}\left\|{u_{2k}(t)}\right\|_{C[0,T]})^{2}}}\right\\}^{\frac{1}{2}}<+\infty.$

Actually, the functions $X_{k}(x)\ (k=0,1,\ldots)$ defined by the relation
(2.1). The norm on the set $J(u)$ is established as follows

$\left\|{u(x,t)}\right\|_{B_{2,T}^{3}}=J(u).$

Let $E_{T}^{3}$ denote the space consisting of the topological product
$B_{2,T}^{3}\times C[0,T]$, which is the norm of the element $z=\\{u,a\\}$
defined by the formula

$\left\|{z(x,t)}\right\|_{E_{T}^{3}}=\left\|{u(x,t)}\right\|_{B_{2,T}^{3}}+\left\|{a(t)}\right\|_{C[0,T]}.$

##  3\. Classical solvability of inverse boundary value problem

Suppose that the data of problem (1.1)–(1.3),(1.6),(1.7) satisfies the
following conditions:

  * $C_{1}$.

$\delta_{1},\delta_{2}\geq 0,\ \
1+\delta_{1}\delta_{2}\geq\delta_{1}+\delta_{2};$

  * $C_{2}$.

$\varphi(x)\in C^{2}[0,1],\ \varphi^{\prime\prime\prime}(x)\in L_{2}(0,1),\
\varphi(0)=\beta\varphi(1),\\\ \varphi^{\prime}(0)=\varphi^{\prime}(1),\
\varphi^{\prime\prime}(0)=\beta\varphi^{\prime\prime}(1);$

  * $C_{3}$.

$\psi(x)\in C^{1}[0,1],\ \psi^{\prime\prime}(x)\in L_{2}(0,1),\
\psi(0)=\beta\psi(1),\ \psi^{\prime}(0)=\psi^{\prime}(1);$

  * $C_{4}$.

$f(x,t),f_{x}(x,t)\in C(D_{T}),\ f_{xx}(x,t)\in L_{2}(D_{T}),\ f(0,t)=\beta
f(1,t),\\\ f_{x}(0,t)=f_{x}(1,t),\ 0\leq t\leq T;$

  * $C_{5}$.

$h(t)\in C^{2}[0,T],\ h(t)\neq 0,\ 0\leq t\leq T.$

Since the system (2.1) forms a Riesz basis in $L_{2}(0,1)$. Then each solution
to problem (1.1)–(1.3),(1.6),(1.7) can be sought in the form:

(3.1) $u(x,t)=\sum\limits_{k=0}^{\infty}{u_{k}(t)X_{k}(x)},$

where

$u_{k}(t)=\int\limits_{0}^{1}{u(x,t)Y_{k}(x)dx},\ k=0,1,....$

Moreover, $X_{k}(x)\ (k=0,1,...)$ and $Y_{k}(x)\ (k=0,1,...)$ are defined by
relations (2.1) and (2.2), respectively.

To determine of the desired functions $u_{k}(t)\ (k=0,1,2,...)$, using
separation of variables and by (1.1) and (1.2), we get

(3.2) $u^{\prime\prime}_{0}(t)=F_{0}(t;u,a),\ 0\leq t\leq T,$ (3.3)
$u^{\prime\prime}_{2k-1}(t)+\lambda_{k}^{2}u_{2k-1}(t)=F_{2k-1}(t;u,a),\
k=1,2,...;\ 0\leq t\leq T,$ (3.4)
$u^{\prime\prime}_{2k}(t)+\lambda_{k}^{2}u_{2k}(t)=F_{2k}(t;u,a)-2p\lambda_{k}u_{2k-1}(t),\
k=1,2,...;\ 0\leq t\leq T,$ (3.5) $u_{k}(0)+\delta_{1}u_{k}(T)=\varphi_{k},\ \
u^{\prime}_{k}(0)+\delta_{2}u^{\prime}_{k}(T)=\psi_{k},\ k=0,1,2,\ldots,$

where

$F_{k}(t;u,a)=a(t)u_{k}(t)+f_{k}(t),\ \
f_{k}(t)=\int\limits_{0}^{1}{f(x,t)Y_{k}(x)dx},\ k=0,1,....$
$\varphi_{k}=\int\limits_{0}^{1}{\varphi(x)Y_{k}(x)dx},\ \
\psi_{k}=\int\limits_{0}^{1}{\psi(x)Y_{k}(x)dx},\ k=0,1,2....$

Solving problem (3.2)–(3.5), we find

(3.6)
$u_{0}(t)=\frac{{\varphi_{0}}}{{1+\delta_{1}}}+\frac{{t-\delta_{1}(T-t)}}{{(1+\delta_{1})(1+\delta_{2})}}\psi_{0}+\int\limits_{0}^{T}{G_{0}(t,\tau)F_{0}(\tau;u,a)d\tau},$
$u_{2k-1}(t)=\frac{1}{{\rho_{k}(T)}}\left[{\varphi_{2k-1}(\cos\lambda_{k}t+\delta_{2}\cos\lambda_{k}(T-t))}\right.$
(3.7)
$+\left.{\frac{{\psi_{2k-1}}}{{\lambda_{k}}}(\sin\lambda_{k}t-\delta_{1}\sin\lambda_{k}(T-t))}\right]+\int\limits_{0}^{T}{G_{k}(t,\tau)F_{2k-1}(\tau;u,a)d\tau},$
$u_{2k}(t)=\frac{1}{{\rho_{k}(T)}}\left[{\varphi_{2k}(\cos\lambda_{k}t+\delta_{2}\cos\lambda_{k}(T-t))+\frac{{\psi_{2k}}}{{\lambda_{k}}}(\sin\lambda_{k}t-\delta_{1}\sin\lambda_{k}(T-t))}\right]$
$+\int\limits_{0}^{T}{G_{k}(t,\tau)(F_{2k}(\tau;u,a))d\tau}$
$-\varphi_{2k-1}\left\\{{-\frac{1}{{\rho_{k}^{2}(T)}}\frac{1}{{\lambda_{k}}}}\right.\left[{\delta_{1}\cos\lambda_{k}t\left({\frac{T}{2}\sin\lambda_{k}T+\delta_{2}\frac{1}{4}(1-\cos
2\lambda_{k}T)}\right)}\right.$
$+\delta_{2}\sin\lambda_{k}t\left({\frac{1}{{2\lambda_{k}}}\sin\lambda_{k}T+\frac{T}{2}\cos\lambda_{k}T+\delta_{2}\left({\frac{T}{2}+\frac{1}{{4\lambda_{k}}}\sin
2\lambda_{k}T}\right)}\right)$
$+\delta_{1}\delta_{2}\left({\frac{1}{{4\lambda_{k}}}(\cos\lambda_{k}(2T-t)-\cos\lambda_{k}t)+\frac{T}{2}\sin\lambda_{k}t}\right.$
$\left.{\left.{+\delta_{2}\left({-\frac{T}{2}\sin\lambda_{k}(T-t)+\frac{1}{{4\lambda_{k}}}(\cos\lambda_{k}(T-t)-\cos\lambda_{k}(T+t))}\right)}\right)}\right]$
$\left.{+\frac{t}{2}\sin\lambda_{k}t+\delta_{2}\left({-\frac{t}{2}\sin\lambda_{k}(T-t)+\frac{1}{{4\lambda_{k}}}(\cos\lambda_{k}(T-t)-\cos\lambda_{k}(T+t))}\right)}\right\\}$
$+\frac{{\psi_{2k-1}}}{{\lambda_{k}}}\left\\{{-\frac{1}{{\rho_{k}^{2}(T)}}\cdot\frac{1}{{\lambda_{k}}}}\right.\left[{\delta_{1}\cos\lambda_{k}t\left({\frac{1}{{2\lambda_{k}}}\sin\lambda_{k}T-\frac{T}{2}\cos\lambda_{k}T}\right)}\right.$
$-\delta_{1}\left({\frac{T}{2}-\frac{1}{{4\lambda_{k}}}\sin
2\lambda_{k}T}\right)+\delta_{2}\sin\lambda_{k}t\left({\frac{T}{2}\sin\lambda_{k}T-\frac{{\delta_{1}}}{{4\lambda_{k}}}(1-\cos
2\lambda_{k}T)}\right)$
$+\delta_{1}\delta_{2}\left({\frac{1}{{4\lambda_{k}}}(\sin\lambda_{k}(2T-t)+\sin\lambda_{k}t)-\frac{T}{2}\cos\lambda_{k}t}\right.$
$\left.{\left.{-\delta_{1}\left({\frac{T}{2}\cos\lambda_{k}(T-t)-\frac{1}{{4\lambda_{k}}}(\sin\lambda_{k}(T-t)+\sin\lambda_{k}(T+t))}\right)}\right)}\right]$
$+\left({\frac{1}{{2\lambda_{k}}}\sin\lambda_{k}t-\frac{t}{2}\cos\lambda_{k}t-\delta_{1}\left({\frac{t}{2}\cos\lambda_{k}(T-t)+}\right.}\right.\frac{1}{{4\lambda_{k}}}(\sin\lambda_{k}(T-t)$
(3.8)
$\left.{\left.{-\sin\lambda_{k}(T+t))}\right)}\right\\}+\int\limits_{0}^{T}{G_{k}(t,\tau)\left({\int\limits_{0}^{T}{G_{k}(\tau,\xi)F_{2k-1}(\xi;u,a)d\xi}}\right)}d\tau,$

where

$G_{0}(t,\tau)=\begin{cases}-\frac{{\delta_{2}t+\delta_{1}(T-\tau)+\delta_{1}\delta_{2}(t-\tau)}}{{(1+\delta_{1})(1+\delta_{2})}},\
\ \ \ \ \ \ \ \ \ \ t\in[0,\tau],\\\
-\frac{{\delta_{2}t+\delta_{1}(T-\tau)-(1+\delta_{1}+\delta_{2})(t-\tau)}}{{(1+\delta_{1})(1+\delta_{2})}},\
\ \ \ t\in[\tau,T],\end{cases}$

$G_{k}(t,\tau)=\begin{cases}-\frac{1}{{\rho_{k}(T)}}\cdot\frac{1}{{\lambda_{k}}}[\delta_{1}\sin\lambda_{k}(T-\tau)\cos\lambda_{k}t+\delta_{2}\cos\lambda_{k}(T-\tau)\sin\lambda_{k}t\\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\delta_{1}\delta_{2}\sin\lambda_{k}(t-\tau)],\ t\in[0,\tau],\\\
-\frac{1}{{\rho_{k}(T)}}\cdot\frac{1}{{\lambda_{k}}}[\delta_{1}\sin\lambda_{k}(T-\tau)\cos\lambda_{k}t+\delta_{2}\cos\lambda_{k}(T-\tau)\sin\lambda_{k}t\\\
\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
+\delta_{1}\delta_{2}\sin\lambda_{k}(t-\tau)]+\frac{1}{{\lambda_{k}}}\sin\lambda_{k}(t-\tau),\
t\in[\tau,T],\\\ \end{cases}$

and

$\rho_{k}(T)=1+(\delta_{1}+\delta_{2})\cos\lambda_{k}T+\delta_{1}\delta_{2}.$

Substituting the expressions of (3.6), (3.7), and (3.8) into (3.1), we find
the component $u(x,t)$ of the classical solution to problem (1.1)–(1.3),
(1.6), (1.7) to be

$u(x,t)=\left\\{{\frac{{\varphi_{0}}}{{1+\delta_{1}}}+\frac{{t-\delta_{1}(T-t)}}{{(1+\delta_{1})(1+\delta_{2})}}\psi_{0}+\int\limits_{0}^{T}{G_{0}(t,\tau)F_{0}(\tau;u,a)d\tau}}\right\\}X_{0}(x)$
$+\sum\limits_{k=1}^{\infty}{\left\\{{\frac{1}{{\rho_{k}(T)}}\left[{\varphi_{2k-1}(\cos\lambda_{k}t+\delta_{2}\cos\lambda_{k}(T-t))}\right.}\right.}$
$\left.{\left.{+\frac{{\psi_{2k-1}}}{{\lambda_{k}}}(\sin\lambda_{k}t-\delta_{1}\sin\lambda_{k}(T-t))}\right]+\int\limits_{0}^{T}{G_{k}(t,\tau)F_{2k-1}(\tau;u,a)d\tau}}\right\\}X_{2k-1}(x)$
$+\sum\limits_{k=1}^{\infty}{\left\\{{\frac{1}{{\rho_{k}(T)}}[\varphi_{2k}(\cos\lambda_{k}t+\delta_{2}\cos\lambda_{k}(T-t))+}\right.}$
$+\left.{\frac{{\psi_{2k}}}{{\lambda_{k}}}(\sin\lambda_{k}t-\delta_{1}\sin\lambda_{k}(T-t))}\right]+\int\limits_{0}^{T}{G_{k}(t,\tau)F_{2k}(\tau;u,a)d\tau}$
$-\varphi_{2k-1}\left\\{{-\frac{1}{{\rho_{k}^{2}(T)}}\cdot\frac{1}{{\lambda_{k}}}}\right.\left[{\delta_{1}\cos\lambda_{k}t\left({\frac{T}{2}\sin\lambda_{k}T+\delta_{2}\frac{1}{4}(1-\cos
2\lambda_{k}T)}\right)}\right.$
$+\delta_{2}\sin\lambda_{k}t\left({\frac{1}{{2\lambda_{k}}}\sin\lambda_{k}T+\frac{T}{2}\cos\lambda_{k}T+\delta_{2}\left({\frac{T}{2}+\frac{1}{{4\lambda_{k}}}\sin
2\lambda_{k}T}\right)}\right)$
$+\delta_{1}\delta_{2}\left({\frac{1}{{4\lambda_{k}}}(\cos\lambda_{k}(2T-t)-\cos\lambda_{k}t)+\frac{T}{2}\sin\lambda_{k}t}\right.$
$\left.{\left.{+\delta_{2}\left({-\frac{T}{2}\sin\lambda_{k}(T-t)+\frac{1}{{4\lambda_{k}}}(\cos\lambda_{k}(T-t)-\cos\lambda_{k}(T+t))}\right)}\right)}\right]$
$\left.{+\frac{t}{2}\sin\lambda_{k}t+\delta_{2}\left({-\frac{t}{2}\sin\lambda_{k}(T-t)+\frac{1}{{4\lambda_{k}}}(\cos\lambda_{k}(T-t)-\cos\lambda_{k}(T+t))}\right)}\right\\}$
$+\frac{{\psi_{2k-1}}}{{\lambda_{k}}}\left\\{{-\frac{1}{{\rho_{k}^{2}(T)}}\cdot\frac{1}{{\lambda_{k}}}}\right.\left[{\delta_{1}\cos\lambda_{k}t\left({\frac{1}{{2\lambda_{k}}}\sin\lambda_{k}T-\frac{T}{2}\cos\lambda_{k}T}\right)}\right.$
$-\delta_{1}\left({\frac{T}{2}-\frac{1}{{4\lambda_{k}}}\sin
2\lambda_{k}T}\right)+\delta_{2}\sin\lambda_{k}t\left({\frac{T}{2}\sin\lambda_{k}T-\frac{{\delta_{1}}}{{4\lambda_{k}}}(1-\cos
2\lambda_{k}T)}\right)$
$+\delta_{1}\delta_{2}\left({\frac{1}{{4\lambda_{k}}}(\sin\lambda_{k}(2T-t)+\sin\lambda_{k}t)-\frac{T}{2}\cos\lambda_{k}t}\right.$
$\left.{\left.{-\delta_{1}\left({\frac{T}{2}\cos\lambda_{k}(T-t)-\frac{1}{{4\lambda_{k}}}(\sin\lambda_{k}(T-t)+\sin\lambda_{k}(T+t))}\right)}\right)}\right]$
$+\left({\frac{1}{{2\lambda_{k}}}\sin\lambda_{k}t-\frac{t}{2}\cos\lambda_{k}t-\delta_{1}\left({\frac{t}{2}\cos\lambda_{k}(T-t)+}\right.}\right.\frac{1}{{4\lambda_{k}}}(\sin\lambda_{k}(T-t)$
(3.9)
$-\sin\lambda_{k}(T+t)))\\}+\left.{\int\limits_{0}^{T}{G_{k}(t,\tau)\left({\int\limits_{0}^{T}{G_{k}(\tau,\xi)F_{2k-1}(\xi;u,a)d\xi}}\right)}d\tau}\right\\}X_{2k}(x).$

Now, using (3.1) from (1.7) we have

(3.10)
$a(t)=[h(t)]^{-1}\left\\{{h^{\prime\prime}(t)-f\left({\frac{1}{2},t}\right)}\right.-\frac{1}{2}\left.{\sum\limits_{k=1}^{\infty}{(-1)^{k}\lambda_{k}^{2}}u_{2k-1}(t)}\right\\}.$

Substituting expressions $u_{2k-1}(t)\ (k=1,2,...)$ from (3.8) into (3.10),
immediately yields:

$a(t)=[h(t)]^{-1}$
$\times\left\\{{h^{\prime\prime}(t)-f\left({\frac{1}{2},t}\right)}\right.-\frac{1}{2}\sum\limits_{k=1}^{\infty}{(-1)^{k}\lambda_{k}^{2}\left\\{{\frac{1}{{\rho_{k}(T)}}\left[{\varphi_{2k-1}(\cos\lambda_{k}t+\delta_{2}\cos\lambda_{k}(T-t))}\right.}\right.}$
(3.11)
$+\left.{\left.{\left.{\frac{{\psi_{2k-1}}}{{\lambda_{k}}}(\sin\lambda_{k}t-\delta_{1}\sin\lambda_{k}(T-t))}\right]+\int\limits_{0}^{T}{G_{k}(t,\tau)F_{2k-1}(\tau;u,a)d\tau}}\right\\}}\right\\}.$

Thus, the solution of problem (1.1)–(1.3), (1.6), (1.7) is reduced to the
solution of system (3.9), (3.11) with respect to unknown functions $u(x,t)$
and $a(t)$.

Using the same discussion in [12], one can prove

######  Lemma 3.

If $\\{u(x,t),a(t)\\}$ is any solution to problem (1.1)–(1.3), (1.6), (1.7),
then the functions

$u_{k}(t)=\int\limits_{0}^{1}{u(x,t)Y_{k}(x)dx},\ \ k=0,1,...,$

satisfy the countable system (3.6), (3.7) and (3.8) on an interval $[0,T]$.

Obviously, if $u_{k}(t)=\int\limits_{0}^{1}{u(x,t)Y_{k}(x)dx},\ k=0,1,...$ is
a solution to system (3.6), (3.7) and (3.8), then the pair $\\{u(x,t),a(t)\\}$
of functions $u(x,t)=\sum\limits_{k=0}^{\infty}{u_{k}(t)}X_{k}(x)$ and $a(t)$
is also a solution to system (3.9), (3.11).

It follows from the Lemma 3 that

######  Corollary 1.

Let system (3.9), (3.11) have a unique solution. Then problem (1.1)–(1.3),
(1.6), (1.7) cannot have more than one solution, i.e. if the problem
(1.1)–(1.3), (1.6), (1.7) has a solution, then it is unique.

Now consider the operator

$\Phi(u,a)=\\{\Phi_{1}(u,a),\Phi_{2}(u,a)\\},$

in the space $E_{T}^{3}$, where

$\Phi_{1}(u,a)=\tilde{u}(x,t)\equiv\tilde{u}_{0}(t)X_{0}(x)+\sum\limits_{k=0}^{\infty}{\tilde{u}_{2k-1}(t)X_{2k-1}(x)}+\sum\limits_{k=1}^{\infty}{\tilde{u}_{2k}(t)X_{2k}(x)},$
$\Phi_{2}(u,a)=\tilde{a}(t),$

and the functions $\tilde{u}_{0}(t),\tilde{u}_{2k-1}(t),\tilde{u}_{2k}(t)\
(k=1,2,...)$, and $\tilde{a}(t)$ are equal correspondingly to the right sides
of (3.6), (3.7), (3.8), and (3.11).

It is clear that under conditions $\delta_{1},\delta_{2}\geq 0,\ \
1+\delta_{1}\delta_{2}>\delta_{1}+\delta_{2}$ and using the inequality

$\rho_{k}(T)\geq 1-(\delta_{1}+\delta_{2})+\delta_{1}\delta_{2}$

we can write

$\frac{1}{{\rho_{k}(T)}}\leq\frac{1}{{1-(\delta_{1}+\delta_{2})+\delta_{1}\delta_{2}}}\equiv\rho>0.$

Using this relation, from (3.6), (3.7), (3.8), and (3.11) we obtain

$\left\|{\tilde{u}_{0}(t)}\right\|_{C[0,T]}\leq\frac{1}{{1+\delta_{1}}}\left|{\varphi_{0}}\right|+\frac{T}{{1+\delta_{2}}}\psi_{0}+\frac{{1+3\delta_{1}+3\delta_{2}}}{{(1+\delta_{1})(1+\delta_{2})}}T$
(3.12)
$\times\left({\sqrt{T}\left({\int\limits_{0}^{T}{\left|{f_{0}(\tau)}\right|^{2}d\tau}}\right)^{\frac{1}{2}}+T\left\|{a(t)}\right\|_{C[0,T]}\left\|{u_{0}(t)}\right\|_{C[0,T]}}\right),$
$\left\\{{\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{3}\left\|{\tilde{u}_{2k-1}(t)}\right\|_{C[0,T]})^{2}}}\right\\}^{\frac{1}{2}}$
$\leq
2\rho(1+\delta_{2})\left({\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{3}\left|{\varphi_{2k-1}}\right|)^{2}}}\right)^{\frac{1}{2}}+2\rho(1+\delta_{1})\left({\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{2}\left|{\psi_{2k-1}}\right|)^{2}}}\right)^{\frac{1}{2}}$
$+2(1+2\rho(\delta_{1}+\delta_{2}+\delta_{1}\delta_{2}))\sqrt{T}\left({\int\limits_{0}^{T}{\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{2}\left|{f_{2k-1}(\tau)}\right|)^{2}d\tau}}}\right)^{\frac{1}{2}}$
(3.13)
$+2(1+2\rho(\delta_{1}+\delta_{2}+\delta_{1}\delta_{2}))T\left\|{a(t)}\right\|_{C[0,T]}\left({\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{3}\left\|{u_{2k-1}(t)}\right\|_{C[0,T]})^{2}}}\right)^{\frac{1}{2}},$
$\left\\{{\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{3}\left\|{\tilde{u}_{2k}(t)}\right\|_{C[0,T]})^{2}}}\right\\}^{\frac{1}{2}}$
$\leq
2\sqrt{2}\rho(1+\delta_{2})\left({\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{3}\left|{\varphi_{2k}}\right|)^{2}}}\right)^{\frac{1}{2}}+2\sqrt{2}\rho(1+\delta_{1})\left({\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{2}\left|{\psi_{2k}}\right|)^{2}}}\right)^{\frac{1}{2}}$
$+2\sqrt{2}(1+2\rho(\delta_{1}+\delta_{2}+\delta_{1}\delta_{2}))\sqrt{T}\left({\int\limits_{0}^{T}{\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{2}\left|{f_{2k}(\tau)}\right|)^{2}d\tau}}}\right)^{\frac{1}{2}}$
$+2\sqrt{2}(1+2\rho(\delta_{1}+\delta_{2}+\delta_{1}\delta_{2}))T\left\|{a(t)}\right\|_{C[0,T]}\left({\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{3}\left\|{u_{2k}(t)}\right\|_{C[0,T]})^{2}}}\right)^{\frac{1}{2}}$
$+2\sqrt{2}\rho_{1}\left({\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{3}\left|{\varphi_{2k-1}}\right|)^{2}}}\right)^{\frac{1}{2}}+2\sqrt{2}\rho_{2}\left({\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{2}\left|{\psi_{2k-1}}\right|)^{2}}}\right)^{\frac{1}{2}}$
$+2\sqrt{2}(1+2\rho(\delta_{1}+\delta_{2}+\delta_{1}\delta_{2}))^{2}T\left[{\sqrt{T}\left({\int\limits_{0}^{T}{\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{2}\left|{f_{2k-1}(\tau)}\right|)^{2}d\tau}}}\right)^{\frac{1}{2}}}\right.$
(3.14)
$\left.{+T\left\|{a(t)}\right\|_{C[0,T]}\left({\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{3}\left\|{u_{2k-1}(t)}\right\|_{C[0,T]})^{2}}}\right)^{\frac{1}{2}}}\right],$
$\left\|{\tilde{a}(t)}\right\|_{C[0,T]}\leq\left\|{[h(t)]^{-1}}\right\|_{C[0,T]}\left\\{{\left\|{h^{\prime\prime}(t)-f\left({\frac{1}{2},t}\right)}\right\|_{C[0,T]}+\frac{1}{2}\left({\sum\limits_{k=1}^{\infty}{\lambda_{k}^{-2}}}\right)^{\frac{1}{2}}}\right.$
$\times\left[{\rho(1+\delta_{2})\left({\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{3}\left|{\varphi_{2k-1}}\right|)^{2}}}\right)^{\frac{1}{2}}+\rho(1+\delta_{1})\left({\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{2}\left|{\psi_{2k-1}}\right|)^{2}}}\right)^{\frac{1}{2}}}\right.$
$+(1+2\rho(\delta_{1}+\delta_{2}+\delta_{1}\delta_{2}))\sqrt{T}\left({\int\limits_{0}^{T}{\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{2}\left|{f_{2k-1}(\tau)}\right|)^{2}d\tau}}}\right)^{\frac{1}{2}}$
(3.15)
$\left.{\left.{+(1+2\rho(\delta_{1}+\delta_{2}+\delta_{1}\delta_{2}))T\left\|{a(t)}\right\|_{C[0,T]}\left({\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{3}\left\|{u_{2k-1}(t)}\right\|_{C[0,T]})^{2}}}\right)^{\frac{1}{2}}}\right]}\right\\},$

where

$\rho_{1}=\rho^{2}\left[{\delta_{1}\left({\frac{{\delta_{2}}}{2}+\frac{T}{2}}\right)+\delta_{2}\left({\frac{1}{2}+\frac{T}{2}+\delta_{2}\left({\frac{1}{4}+\frac{T}{2}}\right)}\right)}\right.$
$\left.{+\delta_{1}\delta_{2}\left({\frac{1}{2}+\frac{T}{2}+\delta_{2}\left({\frac{1}{2}+\frac{T}{2}}\right)}\right)}\right]+\frac{T}{2}+\delta_{2}\left({\frac{1}{2}+\frac{T}{2}}\right),$
$\rho_{2}=\rho^{2}\left[{\delta_{1}\left({\frac{1}{2}+\frac{T}{2}}\right)+\delta_{1}\left({\frac{1}{4}+\frac{T}{2}}\right)+\delta_{2}\left({\frac{{\delta_{1}}}{2}+\frac{T}{2}}\right)}\right.$
$\left.{+\delta_{1}\delta_{2}\left({\frac{1}{2}+\frac{T}{2}+\delta_{1}\left({\frac{1}{2}+\frac{T}{2}}\right)}\right)}\right]+\frac{1}{2}+\frac{T}{2}+\delta_{1}\left({\frac{1}{2}+\frac{T}{2}}\right).$

Then from (3.12), (3.13), (3.14), and (3.15), taking into account (2.3),
(2.4), respectively, we find:

$\left\|{\tilde{u}_{0}(t)}\right\|_{C[0,T]}\leq\frac{2}{{1+\delta_{1}}}\left\|{\varphi(x)}\right\|_{L_{2}(0,1)}+\frac{{2T}}{{1+\delta_{2}}}\left\|{\psi(x)}\right\|_{L_{2}(0,1)}+\frac{{1+3\delta_{1}+3\delta_{2}}}{{(1+\delta_{1})(1+\delta_{2})}}T$
$\times(2\sqrt{T}\left\|{f(x,t)}\right\|_{L_{2}(D_{T})}+T\left\|{a(t)}\right\|_{C[0,T]}\left\|{u(x,t)}\right\|_{B_{2,T}^{3}}),$
$\left\\{{\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{3}\left\|{\tilde{u}_{2k-1}(t)}\right\|_{C[0,T]})^{2}}}\right\\}^{\frac{1}{2}}$
$\leq
4\sqrt{2}\rho(1+\delta_{2})\left\|{\varphi^{\prime\prime\prime}(x)}\right\|_{L_{2}(0,1)}+4\sqrt{2}\rho(1+\delta_{1})\left\|{\psi^{\prime\prime}(x)}\right\|_{L_{2}(0,1)}$
$+4(1+2\rho(\delta_{1}+\delta_{2}+\delta_{1}\delta_{2}))\sqrt{2T}\left\|{f_{xx}(x,t)}\right\|_{L_{2}(D_{T})}$
$+2(1+2\rho(\delta_{1}+\delta_{2}+\delta_{1}\delta_{2}))T\left\|{a(t)}\right\|_{C[0,T]}\left\|{u(x,t)}\right\|_{B_{2,T}^{3}},$
$\left\\{{\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{3}\left\|{\tilde{u}_{2k}(t)}\right\|_{C[0,T]})^{2}}}\right\\}^{\frac{1}{2}}$
$\leq
8\rho(1+\delta_{2})\left\|{\varphi^{\prime\prime\prime}(x)(1-q-px)-3p\varphi^{\prime\prime}(x)}\right\|_{L_{2}(0,1)}$
$+8\rho(1+\delta_{1})\left\|{\psi^{\prime\prime}(x)(1-q-px)-2p\psi^{\prime}(x)}\right\|_{L_{2}(0,1)}$
$+8(1+2\rho(\delta_{1}+\delta_{2}+\delta_{1}\delta_{2}))\sqrt{T}\left\|{f_{xx}(x,t)(1-q-px)-2pf_{x}(x,t)}\right\|_{L_{2}(D_{T})}$
$+2\sqrt{2}(1+2\rho(\delta_{1}+\delta_{2}+\delta_{1}\delta_{2}))T\left\|{a(t)}\right\|_{C[0,T]}\left\|{u(x,t)}\right\|_{B_{2,T}^{3}}$
$+8\rho_{1}\left\|{\varphi^{\prime\prime\prime}(x)}\right\|_{L_{2}(0,1)}+8\rho_{2}\left\|{\psi^{\prime\prime}(x)}\right\|_{L_{2}(0,1)}$
$+2\sqrt{2}(1+2\rho(\delta_{1}+\delta_{2}+\delta_{1}\delta_{2}))^{2}T[2\sqrt{2T}\left\|{f_{xx}(x,t)}\right\|_{L_{2}(D_{T})}$
$+T\left\|{a(t)}\right\|_{C[0,T]}\left\|{u(x,t)}\right\|_{B_{2,T}^{3}}],$
$\left\|{\tilde{a}(t)}\right\|_{C[0,T]}\leq\left\|{[h(t)]^{-1}}\right\|_{C[0,T]}$
$\times\left\\{{\left\|{h^{\prime\prime}(t)-f\left({\frac{1}{2},t}\right)}\right\|_{C[0,T]}}\right.+\frac{1}{2}\left({\sum\limits_{k=1}^{\infty}{\lambda_{k}^{-2}}}\right)^{\frac{1}{2}}[2\sqrt{2}\rho(1+\delta_{2})\left\|{\varphi^{\prime\prime\prime}(x)}\right\|_{L_{2}(0,1)}$
$+2\sqrt{2}\rho(1+\delta_{1})\left\|{\psi^{\prime\prime}(x)}\right\|_{L_{2}(0,1)}+(1+2\rho(\delta_{1}+\delta_{2}+\delta_{1}\delta_{2}))2\sqrt{2T}\left\|{f_{xx}(x,t)}\right\|_{L_{2}(D_{T})}$
$+(1+2\rho(\delta_{1}+\delta_{2}+\delta_{1}\delta_{2}))T\left\|{a(t)}\right\|_{C[0,T]}\left\|{u(x,t)}\right\|_{B_{2,T}^{3}}]\\},$

or

(3.16) $\left\|{\tilde{u}_{0}(t)}\right\|_{C[0,T]}\leq
A_{1}(T)+B_{1}(T)\left\|{a(t)}\right\|_{C[0,T]}\left\|{u(x,t)}\right\|_{B_{2,T}^{3}},$
(3.17)
$\left\\{{\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{3}\left\|{\tilde{u}_{2k-1}(t)}\right\|_{C[0,T]})^{2}}}\right\\}^{\frac{1}{2}}\leq
A_{2}(T)+B_{2}(T)\left\|{a(t)}\right\|_{C[0,T]}\left\|{u(x,t)}\right\|_{B_{2,T}^{3}},$
(3.18)
$\left\\{{\sum\limits_{k=1}^{\infty}{(\lambda_{k}^{3}\left\|{\tilde{u}_{2k}(t)}\right\|_{C[0,T]})^{2}}}\right\\}^{\frac{1}{2}}\leq
A_{3}(T)+B_{3}(T)\left\|{a(t)}\right\|_{C[0,T]}\left\|{u(x,t)}\right\|_{B_{2,T}^{3}},$
(3.19) $\left\|{\tilde{a}(t)}\right\|_{C[0,T]}\leq
A_{4}(T)+B_{4}(T)\left\|{a(t)}\right\|_{C[0,T]}\left\|{u(x,t)}\right\|_{B_{2,T}^{3}},$

where

$A_{1}(T)=\frac{2}{{1+\delta_{1}}}\left\|{\varphi(x)}\right\|_{L_{2}(0,1)}$
$+\frac{{2T}}{{1+\delta_{2}}}\left\|{\psi(x)}\right\|_{L_{2}(0,1)}+\frac{{2(1+3\delta_{1}+3\delta_{2})}}{{(1+\delta_{1})(1+\delta_{2})}}T\sqrt{T}\left\|{f(x,t)}\right\|_{L_{2}(D_{T})},$
$B_{1}(T)=\frac{{1+3\delta_{1}+3\delta_{2}}}{{(1+\delta_{1})(1+\delta_{2})}}T^{2},$
$A_{2}(T)=4\sqrt{2}\rho(1+\delta_{2})\left\|{\varphi^{\prime\prime\prime}(x)}\right\|_{L_{2}(0,1)}+4\sqrt{2}\rho(1+\delta_{1})\left\|{\psi^{\prime\prime}(x)}\right\|_{L_{2}(0,1)}$
$+4(1+2\rho(\delta_{1}+\delta_{2}+\delta_{1}\delta_{2}))\sqrt{2T}\left\|{f_{xx}(x,t)}\right\|_{L_{2}(D_{T})},$
$B_{2}(T)=2(1+2\rho(\delta_{1}+\delta_{2}+\delta_{1}\delta_{2}))T,$
$A_{3}(T)=8\rho(1+\delta_{2})\left\|{\varphi^{\prime\prime\prime}(x)(1-q-px)-3p\varphi^{\prime\prime}(x)}\right\|_{L_{2}(0,1)}$
$+8\rho(1+\delta_{1})\left\|{\psi^{\prime\prime}(x)(1-q-px)-2p\psi^{\prime}(x)}\right\|_{L_{2}(0,1)}$
$+8(1+2\rho(\delta_{1}+\delta_{2}+\delta_{1}\delta_{2}))\sqrt{T}\left\|{f_{xx}(x,t)(1-q-px)-2pf_{x}(x,t)}\right\|_{L_{2}(D_{T})}$
$+8\rho_{1}\left\|{\varphi^{\prime\prime\prime}(x)}\right\|_{L_{2}(0,1)}+8\rho_{2}\left\|{\psi^{\prime\prime}(x)}\right\|_{L_{2}(0,1)}$
$+8(1+2\rho(\delta_{1}+\delta_{2}+\delta_{1}\delta_{2}))^{2}T\sqrt{T}\left\|{f_{xx}(x,t)}\right\|_{L_{2}(D_{T})},$
$B_{3}(T)=2\sqrt{2}(1+2\rho(\delta_{1}+\delta_{2}+\delta_{1}\delta_{2}))T+2\sqrt{2}(1+2\rho(\delta_{1}+\delta_{2}+\delta_{1}\delta_{2}))^{2}T^{2},$
$A_{4}(T)=\left\|{[h(t)]^{-1}}\right\|_{C[0,T]}$
$\times\left\\{{\left\|{h^{\prime\prime}(t)-f\left({\frac{1}{2},t}\right)}\right\|_{C[0,T]}+}\right.\frac{1}{2}\left({\sum\limits_{k=1}^{\infty}{\lambda_{k}^{-2}}}\right)^{\frac{1}{2}}[2\sqrt{2}\rho(1+\delta_{2})\left\|{\varphi^{\prime\prime\prime}(x)}\right\|_{L_{2}(0,1)}$
$+2\sqrt{2}\rho(1+\delta_{1})\left\|{\psi^{\prime\prime}(x)}\right\|_{L_{2}(0,1)}+(1+2\rho(\delta_{1}+\delta_{2}+\delta_{1}\delta_{2}))2\sqrt{2T}\left\|{f_{xx}(x,t)}\right\|_{L_{2}(D_{T})}]\\},$
$B_{4}(T)=\frac{1}{2}\left\|{[h(t)]^{-1}}\right\|_{C[0,T]}\left({\sum\limits_{k=1}^{\infty}{\lambda_{k}^{-2}}}\right)^{\frac{1}{2}}(1+2\rho(\delta_{1}+\delta_{2}+\delta_{1}\delta_{2}))T.$

Now, from (3.16)–(3.18) we obtain:

(3.20) $\left\|{\tilde{u}(x,t)}\right\|_{B_{2,T}^{3}}\leq
A_{5}(T)+B_{5}(T)\left\|{a(t)}\right\|_{C[0,T]}\left\|{u(x,t)}\right\|_{B_{2,T}^{3}},$

where

$A_{5}(T)=A_{1}(T)+A_{2}(T)+A_{3}(T),\ \ B_{5}(T)=B_{1}(T)+B_{2}(T)+B_{3}(T).$

Finally, from (3.19) and (3.20) we conclude:

(3.21)
$\left\|{\tilde{u}(x,t)}\right\|_{B_{2,T}^{3}}+\left\|{\tilde{a}(t)}\right\|_{C[0,T]}\leq
A(T)+B(T)\left\|{a(t)}\right\|_{C[0,T]}\left\|{u(x,t)}\right\|_{B_{2,T}^{3}},$

where

$A(T)=A_{4}(T)+A_{5}(T),\ \ B(T)=B_{4}(T)+B_{5}(T).$

So, let us prove the following theorem.

######  Theorem 2.

Let conditions $C_{1}$ \- $C_{5}$ be satisfied and the inequality

(3.22) $B(T)(A(T)+2)^{2}<1$

holds. Then problem (1.1)–(1.3), (1.6), (1.7) has a unique solution in the
ball $K=K_{R}(\left\|z\right\|_{E_{T}^{3}}\leq R\leq A(T)+2)$ of the space
$E_{T}^{3}$.

######  Remark 1.

Inequality (3.22) is satisfied for sufficiently small values of $T$.

###### Proof.

Let us denote $z=[u(x,t),a(t)]^{T}$ and rewrite the system of equations (3.9)
and (3.11) in the following operator equation

(3.23) $z=\Phi z,$

where $\Phi=[\varphi_{1},\varphi_{2}]^{T}$ and $\varphi_{1}(z),\varphi_{2}(z)$
defined by the right sides of (3.9) and (3.11), respectively.

Consider the operator $\Phi(u,a)$ in the ball $K=K_{R}$ of the space
$E_{T}^{3}$. Similar to (3.21) we obtain that for any $z_{1},z_{2},z\in K_{R}$
the following estimates hold:

(3.24) $\left\|{\Phi z}\right\|_{E_{T}^{3}}\leq
A(T)+B(T)\left\|{a(t)}\right\|_{C[0,T]}\left\|{u(x,t)}\right\|_{B_{2,T}^{3}}\leq
A(T)+B(T)(A(T)+2)^{2},$ (3.25) $\left\|{\Phi z_{1}-\Phi
z_{2}}\right\|_{E_{T}^{3}}\leq
B(T)R(\left\|{a_{1}(t)-a_{2}(t)}\right\|_{C[0,T]}+\left\|{u_{1}(x,t)-u_{2}(x,t)}\right\|_{B_{2,T}^{3}}).$

Then taking into account (3.22) in (3.24) and (3.25), it follows that the
operator $\Phi$ acts in the ball $K=K_{R}$ and is contractive. Therefore, in
the ball $K=K_{R}$ the operator $\Phi$ has a unique fixed point
$\\{z\\}=\\{u,a\\}$ that is a unique solution of (3.23) in the ball $K=K_{R}$;
i.e. it is a unique solution of system (3.9), (3.11) in the ball $K=K_{R}$. ∎

Thus, we obtain that the function $u(x,t)$ as an element of the space
$B_{2,T}^{3}$ is continuous and has continuous derivatives $u_{x}(x,t)$ and
$u_{xx}(x,t)$ in $D_{T}$.

Analogous to [11] one can show that the function $u_{tt}(x,t)$ is continuous
in the region $D_{T}$.

It is easy to verify that Eq.(1.1) and conditions (1.2), (1.3), (1.6), and
(1.7) are satisfied in the ordinary sense. Consequently, $\\{u(x,t),a(t)\\}$
is a solution of problem (1.1)–(1.3), (1.6), (1.7) and by Remark 3.1 this
solution is unique in the ball $K=K_{R}$. The theorem is proved.

From Theorem 1 and Theorem 2 immediately imply that the original problem
(1.1)–(1.5) has a unique classical solution.

######  Theorem 3.

Suppose that all the conditions of Theorem 2 are satisfied and

$\int\limits_{0}^{1}{f(x,t)dx}=0,\ 0\leq t\leq T,\ \
\int\limits_{0}^{1}{\varphi(x)dx}=0,\ \ \int\limits_{0}^{1}{\psi(x)dx}=0,$
$h(0)+\delta_{1}h(T)=\varphi\left({\frac{1}{2}}\right),\ \
h^{\prime}(0)+\delta_{2}h^{\prime}(T)=\psi\left({\frac{1}{2}}\right),$
$\frac{{(1+2\delta_{1}+3\delta_{2}+\delta_{1}\delta_{2})T^{2}(A(T)+2)}}{{2(1+\delta_{1})(1+\delta_{2})}}<1.$

Then problem (1.1)–(1.5) has a unique classical solution in the ball
$K=K_{R}(\left\|z\right\|_{E_{T}^{3}}\\\ \leq R\leq A(T)+2)$ of the space
$E_{T}^{3}$.

##  4\. Conclusion

In the work, the classical solvability of a nonlinear inverse boundary value
problem for a second-order hyperbolic equation with nonlocal conditions was
studied. The considered problem was reduced to an auxiliary inverse boundary
value problem in a certain sense and its equivalence to the original problem
is shown. Then using the Fourier method and contraction mappings principle,
the existence and uniqueness theorem for auxiliary problem is proved. Further,
on the basis of the equivalency of these problems, the existence and
uniqueness theorem for the classical solution of the original inverse
coefficient problem is established for the smaller value of time.

## Acknowledgements

The authors express their profound thanks to the anonymous reviewers for their
insightful comments and helpful suggestions.

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# Low-Resource Compositional Semantic Parsing with Concept Pretraining

Subendhu Rongali
Amazon Alexa AI
<EMAIL_ADDRESS>
Mukund Sridhar∗
Google
<EMAIL_ADDRESS>
Haidar Khan
Amazon Alexa AI
<EMAIL_ADDRESS>
Konstantine Arkoudas∗
Dyania Health
<EMAIL_ADDRESS>
Wael Hamza
Amazon Alexa AI
<EMAIL_ADDRESS>
Andrew McCallum
UMass Amherst
<EMAIL_ADDRESS>
work done when SR was at UMass and MS and KA were at Amazon; it does not
relate to their current positions.

###### Abstract

Semantic parsing plays a key role in digital voice assistants such as Alexa,
Siri, and Google Assistant by mapping natural language to structured meaning
representations. To extend the capabilities of a voice assistant for a new
domain, the underlying semantic parsing model needs to be retrained using
thousands of annotated examples from the new domain, which is time-consuming
and expensive. In this work, we present an architecture to perform such
_domain adaptation_ automatically, with only a small amount of metadata about
the new domain and without any new training data (zero-shot) or with very few
examples (few-shot). We use a base seq2seq (sequence-to-sequence) architecture
and augment it with a _concept_ encoder that encodes intent and slot tags from
the new domain. We also introduce a novel decoder-focused approach to pretrain
seq2seq models to be concept aware using Wikidata. This pretraining helps our
model learn important concepts and perform well in low-resource settings. We
report few-shot and zero-shot results for compositional semantic parsing on
the TOPv2 dataset and show that our model outperforms prior approaches in few-
shot settings for the TOPv2 and SNIPS datasets.

Figure 1: The architecture of Concept-Seq2Seq for low resource domain
adaptation. The concept encoder encodes descriptions of each of the concept
tags into an embedding and incorporates them into the decoded parse.

##  1 Introduction

Voice assistants such as Alexa, Siri, and Google Assistant often rely on
semantic parsing to understand requests made by their users. The underlying
semantic parsing model converts natural language user utterances into logical
forms consisting of actions requested by the user (play music, check weather),
called _intents_ , and relevant entities in the request (which song? which
location?), called _slots_. The model is built to process requests in a fixed
set of domains, such as music, weather, shopping, and so on. With voice
assistants increasingly pervading more aspects of daily life, systems need to
be continuously updated to comprehend new intents and slots across an ever-
growing number of domains.

Current semantic parsing models are trained on large amounts of annotated data
from a predetermined set of domains. Extending these models to learn new
intents or slots typically involves collecting and annotating large amounts of
new data. This process is expensive and time-consuming. To combat this
problem, researchers have proposed semantic parsing models that can be
efficiently trained with fewer examples (few-shot) from new domains
(Shrivastava et al., 2021; Mansimov and Zhang, 2021; Ghoshal et al., 2020;
Shin et al., 2021; Desai et al., 2021; Rongali et al., 2022; Shrivastava et
al., 2022). While these methods facilitate few-shot learning, they have
limitations. Some of them rely on hand-crafted knowledge such as intermediate
grammars or logical form templates (Shrivastava et al., 2022; Shin et al.,
2021; Rongali et al., 2022). Others rely on very large pretrained language
models, such as GPT-3, to perform in-context learning by appending test
examples with instructional prompts (Shin et al., 2021).

In this work, we explore few-shot domain adaptation for semantic parsing
without any additional hand-crafted knowledge apart from the intent and slot
tag names, and with much smaller architectures that can perform efficient
inference in practical production environments. We also explore zero-shot
domain adaptation, when we have no annotated training data from a new domain.

To that end, we propose Concept-Seq2Seq, a novel architecture based on a
state-of-the-art semantic parsing model, Seq2Seq-Ptr (Rongali et al., 2020),
which uses seq2seq models and a pointer generator network to decode the target
semantic parse. We augment this model with a _concept_ encoder that encodes
intents and slots from the schema and uses those encodings to conditionally
decode the semantic parse. Figure 1 shows the architecture of our proposed
model. We train this model on annotated data from the given domains. During
inference, we simply encode all intents and slots from the schema, including
new, unseen ones, into the learned concept space, and decode the target parse.
This model has the same time complexity as the original Seq2Seq-Ptr model but
comes with the added benefit of now being able to effectively parse utterances
from unseen domains without any additional effort.

There have been a few zero-shot semantic parsing approaches proposed in the
past but they either covered only simple slot-filling style utterances (Bapna
et al., 2017; Lee and Jha, 2019) or compositional utterances that also came
with carefully crafted intermediate representations and context-free grammars
Herzig and Berant (2018); Wu et al. (2021). Our model is capable of performing
zero-shot domain adaptation for compositional semantic parsing, producing
meaning representations with nested intents and slots, but also doesn’t
require any grammars, whose construction effort often exceeds the effort
required to annotate a few examples.

In few-shot scenarios, we fine-tune our zero-shot model checkpoints further on
the small number of available examples. Due to the presence of the concept
encoder in our architecture, we expect to receive better knowledge-transfer
advantages by encoding intent and slot tags from new domains as opposed to
initializing them as new tags. To further improve performance, we propose a
novel decoder-focused pretraining scheme for Concept-Seq2Seq using an entity-
centric processed version of Wikidata (Vrandečić and Krötzsch, 2014) called
WikiWiki (Li et al., 2022), to help it better encode unseen concepts and parse
effectively.

We report the first zero-shot performance numbers for semantic parsing on the
compositional TOPv2 dataset Chen et al. (2020) and show that Concept-Seq2Seq
achieves commendable zero-shot performance on the flat-entity SNIPS dataset
Coucke et al. (2018). We also evaluate in few-shot settings and show that we
match or outperform previous state-of-the-art models while still being
production-viable.

In summary, our contributions are as follows.

  * •

We propose Concept-Seq2Seq, a bi-tower architecture with a seq2seq model and a
concept encoder, that can perform few-shot and zero-shot domain adaptation for
compositional semantic parsing without additional handcrafted knowledge.

  * •

We propose a novel decoder-focused pretraining scheme for Concept-Seq2Seq
using Wikidata that helps it better encode unseen concepts and parse
effectively.

  * •

We report few-shot and zero-shot semantic parsing results on the TOPv2 and
SNIPS datasets and show that our model outperforms or matches previously
proposed approaches on a variety of few-shot settings.

##  2 Methodology

In this section, we describe our proposed model, Concept-Seq2Seq, for low
resource (few-shot and zero-shot) domain adaptation for semantic parsing. It
is based on the Seq2Seq-Ptr model from Rongali et al. (2020), consisting of a
sequence-to-sequence encoder-decoder component, augmented with a pointer
generator network to constrain the target decoding vocabulary. Since our task
at hand is to perform potential zero-shot semantic parsing with just
descriptive metadata about the new domain, we modify the architecture of
Seq2Seq-Ptr to incorporate information about new intents and slots from new
domains by adding a concept encoder. Section 2.2 describes this architecture
in detail. To help our model learn to parse utterances from unseen domains
better, we also propose a novel pretraining scheme to incorporate general
concept parsing knowledge into it. Section 2.3 describes this concept
pretraining scheme. Finally, we describe Concept-Seq2Seq model specifics for
few-shot and zero-shot settings in Section 2.4. Before we get to these
sections, we first describe the source and target sequence formulation for the
semantic parsing task below.

###  2.1 Task Formulation

Our model solves semantic parsing as a sequence-to-sequence task, where the
source sequence is the utterance and the target sequence is a linearized
representation of the semantic parse. Following Rongali et al. (2020), we
modify the target sequence to only contain intent/slot tags or pointers to
utterance tokens. An example source and target sequence from the TOPv2 dataset
are given below.

⬇

Source: How far is the coffee shop

Target: [IN:GET_DISTANCE $@ptr_{0}$ $@ptr_{1}$ $@ptr_{2}$

[SL:DESTINATION [IN:GET_RESTAURANT_LOCATION

$@ptr_{3}$ [SL:TYPE_FOOD $@ptr_{4}$ SL:TYPE_FOOD]

$@ptr_{5}$ IN:GET_RESTAURANT_LOCATION]

SL:DESTINATION] IN:GET_DISTANCE]

Each $@ptr_{i}$ token here points to the $i^{th}$ token in the source
sequence. So $@ptr_{4}$ corresponds to the word _coffee_.

###  2.2 Model Architecture

Concept-Seq2Seq consists of three main components: an encoder, a decoder, and
a concept encoder. Just like in traditional sequence-to-sequence models, the
encoder encodes the source sequence, and the decoder autoregressively decodes
the target sequence. However, to effectively understand new intent and slot
tags in target sequences that the model hasn’t seen during training, our model
needs to be able to incorporate new intents and slots, or concepts, and decode
the target sequence accordingly. The concept encoder helps us do this by
encoding descriptive metadata about new concepts and creating vector
representations that we can use while decoding the target sequence.

Specifically, for an input sequence $[x_{1}\dots x_{n}]$, we first encode it
using the encoder into a sequence of hidden states $e_{1}\dots e_{n}$. Then,
having generated the first $t-1$ tokens, the decoder generates the token at
step $t$ as follows. It first produces the decoder hidden state at time $t$,
$d_{t}$ by building a multi-layer, multi-head self-attention on the encoder
hidden states and the decoder states so far. This step is based on the
transformer decoder from Vaswani et al. (2017). In a traditional sequence-to-
sequence generation task, $d_{t}$ is then fed into a dense layer to produce
scores over the target vocabulary.

Our target vocabulary consists of pointer tokens and the concept tags. Since
we do not have access to all concept tags at the time of training, we train
our model to incorporate descriptive information about concepts instead of
using a fixed-size dense layer. To do this, we encode intent and slot concepts
using a concept encoder. The descriptions we use in this work are simply rule-
based naturalized versions of the intent and slot names in the dataset. For
example, the description for the intent tag token [IN:GET_DISTANCE is set as “
_begin get distance intent_ ”. Similarly, for SL:DESTINATION], it is set to “
_end destination slot_ ”. We purposely use just this information and no
additional hand-crafted knowledge to remove any additional user input and to
compare to previous approaches in the same setting.

Given $m$ concept tokens (both begin and end) and their descriptions, the
concept encoder encodes each of them to produce concept vector representations
$[c_{1}\dots c_{m}]$. We then use the computed decoder hidden state at $t$,
$d_{t}$, as the query and compute unnormalized attention scores $[s_{1}\dots
s_{m}]$ with $[c_{1}\dots c_{m}]$, and $[a_{1}\dots a_{n}]$ with $[e_{1}\dots
e_{n}]$. Concatenating all these scores, we obtain an unnormalized
distribution over $m+n$ tokens, the first $m$ of which are the intent and slot
tagging tokens from the concepts, and the last $n$ of which are the
$@ptr_{i}(0<i<n)$ tokens pointing to the source sequence. We feed this through
a softmax layer to obtain the final probability distribution. This probability
distribution is used in the loss function during training and to choose the
next token to generate during inference. For the target token embeddings in
the decoder, we use a set of special embeddings to represent the $@ptr_{i}$
tokens and $[c_{1}\dots c_{m}]$ to represent the concept embeddings.

Figure 1 shows this process in action on a toy example. The model is decoding
the next token after SL:genre at step 5. To do this, the model computes the
pointer attention scores $[a_{1}\dots a_{n}]$ (blue, left) and the concept
token attention scores $[s_{1}\dots s_{n}]$ (green, right). The highest
overall score is for the token $@ptr_{2}$, corresponding to the word _country_
in the source sequence, so the next predicted token is _country_.

Figure 2: An example sentence from the Wikiwiki dataset with the associated
mention, entity, and type fields. The full hyperlinked sub-span is extracted
as the mention and the entity and type are extracted from the target page.

###  2.3 Concept Pretraining

Concept-Seq2Seq has the ability to incorporate new, unseen concepts while
parsing using the concept encoder and transfer knowledge on similar concepts.
In order to produce these unseen concepts or types, and have our model be
robust in low-data settings, it is important for our decoder to be type aware.
Conventional seq2seq pretraining schemes such as Lewis et al. (2020); Raffel
et al. (2020); Soltan et al. (2022) pretrain the decoder using language
modeling criterion. Li et al. (2022) extend the language modeling task to
induce entity-type information by treating it as a question answering task. We
pretrain the seq2seq model on the semantic parsing task using the WikiWiki (Li
et al., 2022) dataset. We explain how to achieve this by keeping an open-
domain, extensible output space for the semantic parse.

The Wikiwiki dataset curates mentions, entities, and entity types from 10M
Wikipedia documents using hyperlink information linking sub-spans of text in
sentences to other Wikipedia pages. The hyperlink is considered as the
mention, and the entity and the type information are extracted from the new
page. For further details on this processing, please refer to Li et al.
(2022). This dataset contains around 2m entities and 40K entity types.

Each example in the Wikiwiki dataset consists of a _context_ , which is a
paragraph from a wiki page, _mentions_ , which are sub-spans of text that link
to another page, _entities_ , which correspond to each mention, and _entity
types_ , which describe the type of the entity. We extract individual
sentences from this dataset and use them to train Concept-Seq2Seq to learn to
encode a wide variety of concepts using the entity type fields as descriptions
and tag the relevant mentions in the sentence. Figure 2 shows an example
sentence from this dataset and the different fields. The source and target
sequences for pretraining, and the descriptions of the concept tags for this
example are given below.

⬇

Source: He is a member of The Soul Seekers

Target: $@ptr_{0}$ $@ptr_{1}$ $@ptr_{2}$ $@ptr_{3}$ $@ptr_{4}$

[Q215380 $@ptr_{5}$ $@ptr_{6}$ $@ptr_{7}$ Q215380]

Concept descriptions:

[Q215380: begin musical group

Q215380]: end musical group

During training, we collect all the concept tokens within a training batch and
use them to create in-batch negatives (denominator of the softmax calculation)
for the decoding task. We do this since it is extremely inefficient to encode
all 40K $\times$ 2 concept token descriptions (begin and end) from Wikiwiki in
every step.

| Alarm | Event | Timer | Weather | Alarm | Event | Timer | Weather
---|---|---|---|---|---|---|---|---
| F1 score | EM Accuracy
Concept-Seq2Seq w/o pretraining | 62.53 | 30.25 | 55.68 | 51.23 | 45.94 | 0.00 | 0.51 | 8.62
Concept-Seq2Seq | 71.00 | 70.47 | 58.48 | 66.29 | 53.64 | 20.21 | 3.86 | 26.63

Table 1: Zero-shot performance of Concept-Seq2Seq on domains in TOPv2. We
observe notable scores in the alarm and weather domains and improvements
across all the domains after the concept pretraining step.

###  2.4 Few-shot and Zero-shot Specifics

Concept-Seq2Seq is primarily designed to perform low resource domain
adaptation for semantic parsing by effectively encoding the output space via a
concept encoder.

In the zero-shot setting, we deploy the following procedure to build our
models. We first perform concept pretraining on Concept-Seq2Seq using Wikiwiki
example sequences. We take this checkpoint and train on a set of known domains
to then obtain the zero-shot model. During inference, we encode all the intent
and slot tags from the new unknown domain using the concept encoder of the
obtained zero-shot model and set the appropriate decoder parameters to reduce
the architecture to a simple encoder-decoder setting.

In few-shot settings, we further fine-tune the zero-shot checkpoint on the
available handful of training examples. Since we explore extremely low-
resource settings (1, 5, 25 samples per intent/slot), we run the risk of over-
fitting and instability during training. To account for these risks and smooth
training, we augment the finetuning loss at every step with the loss from a
randomly sampled batch of training data from the known domains. We scale the
loss from the random known domain batch down using a multiplier before adding
to the loss. This scheme is akin to rehearsal (Ratcliff, 1990), a popular
technique in domain adaptation.

| Music | Book | Creative | Weather | Restaurant | Playlist | Screening
---|---|---|---|---|---|---|---
F1 score
Slot-filling (Lee and Jha, 2019; Bapna et al., 2017) | 31.20 | 34.13 | 86.21 | 65.64 | 51.40 | 59.96 | 44.50
Concept-Seq2Seq | 50.00 | 30.26 | 88.75 | 74.58 | 57.78 | 57.11 | 45.78
EM Accuracy
Slot-filling (Lee and Jha, 2019; Bapna et al., 2017) | 11.00 | 1.00 | 69.00 | 37.00 | 12.00 | 21.00 | 24.00
Concept-Seq2Seq | 20.00 | 2.00 | 69.00 | 42.00 | 13.00 | 19.00 | 26.00

Table 2: Zero-shot performance of Concept-Seq2Seq on domains in SNIPS. Our
model matches or outperforms the slot-filling style baseline on most domains.

##  3 Experimental Setup

We evaluate Concept-Seq2Seq on few-shot and zero-shot domain adaptation using
two popular English task-oriented semantic parsing datasets - TOPv2 (Chen et
al., 2020) and SNIPS (Coucke et al., 2018). Both datasets have utterances
grouped into multiple domains; TOPv2 has eight domains and SNIPS has seven
intents from seven different areas, which we consider domains. TOPv2 is a
large dataset consisting of 10k-20k training and 3k-7k test examples per
domain. It is also comprised of compositional examples with nested intents and
slots. We exclude the _unsupported_ utterances from the training and test sets
in TOPv2 for the zero-shot experiments (we use the full sets in few-shot).
_Unsupported_ utterances consist of utterances that belong to a domain but are
not supported, which is impossible to learn in zero-shot. SNIPS is a smaller
and simpler dataset with flat, disjoint slots. It has 2k training and 100 test
examples per domain.

For zero-shot, we use a leave-one-out approach where given $n$ domains, we
train models on annotated data from $n-1$ of them and evaluate on utterances
from the left-out domain. For few-shot settings, where the model has access to
a few annotated examples from the left-out domain, we further fine-tune using
1, 5, and 25 samples-per-intent/slot (SPIs), which we randomly sample from the
training data of the left-out domain. We fine-tune Concept-Seq2Seq on three
different randomly sampled training sets per domain per SPI setting and report
the average performance score of the three runs.

We use a transformer encoder, initialized from a _roberta-base_ checkpoint,
for Concept-Seq2Seq. The decoder is a transformer decoder initialized from
scratch and it contains 6 layers, 8 heads, and a hidden state size of 768. The
concept encoder is also a transformer encoder and it is initialized from a
_bert-base-uncased_ checkpoint. We choose a BERT-based model here since it is
pretrained to compute a vector for the whole sequence using the CLS token,
which is what we need for encoding a concept consisting of a multi-word
description. We also choose all _base_ -size components to keep the overall
model size small and expect the relative improvements shown by our model to
generalize.

We train our zero-shot models using sequence cross entropy loss. We use the
Adam optimizer with learning rate $2e^{-5}$ and $\epsilon=1e^{-8}$, warm-up
proportion $0.1$, weight decay $0.01$, and batch-size 128. The number of
epochs is set to 100 and we evaluate after every epoch and early stop with a
patience of 5. For the Wikiwiki pretraining step, we use the same hyper-
parameters but stop the model training after 2 epochs on the entire Wikiwiki
dataset. We did not perform explicit hyper-parameter tuning.

For the few-shot experiments, we take the zero-shot model trained by excluding
the given few-shot domain and fine-tune it on small set of annotated examples
for 1000 epochs, evaluating after every 25 epochs. All other parameters are
set to the same values as in the initial zero-shot training. We use a
multiplier of 0.1 for the augmented loss from the random known-domain data
batch. To speed up evaluation during training, we use teacher-forced sequence
accuracy as our validation metric, which doesn’t require us to perform any
beam search. During inference, we use beam search decoding with a beam size of
4.

We report exact match (EM) accuracy for the few-shot experiments, meaning the
entire predicted parse has to match exactly with the gold parse. For zero-
shot, we report both EM and F1 score since the task is more difficult and the
performance is generally lower. At these lower numbers, F1 score, which awards
partial credit to correctly tagged spans provides a better picture for
improvements than EM accuracy, which requires the entire predicted parse to be
correct for credit. For comparison wherever applicable, we use prior state-of-
the-art models as baselines. In addition, we also report the performance of a
vanilla Seq2Seq-Ptr model without concept pretraining.

##  4 Results and Discussion

In this section, we report and discuss the performance of Concept-Seq2Seq on
zero-shot and few-shot domain adaptation for semantic parsing. We first
briefly describe our findings in zero-shot setting and then describe findings
in a variety of few-shot scenarios.

###  4.1 Zero-shot Domain Adaptation

We report the first zero-shot performance numbers for domain adaptation on the
TOPv2 dataset. Table 1 contains these numbers. We observed that our model
produced decent predictions on four of the eight domains in the dataset, which
we document in the table. For the other domains, the scores were very low.
Concept-Seq2Seq achieves good F1 and EM Accuracy scores on the alarm domain
(71.00% F1 and 53.64% EM). On the weather and event domains, the concept
pretraining step helps it achieve decent EM scores around 20%. On the timer
domain, Concept-Seq2Seq achieves a fairly high F1 score (58.48%) but a very
low EM score (3.86%). Upon manual examination, we found that this was because
our model always skipped a certain tag. In the timer domain, there is a slot
tag called SL:METHOD_TIMER which tags the kind of timer such as _timer_ or
_stopwatch_. Our model never learns to tag these words with that slot. We
believe this is probably due to the description being inadequate for
performing the requisite task.

Overall, we believe the task at hand here is difficult due to the combination
of the zero-shot setting and the presence of specific nesting/parsing rules in
a compositional semantic parsing task. While a good amount of information can
be gleaned from the intent and slot names, our model has no access to any new
kinds of tagging rules since it has no annotated data or any descriptions of
those rules within the concept descriptions. The descriptions themselves are
sometimes inadequate as described with the timer domain above. We simply use
the descriptions from the dataset and they weren’t really designed to be used
to describe the entity being tagged. We leave exploration into better
descriptions and incorporating parsing rules without explicit annotations for
future work.

We also evaluated our model on the SNIPS dataset to compare Concept-Seq2Seq to
prior zero-shot approaches for flat slot-filling style datasets. We created a
strong baseline using recent NLP advancements such as pretrained transformers
and attention mechanisms on the slot-filling style zero-shot model proposed by
Bapna et al. (2017) and Lee and Jha (2019). Table 2 compares the performance
of Concept-Seq2Seq to this baseline. We observe that our model matches or
outperforms the slot-filling baseline on most domains while also being
adaptable to compositional datasets.

| Alarm | Event | Messaging | Music | Navigation | Reminder | Timer | Weather
---|---|---|---|---|---|---|---|---
Few-shot 1 SPIs
Seq2Seq-Ptr | 20.41 | 31.85 | 38.12 | 25.58 | 19.96 | 23.66 | 16.62 | 47.24
Inventory Desai et al. (2021) | 62.13 | 46.57 | 46.54 | 23.00 | 21.16 | 28.58 | 28.92 | 54.53
RAF Shrivastava et al. (2022) | 62.71 | - | - | 35.47 | - | - | 55.06 | 61.05
Concept-Seq2Seq w/o pretraining | 61.72 | 44.28 | 34.24 | 20.66 | 20.82 | 35.39 | 44.75 | 52.24
Concept-Seq2Seq | 64.71 | 54.42 | 46.13 | 36.30 | 30.00 | 36.93 | 53.44 | 54.68
Few-shot 5 SPIs
Seq2Seq-Ptr | 45.50 | 38.31 | 52.79 | 48.75 | 43.38 | 36.37 | 54.79 | 49.94
Inventory Desai et al. (2021) | 71.81 | 58.87 | 63.72 | 53.59 | 42.59 | 48.88 | 55.54 | 65.09
Concept-Seq2Seq w/o pretraining | 71.32 | 53.73 | 51.52 | 45.96 | 50.71 | 50.83 | 58.89 | 66.65
Concept-Seq2Seq | 74.17 | 61.72 | 61.20 | 51.24 | 56.76 | 54.36 | 63.13 | 68.54
Few-shot 25 SPIs
Seq2Seq-Ptr | - | - | - | - | - | 55.7 | - | 71.6
RINE Mansimov and Zhang (2021) | - | - | - | - | - | 68.71 | - | 74.53
Concept-Seq2Seq w/o pretraining | 78.16 | 68.21 | 75.28 | 65.54 | 67.67 | 67.92 | 70.72 | 74.30
Concept-Seq2Seq | 79.87 | 72.96 | 80.45 | 67.91 | 70.94 | 67.76 | 72.41 | 76.44
Reference - Fully trained
Concept-Seq2Seq | 88.07 | 83.23 | 93.11 | 79.47 | 81.63 | 79.57 | 77.33 | 90.73

Table 3: EM Accuracy scores of various models in few-shot settings on TOPv2.
We see that our model outperforms prior approaches on many domains and
settings, most notably in the 1 SPIs setting.

###  4.2 Few-shot Domain Adaptation

We evaluated Concept-Seq2Seq in few-shot settings of 1, 5, and 25 samples per
intent/slot. Table 3 reports the EM accuracy scores of Concept-Seq2Seq and
other recent baselines on TOPv2. We also report the performance of a fully
trained Concept-Seq2Seq model on all the training data for reference and to
show that the architecture of Concept-Seq2Seq is competitive with other state-
of-the-art methods in the full-resource setting.

We evaluated with a range of SPIs to allow for comparison with a wide range of
models focused on both extremely low-resource (1, 5 SPIs) and medium low-
resource (25 SPIs) settings. We report numbers for the baselines from their
original papers, so they are missing for some domains.

As shown in the table, Concept-Seq2Seq outperforms a vanilla Seq2Seq-Ptr,
Inventory (Desai et al., 2021), and Retrieve-and-Fill (RAF) Shrivastava et al.
(2022) models on most domains in the 1 SPIs setting. RAF scores are very close
and the approach outperforms our model on two domains but it uses additional
hand-crafted information such as handmade descriptions and examples for
intents and slots, as well as an intermediate scenario-bank to retrieve
templates from. Concept-Seq2Seq simply works off of the existing information
in the dataset. In the 5 SPIs setting, it again outperforms the vanilla
Seq2Seq-Ptr and Inventory models on most domains and on average. Inventory is
a similar model to ours where the lexical information from intents and slots
is used to help better transfer knowledge in the low-resource setting.
However, this information is prepended to the input sequence and this might
cause input size issues for large inventories. In the slightly higher resource
setting of 25 SPIs, Concept-Seq2Seq beats a vanilla Seq2Seq-Ptr model and
matches the performance of RINE Mansimov and Zhang (2021), reported on two
domains.

Across all the SPIs settings, we see that there is a noticeable drop in
performance of Concept-Seq2Seq without the Wikiwiki concept pretraining. This
shows the effectiveness of the pretraining step in helping the model
generalize to unseen concepts and domains better.

To wrap up our evaluation, we also report the performance of Concept-Seq2Seq
on SNIPS in the few-shot settings described above. Table 4 shows these
numbers. We can see that we almost catch up to a fully trained model by
training with just 25 SPIs in this dataset on most domains except music. We
believe the music domain probably requires a lot of samples to effectively
identify the diverse set of entities in the domain.

Overall, we find Concept-Seq2Seq to be a very promising approach which
achieves high performance scores in low resource domain adaptation. It is
capable of doing this in both compositional and flat semantic parsing
settings, without any additional hand-crafted information apart from the
little documentation in the dataset, and with the memory and inference latency
footprint of a vanilla Seq2Seq-Ptr model.

| Music | Book | Creative | Weather | Restaurant | Playlist | Screening
---|---|---|---|---|---|---|---
| F1 | EM | F1 | EM | F1 | EM | F1 | EM | F1 | EM | F1 | EM | F1 | EM
1 SPIs | 67.10 | 39.00 | 86.74 | 61.67 | 92.39 | 80.33 | 88.14 | 71.00 | 89.64 | 72.00 | 78.31 | 47.00 | 79.19 | 61.00
5 SPIs | 84.33 | 68.00 | 96.83 | 89.33 | 92.58 | 84.33 | 94.64 | 86.33 | 93.27 | 81.33 | 87.74 | 68.33 | 94.16 | 87.67
25 SPIs | 85.90 | 72.67 | 99.32 | 97.33 | 98.10 | 95.33 | 98.27 | 94.67 | 96.38 | 89.33 | 91.73 | 80.67 | 95.02 | 89.33
Fully-trained | 90.78 | 83.00 | 99.18 | 97.00 | 100.00 | 100.00 | 98.55 | 96.00 | 96.89 | 90.00 | 94.53 | 87.00 | 98.35 | 97.00

Table 4: Few-shot performance of Concept-Seq2Seq on domains in SNIPS. Our 25
SPIs model almost catches up to a fully trained model. Numbers are an average
of three runs with different random samples of SPIs.

##  5 Related Work

Zero-shot domain adaptation for task-oriented semantic parsing has been
previously explored for simple flat queries with single intents and disjoint,
non-overlapping slots. Bapna et al. (2017) and Lee and Jha (2019) encode the
lexical tag features and create a token-tagging schema to create the final
semantic parses. Yu et al. (2021) solve the task using a retrieve-and-fill
mechanism. Our baseline model for simple queries is based on these approaches.

For complex utterances with nested structures, zero-shot semantic parsing has
been explored using intermediate, concept-agnostic logical forms (Herzig and
Berant, 2018; Dong and Lapata, 2018; Reddy et al., 2017) or natural language
canonical forms (Wu et al., 2021). These approaches apply to semantic parsing
datasets which have context free grammars and specified rules, such as
database or knowledge graph queries. The effort to craft these grammars for
task-oriented semantic parsing in a voice assistant setting could quite
possibly be greater than annotating utterances.

A more relevant class of approaches for this work are ones that solve task-
oriented semantic parsing for complex utterances in a few-shot setting using
lexical tag features. Shrivastava et al. (2021) and Mansimov and Zhang (2021)
modify the seq2seq architecture from Rongali et al. (2020) to perform non-
autoregressive style decoding and show that their models perform better in a
few-shot setting. Ghoshal et al. (2020) use adaptive label smoothing, a model-
agnostic technique. Shin et al. (2021) proposed a prompting-style approach
where custom instructional prompts filled with handful of annotated examples
and an unsolved utterance are fed as input to GPT-3 to directly produce a
semantic parse. Their approach is extremely slow and cannot be easily adapted
into a zero-shot framework. Shrivastava et al. (2022) explore a retrieve-and-
fill style approach where they retrieve the best _scenario_ , an intermediate
logical form consisting of the semantic frame and abstracted out tags, from a
scenario bank of all supported semantic parses. Their approach is contingent
on the availability of this scenario bank which could possibly entail more
effort than annotating utterances. Mueller et al. (2022) and Desai et al.
(2021) use lexical features from intent and slot names to create an
_inventory_ and use it as input to train semantic parsers for new domains.
Mueller et al. (2022) also pretrain their model to improve generalizability
but only evaluate it on an intent classification task. Desai et al. (2021)
evaluate their model for full sequences and our model is similar to theirs.
However, we use our inventory to create custom decoder embeddings in a seq2seq
model, which removes any input size issues that their model will encounter
with large inventories. We also pretrain our model with Wikidata and evaluate
it in a completely zero-shot setting, in addition to few-shot. Zhao et al.
(2022) is another recent question-answering-based approach that uses lexical
features from the intent and slot tags by using them as context and posing
questions but it has a similar input size issue with large inventories.

##  6 Conclusion

We propose a model called Concept-Seq2Seq to perform low-resource domain
adaptation for compositional semantic parsing. Our model is built on the
Seq2Seq-Ptr framework and is augmented with a concept encoder to transfer
knowledge and encode unseen intents and slots from new domains through their
text definitions. We also propose a novel concept pretraining scheme to
incorporate general concept knowledge into our model using an entity-centric
Wikipedia dataset called Wikiwiki.

We evaluate our model in zero-shot and multiple few-shot settings on Facebook
TOPv2 and SNIPS datasets. We show that our model is capable of performing
zero-shot domain adaptation on some domains of the TOPv2 dataset and beats a
strong slot-filling baseline on the SNIPS dataset. In few-shot, over multiple
dataset sizes of 1, 5, and 25 SPIs, we show that our model outperforms many
strong prior models on TOPv2. Using the SNIPS dataset, we also demonstrate how
our model catches up to a fully-trained semantic parsing model using just 25
SPIs on most domains. Our model is capable of low-resource domain adaptation
in both compositional and flat parsing settings, without additional hand-
crafted information, and with the inference behavior of a vanilla Seq2Seq-Ptr
model.

## Acknowledgements

This work was supported in part by Amazon Alexa AI, in part by the Chan
Zuckerberg Initiative, in part by IBM Research AI through the AI Horizons
Network, and in part by the National Science Foundation (NSF) grants
IIS-1763618 and IIS-1955567. Any opinions, findings, conclusions, or
recommendations expressed in this material are those of the authors and do not
necessarily reflect those of the sponsors.

## Limitations

Our models were trained on GPUs that had at least 20GB on-board memory since
in addition to the traditional encoder and decoder components in a seq2seq
model, we also train a concept encoder, which is around the same size as the
encoder. This eliminates the use of popular GPUs such as 1080-ti and 2080-ti,
unless parameter freezing or other tricks are employed during training. During
inference however, once the concepts are trained, we can simply encode the
target tags and this reduces to the size and performance of a traditional
Seq2Seq-Ptr model.

We also report all results on models (ours and baselines) with _base_ -size
components such as _roberta-base_. We do this since these models are more
likely to be used in production than the much bigger _large_ -size models.
Results and comparison with _large_ -sized models is missing from this work
(we expect the trends shown to generalize) and we leave this to future work.

Finally, to simulate our low-resource experiments, we randomly sample a few
examples from the existing training datasets. While this is useful for
experimentation, it doesn’t truly mimic a real low-resource workflow where
these few examples could be carefully crafted by developers to ensure better
semantic coverage in terms of the language of the utterances. This work
doesn’t include any analysis on the influence of the content of the few
selected examples; it just focuses on their number.

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