# Neural Operators of Backstepping Controller and Observer
Gain Functions for Reaction-Diffusion PDEs

Miroslav Krstic<EMAIL_ADDRESS>Luke Bhan<EMAIL_ADDRESS>Yuanyuan Shi
<EMAIL_ADDRESS>University of California, San Diego, USA

###### Abstract

Unlike ODEs, whose models involve system matrices and whose controllers
involve vector or matrix gains, PDE models involve functions in those
roles—functional coefficients, dependent on the spatial variables, and gain
functions dependent on space as well. The designs of gains for controllers and
observers for PDEs, such as PDE backstepping, are mappings of system model
functions into gain functions. These infinite-dimensional nonlinear operators
are given in an implicit form through PDEs, in spatial variables, which need
to be solved to determine the gain function for each new functional
coefficient of the PDE. The need for solving such PDEs can be eliminated by
learning and approximating the said design mapping in the form of a neural
operator. Learning the neural operator requires a sufficient number of prior
solutions for the design PDEs, offline, as well as the training of the
operator. In recent work, we developed the neural operators for PDE
backstepping designs for first-order hyperbolic PDEs. Here we extend this
framework to the more complex class of parabolic PDEs. The key theoretical
question is whether the controllers are still stabilizing, and whether the
observers are still convergent, if they employ the approximate functional
gains generated by the neural operator. We provide affirmative answers to
these questions, namely, we prove stability in closed loop under gains
produced by neural operators. We illustrate the theoretical results with
numerical tests and publish our code on github. The neural operators are three
orders of magnitude faster in generating gain functions than PDE solvers for
such gain functions. This opens up the opportunity for the use of this neural
operator methodology in adaptive control and in gain scheduling control for
nonlinear PDEs.

, ,

##  1 Introduction

### ML as a tool for learning control methodologies.

In the recent manuscript [10] we introduced a learning-based control framework
which devises a new role for machine learning (ML): learn an entire control
design methodology, in the form of a mapping from the plant model to the
controller gains, or even to the control inputs.

This framework is neither model-free nor methodology-agnostic. On the
contrary, it is method-specific. For a particular method (LQR, pole placement,
MPC, backstepping, etc.), after a large number of training calculations of the
controller gains on a sample set of plant models, an ML-approximated mapping
of the method is learned. Once learned as a plant-to-gains mapping, the
control design for a new/next plant (outside of the training set) does not
require another solution of the design equations (Riccati, Bezout, etc.) but
merely entails an “evaluation of the learnt map” to obtain the control gains.

One would argue that no dire need exists for LQR or other linear finite-
dimensional designs for such a learning-based capability, where an entire
methodology is “encoded” into a neural mapping. The cost of the solution of a
design problem (say, a Riccati equation, even of a high dimension) is not
prohibitive, even online with current technology. Indeed, we are not motivated
by design challenges in finite dimensions but for PDEs.

In PDE control, the design problems are not matrix equations. They are PDEs
themselves (or harder problems, such as operator Riccati equations). Since the
infinite-dimensional state of a PDE is a function of spatial variables, the
controller gain is also a function of spatial variables. Finding the gain
typically entails solving a PDE in space (but not in time). It is therefore of
interest, in PDE control, to have a capability where producing the control
gain functions is just an evaluation of a neural mapping that has already
learned the design methodology on a large set of previously offline-solved
control design problems for a sample set of PDEs in a certain class.

### Neural operators for approximating mappings of functions into functions.

Just as the control designs for linear finite-dimensional systems are matrix-
to-matrix mappings ($A,B$ into gain $K$), control designs for PDEs are
function-to-function mappings (spatially-dependent coefficients into gains).
Our inspiration for encoding PDE control methodologies into machine learning
comes from recent advances in the mathematics of machine learning. Motivated
by the tasks of finding solution/flow maps (from the initial conditions into
future states) for physical PDEs (such as the difficult Navier-Stokes PDEs),
research teams led by George Karniadakis [55, 56], Anima Anandkumar and Andrew
Stuart [53, 54], and George Pappas and Manfred Morari [41, 73], have developed
neural approximation methods, termed “neural operators,” with provable
properties for nonlinear operators acting on functions and producing
functions. These approaches are not simply discretizing PDEs and finding
solution maps to the resulting large ODE solution problems. In the language of
distributed parameter systems, they are not “early lumping” methods of
learning solution maps. They approximate (non-discretized) function-to-
function nonlinear operators and provide guarantees of the accuracy of
approximation in terms of the required sizes of the training sets and neural
networks.

The value of such a capability in PDE control cannot be understated. With a
theoretically rigorous and numerically powerful capability like this, specific
PDE control methods, for specific classes of PDEs, can be learned once and
encoded as neural operators, ready to produce the control gain functions for
any new functional coefficients of the same classes of PDEs.

In a theoretically rigorous field like PDE control, a computational capability
with rigorous approximation guarantees has a value primarily if it allows the
retention of the theoretical properties proven for the “exact design”. This is
indeed what we show in the paper [10] in which we introduce the framework:
approximate neural operator representations of a particular PDE control
method—PDE backstepping—preserves its stability guarantees in spite of the
control gains not being generated by solving the design PDEs but by the gains
being generated from the learned “neural model” of PDE backstepping.

Figure 1: An algorithmic representation of our design paradigm of employing
neural operators in boundary control of PDEs. Three major step clusters are
performed: (1) derivation of the integral equations for the backstepping
kernels, performed only once; (2) learning of the mapping from the plant
parameter functions into the backstepping kernel functions, also performed
only once; and (3) implementation of the controller for specific plant
parameters. The task in the top box has been completed in [46, 77]. In this
paper, the task in the middle box is introduced and stability guarantees for
the task in the bottom box are provided.

### Extension of PDE backstepping neural operators from hyperbolic [10] to
parabolic PDEs.

Hyperbolic PDEs involve only the first derivatives in space and time. This
makes them (all else being equal) the “simplest” PDE class for control. Delay
systems combine ODEs with delays—the simplest form of a PDE. While the
simplest among PDEs, hyperbolic PDEs are not necessarily easy to control. They
can be unstable, with many unstable eigenvalues, and only one input acting at
the boundary of a domain. This mix of simplicity within the PDE family, with
the non-triviality for control, makes hyperbolic PDEs the ideal entry point
for any new study in PDE control, including the introduction of a new
framework for learning-based control in our [10]. The framework is depicted in
Figure 1. The learning and implementation portions of the framework in Figure
1 are depicted in Figure 2.

Figure 2: Stages (2) and (3) of the framework in Figure 1. TOP: The process of
learning the PDE backstepping design operator $\mathcal{K}:\lambda\mapsto k$
involves many solutions of a kernel PDE $k_{xx}-k_{yy}=\lambda k$ in the
Goursat form, for different functions $\lambda_{i}(x)$ and then training of a
neural operator $\hat{\mathcal{K}}:\lambda\mapsto\hat{k}$. BOTTOM: Feedback
implementation of PDE backstepping control with gain kernel $\hat{k}(1,x)$
generated by the DeepONet $\hat{\mathcal{K}}$.

Control design problems for hyperbolic PDEs are hyperbolic PDEs themselves,
namely, equations with only first derivatives in multiple spatial variables.
Parabolic PDEs, with their first derivative in time but second derivatives in
space, are the natural next challenge for learning the PDE backstepping
methodology using neural operators. This is what we undertake in this paper.
The chief difficulty with learning backstepping kernel operators for parabolic
PDEs is that the kernels are governed by second-order PDEs, which raises the
difficulty for solving such PDEs and for proving the sufficient smoothness of
their solutions so that the neural operator (NO) approximations have guarantee
of sufficient accuracy for preserving stabilization. At the intuitive level,
with more derivatives, and more boundary conditions, the nonlinear operator
from the reaction function to the gain in parabolic PDEs is a more complex
operator than the nonlinear operator from the recirculation function to the
gain in hyperbolic PDEs. There are hyperbolic cases where the kernel mapping
can be written (though not solved) using the Laplace transform. That is never
the case with parabolic PDEs; the design problem is never of spatial dimension
lower than two.

We consider parabolic PDE systems of the form

$\displaystyle u_{t}(x,t)$ $\displaystyle=$ $\displaystyle
u_{xx}(x,t)+\lambda(x)u(x,t),\qquad\mbox{$x\in[0,1)$}$ (1) $\displaystyle
u(0,t)$ $\displaystyle=$ $\displaystyle 0$ (2) $\displaystyle u(1,t)$
$\displaystyle=$ $\displaystyle U(t).$ (3)

Our goal is the design of a PDE backstepping boundary control

$U(t)=\int_{0}^{1}k(1,y)u(y,t)dy,$ (4)

as well as an observer with the (collocated) boundary sensing of $u_{x}(1,t)$.
By “design” we mean to find the gain function $k$ in the control law (4),
namely, to find the output $k$ of the function-to-function mapping
$\mathcal{K}:\lambda\mapsto k$, depicted in Figure 3. This paper’s objective
is to learn the design operator $\mathcal{K}$ with a neural operator
approximation $\hat{\mathcal{K}}$ (top of Figure 2) and to employ the
resulting approximate gain $\hat{k}$ in the control law (bottom of Figure 2).

actuation | opposite boundary | sensing |
---|---|---|---
$u(1,t)=U(t)$ | $u(0,t)=0$ | $u_{x}(0,t)$ | anti-col
$\boldsymbol{u(1,t)=U(t)}$ | $\boldsymbol{u(0,t)=0}$ | $\boldsymbol{u_{x}(1,t)}$ | col
$u(1,t)=U(t)$ | $u_{x}(0,t)=0$ | $u(0,t)$ | anti-col
$u(1,t)=U(t)$ | $u_{x}(0,t)=0$ | $u(1,t)$ | col
$u_{x}(1,t)=U(t)$ | $u(0,t)=0$ | $u_{x}(0,t)$ | anti-col
$u_{x}(1,t)=U(t)$ | $u(0,t)=0$ | $u_{x}(1,t)$ | col
$u_{x}(1,t)=U(t)$ | $u_{x}(0,t)=0$ | $u(0,t)$ | anti-col
$u_{x}(1,t)=U(t)$ | $u_{x}(0,t)=0$ | $u(1,t)$ | col
Table 1: Eight possible combinations of boundary actuation, sensing, and
boundary condition at the opposite end of $[0,1]$. We focus on the simplest
combination—in the second row.

Since parabolic PDEs in one dimension have two boundary conditions, and also
boundary actuation and boundary sensing can be employed at either boundary, a
total of sixteen combinations of boundary actuation, boundary sensing, and
boundary condition on the unactuated boundary are possible. Taking the
symmetry between the boundaries $x=0$ and $x=1$ into account, the total number
of truly distinct combinations is eight. They are listed in Table 1.

We are able to solve all eight problems but, in this paper, pursue the
simplest of the eight combinations for pedagogical reasons. The case with
Dirichlet boundary conditions, $u(0,t)=0,u(1,t)=U(t)$ is, notationally, the
simplest case. It allows the reader to most quickly grasp the utility and the
technical steps in employing neural operators in the control of parabolic
PDEs.

Figure 3: The PDE backstepping design operator ${\mathcal{K}}:\lambda\mapsto
k$, where $\lambda(x)$ is the spatially-varying reaction coefficient of the
PDE, whereas $k(x,y)$ is the kernel function of the backstepping
transformation, producing the feedback gain function $k(1,y)$ in the feedback
law $U(t)=\int_{0}^{1}k(1,y)u(y,t)dy$.

All the results in the paper—a full-state controller, an observer, and an
output-feedback law (as well as the seven additional combinations not pursued
in the paper)—can be extended to the more general class of parabolic PDE
systems,

$\displaystyle v_{t}(x,t)$ $\displaystyle=$
$\displaystyle\varepsilon(x)v_{xx}(x,t)+b(x)v_{x}(x,t)$ (5)
$\displaystyle+\lambda(x)v(x,t)+g(x)v(0,t)$
$\displaystyle+\int_{0}^{x}f(x,y)v(y,t)dy,\quad x\in(0,L).$

The domain $[0,L]$ is easily normalized to $[0,1]$, the diffusion
$\varepsilon(x)$ is easily normalized to unity, and the advection term
$b(x)v_{x}(x,t)$ is easily eliminated with a scaling transformation. We forego
pursuing this myriad of alternatives for pedagogical reasons—they are
overwhelming but standard. Likewise, we forgo the treatment of the terms
$g(x)v(0,t)+\int_{0}^{x}f(x,y)v(y,t)dy$ because it adds complications but it
is standard as well.

The parabolic PDE with unity diffusion and with spatially-varying reaction
$\lambda(x)$ is a perfect introduction to the possibilities that neural
operators present for PDE backstepping control where the computation of the
gain kernel $k(x,y)$ for each new $\lambda(x)$ can be avoided by developing a
neural operator approximation of the functional mapping (nonlinear operator)
${\mathcal{K}}:\lambda\mapsto k$, which is depicted in Figure 3.

We opt to present in the paper the results for the combination in the second
row of Table 1 because this combination allows us to “kill two birds with one
stone” in our exposition. For this particular actuator-sensor combination,
which is collocated (and the simplest of the four collocated combinations),
the same kernel is used to obtain the gain functions for both the controller
and the observer. This relieves the reader of the burden of following multiple
approximations of the kernel, multiple neural operators, multiple training
processes for those operators, and multiple theorems that guarantee the
approximability of those multiple operators. The concept of encoding the
methodologies of controller, observer, and output-feedback design into a
neural operator is grasped through a single operator
${\mathcal{K}}:\lambda\mapsto k$. And the duplication in the exposition is
avoided. A reader who possesses the skills in calculations and the stamina can
work out the remaining combinations in Table 1.

### PDE Backstepping.

Even though PDE backstepping was first developed for parabolic systems [77],
it is best to begin its study from the easier, hyperbolic case [46]. Control
of hyperbolic PDEs has grown into a rich area because, in the hyperbolic case,
one can stabilize a coupled system with fewer inputs than PDEs. A pair of
coupled hyperbolic PDEs was stabilized with a single boundary input in [18],
an extension to $n+1$ hyperbolic PDEs with a single input was introduced in
[28], an extension to $n+m$ PDEs with boundary actuation on $m$
“homodirectional” PDEs in [35, 36], an extension to cascades with ODEs in
[29], and an extension to “sandwiched” ODE-PDE-ODE systems in [94, 95].
Redesigns robust to delays are provided in [3]. PDE backstepping-based output-
feedback regulation with disturbances is proposed in [26, 27].

For parabolic PDEs, backstepping for full-state feedback stabilization was
developed in [77] and for observer design in [78]. A complex extension from
linear to nonlinear parabolic PDEs, using infinite Volterra series, was
provided in [88, 89]. Backstepping was combined with differential flatness in
[61]. The first solutions to the null-controllability problem for parabolic
PDEs were provided, using backstepping, in [19, 31]. Sampled-data and event-
triggered versions of backstepping for parabolic PDEs appeared in [39, 38, 30,
71]. Work on cascades of parabolic PDEs with other systems has included heat-
ODE cascades [43, 5], delay-parabolic cascades [44], and ODE-heat-ODE sandwich
systems [93]. A backstepping design for a moving-boundary PDE-ODE Stefan
system was presented in [42]. Coupled parabolic PDEs introduce special
challenges and have been tackled in [4, 65, 90]. Extensions from multiple 1D
parabolic PDEs to PDEs in 2D and higher dimensions, such as in the book [60]
are arguably even more challenging and have been pursued for Navier-Stokes and
magnetohydrodynamic systems in [87, 92] on channel domains, as well as for
reaction-diffusion systems on balls of arbitrary dimensions [91]. Adaptive
control designs for parabolic PDEs were introduced in [45, 79, 80], extended
in [40], and extended to the hyperbolic case in [8]. For coupled hyperbolic
PDEs with unknown parameters, the parabolic designs in [45, 79, 80] inspired a
comprehensive collection of adaptive control designs in the book [1].
Applications of backstepping to PDE models of traffic are introduced in [96,
97].

### Advances in learning-based control.

What we present here is one more among many directions in learning-based
control. For the benefit of the reader from PDE control, we highlight a few
results from this vast and growing literature. Stability of learning-based MPC
was established in [2, 72] and followed, for nonlinear systems, by efforts on
joint learning of the controller and(or) Lyapunov functions [21, 13, 14, 15,
22]. In addition, [83, 64] have explored how learning-based control affects
systems with known Lyapunov functions, [12, 68, 23] studied learning of
stability certificates and stable controllers from data, and [6] developed a
provably stable data-driven algorithm based on system measurements and prior
system knowledge.

For reinforcement learning (RL) [9], the focus has been on learning the system
dynamics and providing closed-loop guarantees in _finite-time_ for both linear
[24, 16, 48] and nonlinear systems [7, 37, 76, 49]. For model-free RL, [32,
62, 66, 100] proved the convergence of policy optimization to the optimal
controller for LTI systems, [67, 63] for LTV systems, [82] for partially
observed linear systems. For a review of policy optimization (PO) methods for
LQR, $H_{\infty}$ control, risk-sensitive control, LQG, and output feedback
synthesis, see [34]. For nonlinear systems, [17, 20, 75] investigated PO with
stability guarantees from CLFs. In addition to PO, [85, 51, 11, 84] proved
stability and convergence of actor-critic methods [85, 51] and Q-learning
[84]. In CPS, learning-based control was developed for partially observable
systems [57].

Learning-based control in games and for MAS is pursued in [99, 59, 33, 98, 58,
69, 86, 70]. Convergence is shown to Nash equilibria in zero-sum linear
quadratic games [99], continuous games [59], Stackelberg games [33], Markov
games [98, 58], and multi-agent learning over networked systems [70, 69].

We focus on learning-based control for PDE systems. In our previous work [74],
we demonstrate the empirical success of using NOs for accelerating PDE
backstepping observers. Our recent work [10] represents the first step towards
using NOs for _provably_ bypassing gain computations and directly learning the
controller with closed-loop stabilization guarantee, in hyperbolic PDE
systems.

### Neural operators—a brief summary.

Neural operators are neural network-parameterized maps for learning
relationships between function spaces. They consist of three components: an
encoder, an approximator, and a reconstructor [50]. The encoder is an
interpolation from an infinite-dimensional function space to a finite-
dimensional vector representation. The approximator aims to mimic the infinite
map using a finite-dimensional representation of both the domain function
space and the target function space. The reconstructor then transforms the
approximation output into the infinite-dimensional target function space. The
implementation of both the approximator and the reconstructor is generally
coupled and can take many forms. For example, the original DeepONet [56]
contains a “branch” net that represents the approximation network and a
“trunk” net that builds a basis for the target function space. The outputs of
the two networks are then taken in linear combination with each other to form
the operator. FNO [54] utilizes the approximation network in the Fourier
domain where the reconstruction is done on a basis of the trigonometric
polynomials. LOCA [41] integrates the approximation network and reconstruction
step with a unified attention mechanism. NOMAD [73] extends the linear
reconstructor map in DeepONet to a nonlinear map that is capable of learning
on nonlinear submanifolds in function spaces.

With the basic notions and notation for NOs given in Appendix A, we state next
the key technical result that enables our use of NOs to learn the PDE
backstepping kernel mappings. The result is quoted in its general/abstract
form. It is specialized to the PDE control setting in our Theorem 4.

######  Theorem 1.

(DeepONet universal approximation theorem [25, Theorem 2.1]). Let
$X\subset\mathbb{R}^{d_{x}}$ and $Y\subset\mathbb{R}^{d_{y}}$ be compact sets
of vectors $x\in X$ and $y\in Y$, respectively. Let
${\mathcal{U}}:X\rightarrow U\subset\mathbb{R}^{d_{u}}$ and
${\mathcal{V}}:Y\rightarrow V\subset\mathbb{R}^{d_{v}}$ be sets of continuous
functions $u(x)$ and $v(y)$, respectively. Let ${\mathcal{U}}$ be also
compact. Assume the operator
$\mathcal{G}:{\mathcal{U}}\rightarrow{\mathcal{V}}$ is continuous. Then, for
all $\epsilon>0$, there exist $m^{*},p^{*}\in\mathbb{N}$ such that for each
$m\geq m^{*}$, $p\geq p^{*}$, there exist $\theta^{(k)},\vartheta^{(k)}$,
neural networks
$f^{\mathcal{N}}(\cdot;\theta^{(k)}),g^{\mathcal{N}}(\cdot;\vartheta^{(k)}),k=1,\ldots,p$,
and $x_{j}\in X,j=1,\ldots,m$, with corresponding
$\mathbf{u}_{m}=(u(x_{1}),u(x_{2}),\cdots,u(x_{m}))^{\rm T}$, such that

$|\mathcal{G}(u)(y)-\mathcal{G}_{\mathbb{N}}(\mathbf{u}_{m})(y)|<\epsilon$ (6)

for all functions $u\in{\mathcal{U}}$ and all values $y\in Y$ of
${\mathcal{G}}(u)\in{\mathcal{V}}$.

In the sequel, we denote the DeepONet neural operator values
$\mathcal{G}_{\mathbb{N}}(\mathbf{u}_{m})(y)$ compactly as
$\hat{\mathcal{G}}(u)(y)$ and the operators themselves as $\hat{\mathcal{G}}$.

### Paper outline and contributions.

In Section 2 we recap the basic PDE backstepping approach from [77]. Recalling
in Section 3 the twice continuous differentiability of the backstepping kernel
function we establish the existence of a neural operator with an arbitrary
accuracy for a set of continuously differentiable reaction coefficients not
exceeding a certain size in the supremum norm. Sections 4 and 5 contain our
main results. In Section 4 we prove the stability of a feedback law employing
a DeepONet approximation of the backstepping gain. In Section 5 we prove the
convergence of a backstepping observer that employs a DeepONet approximation
of the observer gain. In Section 6 we combine the DeepONet-based full-state
feedback and observer, to obtain a DeepONet-based output feedback controller
with an actuator-sensor pair collocated at the $x=1$ boundary. In Section 7 we
illustrate the theoretical results with numerical tests.

This paper’s contribution relative to the inaugural work on backstepping for
parabolic PDEs [77] is in providing a methodology for capturing the
backstepping design in the form of a neural operator and avoiding the need for
the solution of kernel PDEs, after the neural operator is once synthesized.
This capability is highly valuable in future work in the adaptive control of
parabolic PDEs and gain-scheduling for semilinear parabolic PDEs. In relation
to our recent work on neural operator approximated backstepping control of
hyperbolic PDEs, this paper extends this methodology, including stability
guarantees, to a more difficult class of PDE systems and kernel operators.
Additionally, compared to [10] where only full-state feedback is considered,
in this paper, we solve problems in observer design and output-feedback
control, with a convergence guarantee for the DeepONet-approximated
backstepping observer.

##  2 Basic Backstepping Design for Reaction-Diffusion PDE

We employ the following backstepping transformation,

$\displaystyle w(x,t)=u(x,t)-\int_{0}^{x}k(x,y)u(y,t)dy,$ (7)

to convert (1), (2), (3) into the target system

$\displaystyle w_{t}$ $\displaystyle=$ $\displaystyle w_{xx}$ (8)
$\displaystyle w(0,t)$ $\displaystyle=$ $\displaystyle 0$ (9) $\displaystyle
w(1,t)$ $\displaystyle=$ $\displaystyle 0$ (10)

with the help of feedback (4). We could as well pursue the target system
$w_{t}=w_{xx}-cw,c>0$, but we forego this design flexibility for the sake of
simplicity.

To convert (1), (2), (3) into (8), (9), (10), $k$ needs to satisfy

$\displaystyle k_{xx}(x,y)-k_{yy}(x,y)$ $\displaystyle=$
$\displaystyle\lambda(y)k(x,y),\quad\forall(x,y)\in\breve{\mathcal{T}}$ (11)
$\displaystyle k(x,0)$ $\displaystyle=$ $\displaystyle 0$ (12) $\displaystyle
k(x,x)$ $\displaystyle=$ $\displaystyle-{1\over 2}\int_{0}^{x}\lambda(y)dy$
(13)

where $\breve{\mathcal{T}}=\\{0<y\leq x<1\\}$ and ${\mathcal{T}}=\\{0\leq
y\leq x\leq 1\\}$.

The following assumption is important.

######  Assumption 2.

$\lambda\in C^{1}([0,1])$.

##  3 Accuracy of Approximation of Backstepping Kernel Operator with DeepONet

######  Theorem 3.

(proven in [77, 81]) For every $\lambda\in C^{1}([0,1])$, the PDE system (11),
(12), (13) has a unique $C^{2}({\mathcal{T}})$ solution with the property

$|k(x,y)|\leq\bar{\lambda}{\rm e}^{2\bar{\lambda}x},$ (14)

for all $x\in[0,1]$, where $\bar{\lambda}=\sup_{x\in[0,1]}|\lambda(x)|$.

This theorem is proven by representing the PDE system (11), (12), (13) as an
integral equation

$\displaystyle G(\xi,\eta)$ $\displaystyle=$ $\displaystyle-{1\over
4}\int_{\eta}^{\xi}\lambda\left({s\over 2}\right)ds$ (15)
$\displaystyle+{1\over
4}\int_{\eta}^{\xi}\int_{0}^{\eta}\lambda\left({\sigma-s\over
2}\right)G(\sigma,s)dsd\sigma,$

where

$\displaystyle\xi=x+y,\ \ \eta=x-y,$ $\displaystyle x={\xi+\eta\over 2},\ \
y={\xi-\eta\over 2}\ \ $ (16) $\displaystyle G(\xi,\eta)=k(x,y)$
$\displaystyle=$ $\displaystyle k\left({\xi+\eta\over 2},{\xi-\eta\over
2}\right).$ (17)

The change of variables (16) converts the domain ${\mathcal{T}}$ for $(x,y)$
into the larger triangular domain ${\mathcal{T}}_{1}=\\{0\leq\eta\leq\xi\leq
1\\}\cup\\{1\leq\xi\leq 2-\eta\leq 2\\}$ for $(\xi,\eta)$. The integral
equation (15) is one of the useful approaches in generating solutions for
$k(x,y)$ for the purpose of training the neural approximation of the operator
$\lambda\mapsto k$.

Next, denote the set of functions

$\underline{K}=\left\\{\left.k\in C^{2}({\mathcal{T}})\right|k(x,0)=0,\forall
x\in[0,1]\right\\}$ (18)

and let the operator ${\mathcal{K}}:C^{1}[0,1]\rightarrow\underline{K}$ be
defined by

$k(x,y)=:{\mathcal{K}}(\lambda)(x,y).$ (19)

Additionally, let the operator
${\mathcal{M}}:C^{1}[0,1]\rightarrow\underline{K}\times C^{1}[0,1]\times
C^{0}({\mathcal{T}})$ be defined by

$\left(k(x,y),\kappa_{1}(x),\kappa_{2}(x,y)\right)=:{\mathcal{M}}(\lambda)(x,y),$
(20)

where

$\displaystyle\kappa_{1}(x)$ $\displaystyle=$ $\displaystyle 2{d\over
dx}\left(k(x,x)\right)+\lambda(x)$ (21) $\displaystyle\kappa_{2}(x,y)$
$\displaystyle=$ $\displaystyle k_{xx}(x,y)-k_{yy}(x,y)-\lambda(y)k(x,y).$
(22)

Based on Theorem 3, $\mathcal{M}$ is a continuous operator. By applying
Theorem 1, we get the following key result for the approximation of a
backstepping kernel by a DeepONet (top of Figure 2).

######  Theorem 4.

For all $B_{\lambda},B_{\lambda^{\prime}}>0$ and $\epsilon>0$, there exists a
neural operator $\hat{\mathcal{M}}$ such that, for all
$(x,y)\in{\mathcal{T}}$,

$\left|{\mathcal{M}}(\lambda)(x,y)-\hat{\mathcal{M}}(\lambda)(x,y)\right|<\epsilon$
(23)

holds for all Lipschitz $\lambda$ with the properties that
$\|\lambda\|_{\infty}\leq B_{\lambda},\|\lambda^{\prime}\|_{\infty}\leq
B_{\lambda^{\prime}}$, namely, there exists a neural operator
$\hat{\mathcal{K}}$ such that $\hat{\mathcal{K}}(\lambda)(x,0)\equiv 0$ and

$\displaystyle\left|{\mathcal{K}}(\lambda)(x,y)-\hat{\mathcal{K}}(\lambda)(x,y)\right|$
$\displaystyle+\left|2{d\over
dx}\left({\mathcal{K}}(\lambda)(x,x)-\hat{\mathcal{K}}(\lambda)(x,x)\right)\right|$
$\displaystyle+\left|\left(\partial_{xx}-\partial_{yy}\right)\left({\mathcal{K}}(\lambda)(x,y)-\hat{\mathcal{K}}(\lambda)(x,y)\right)\right.$
$\displaystyle\left.-\lambda(y)\left({\mathcal{K}}(\lambda)(x,y)-\hat{\mathcal{K}}(\lambda)(x,y)\right)\right|<\epsilon.$
(24)

##  4 Stabilization under DeepONet Gain Feedback

The following theorem establishes the properties of the feedback system at the
bottom of Figure 2.

######  Theorem 5.

Let $B_{\lambda},B_{\lambda^{\prime}}>0$ be arbitrarily large and consider the
system (1), (2), (3) with any $\lambda\in C^{1}([0,1])$ whose derivative
$\lambda^{\prime}$ is Lipschitz and which satisfies $\|\lambda\|_{\infty}\leq
B_{\lambda}$ and $\|\lambda^{\prime}\|_{\infty}\leq B_{\lambda^{\prime}}$.
There exists a sufficiently small
$\epsilon^{*}(B_{\lambda},B_{\lambda^{\prime}})>0$ such that the feedback law

$U(t)=\int_{0}^{1}\hat{k}(1,y)u(y,t)dy,$ (25)

with all NO gain kernels $\hat{k}=\hat{\mathcal{K}}(\lambda)$ of approximation
accuracy $\epsilon\in(0,\epsilon^{*})$ in relation to the exact backstepping
kernel $k={\mathcal{K}}(\lambda)$ ensures that the closed-loop system
satisfies the exponential stability bound

$\|u(t)\|\leq M{\rm e}^{-(t-t_{0})/2}\|u_{0}\|,\quad\forall t\geq t_{0},$ (26)

where

$M(\epsilon,\bar{\lambda})=\left(1+\bar{\lambda}{\rm
e}^{2\bar{\lambda}}\right)\left(1+\bar{\lambda}{\rm
e}^{2\bar{\lambda}}+\epsilon\right){\rm e}^{\bar{\lambda}{\rm
e}^{2\bar{\lambda}}+\epsilon}.$ (27)

Approximate backstepping transform and perturbed target system. Take the
backstepping transformation

$\hat{w}(x,t)=u(x,t)-\int_{0}^{x}\hat{k}(x,y)u(y,t)dy,$ (28)

where $\hat{k}=\hat{\mathcal{K}}(\lambda)$. With the control law (25), the
target system becomes

$\displaystyle\hat{w}_{t}(x,t)$ $\displaystyle=$
$\displaystyle\hat{w}_{xx}(x,t)+\delta_{k0}(x)u(x,t)$ (29)
$\displaystyle+\int_{0}^{x}\delta_{k1}(x,y)u(y,t)dy$
$\displaystyle\hat{w}(0,t)$ $\displaystyle=$ $\displaystyle 0$ (30)
$\displaystyle\hat{w}(1,t)$ $\displaystyle=$ $\displaystyle 0,$ (31)

with

$\displaystyle\delta_{k0}(x)$ $\displaystyle=$ $\displaystyle 2{d\over
dx}\left(\hat{k}(x,x)\right)+\lambda(x)$ (32) $\displaystyle=$
$\displaystyle-2{d\over dx}\left(\tilde{k}(x,x)\right)$
$\displaystyle\delta_{k1}(x,y)$ $\displaystyle=$
$\displaystyle\partial_{xx}\hat{k}(x,y)-\partial_{yy}\hat{k}(x,y)-\lambda(y)\hat{k}(x,y)$
(33) $\displaystyle=$
$\displaystyle-\partial_{xx}\tilde{k}(x,y)+\partial_{yy}\tilde{k}(x,y)+\lambda(y)\tilde{k}(x,y),$

where

$\tilde{k}=k-\hat{k}={\mathcal{K}}(\lambda)-\hat{\mathcal{K}}(\lambda).$ (34)

With (4), we get

$\displaystyle\|\delta_{k0}\|_{\infty}$ $\displaystyle\leq$
$\displaystyle\epsilon$ (35) $\displaystyle\|\delta_{k1}\|_{\infty}$
$\displaystyle\leq$ $\displaystyle\epsilon.$ (36)

Inverse approximate backstepping transformation. Since the state $u$ appears
under the integral in the $\hat{w}$-system (29), in the Lyapunov analysis we
need the inverse backstepping transformation

$u(x,t)=\hat{w}(x,t)+\int_{0}^{x}\hat{l}(x,y)\hat{w}(y,t)dy.$ (37)

It is shown in [47] that the direct and inverse backstepping kernels satisfy
in general the relationship

$\hat{l}(x,y)=\hat{k}(x,y)+\int_{y}^{x}\hat{k}(x,\xi)\hat{l}(\xi,y)dy.$ (38)

The inverse kernel satisfies the following conservative bound

$\|\hat{l}\|_{\infty}\leq\|\hat{k}\|_{\infty}{\rm e}^{\|\hat{k}\|_{\infty}}.$
(39)

Since $\|k-\hat{k}\|_{\infty}<\epsilon$, we have that
$\|\hat{k}\|_{\infty}\leq\|k\|_{\infty}+\epsilon$. With (14) we get

$\displaystyle\|\hat{k}\|_{\infty}$ $\displaystyle\leq$
$\displaystyle\bar{k}+\epsilon$ (40) $\displaystyle\bar{k}(\bar{\lambda})$
$\displaystyle:=$ $\displaystyle\bar{\lambda}{\rm e}^{2\bar{\lambda}},$ (41)

and hence

$\|\hat{l}\|_{\infty}\leq\left(\bar{\lambda}{\rm
e}^{2\bar{\lambda}}+\epsilon\right){\rm e}^{\bar{\lambda}{\rm
e}^{2\bar{\lambda}}+\epsilon}.$ (42)

Lyapunov analysis. The Lyapunov functional

$V={1\over 2}\|\hat{w}\|^{2}$ (43)

has a derivative

$\dot{V}=-\|\hat{w}_{x}\|^{2}+\Delta_{0}+\Delta_{1},$ (44)

where

$\displaystyle\Delta_{0}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\hat{w}(x,t)\delta_{k0}(x)u(x,t)dx$ (45)
$\displaystyle\Delta_{1}(t)$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\hat{w}(x,t)\int_{0}^{x}\delta_{k1}(x,y)u(y,t)dydx.$
(46)

With several straightforward majorizations, we get

$\displaystyle\Delta_{0}$ $\displaystyle\leq$
$\displaystyle\|\delta_{k0}\|_{\infty}\left(1+\|\hat{l}\|_{\infty}\right)\|\hat{w}\|^{2}$
(47) $\displaystyle=$
$\displaystyle\|\delta_{k0}\|_{\infty}\left(1+\|\hat{l}\|_{\infty}\right)2V.$

and

$\displaystyle\Delta_{1}$ $\displaystyle=$
$\displaystyle\int_{0}^{1}\hat{w}(x)\int_{0}^{y}\hat{w}(y)\int_{y}^{x}\delta_{k}(x,\sigma)\hat{l}(\sigma,y)d\sigma
dydx$ (48)
$\displaystyle+\int_{0}^{1}\hat{w}(x)\int_{0}^{x}\delta(x,y)\hat{w}(y)dydx$
$\displaystyle\leq$
$\displaystyle\|\delta_{k1}\|_{\infty}\left(1+\|\hat{l}\|_{\infty}\right)\|\hat{w}\|^{2}$
$\displaystyle=$
$\displaystyle\|\delta_{k1}\|_{\infty}\left(1+\|\hat{l}\|_{\infty}\right)2V.$

From (44), (47), (48), (42), and Poincare’s inequality, we get

$\dot{V}\leq-{1\over 2}(1-\delta^{*})V,$ (49)

where

$\delta^{*}(\epsilon,\bar{\lambda})=2\epsilon\left(1+\bar{\lambda}{\rm
e}^{2\bar{\lambda}}+\epsilon\right){\rm e}^{\bar{\lambda}{\rm
e}^{2\bar{\lambda}}+\epsilon}$ (50)

is an increasing function of $\epsilon,\bar{\lambda}$, with the property that
$\delta^{*}(0,\bar{\lambda})=0$. Hence, there exists
$\epsilon^{*}(\bar{\lambda})$ such that, for all
$\epsilon\in[0,\epsilon^{*}]$,

$\dot{V}\leq-{1\over 4}V,$ (51)

namely, $V(t)\leq V_{0}{\rm e}^{-(t-t_{0})/4}$. From the direct and inverse
backstepping transformations it follows that

${1\over
1+\|\hat{l}\|_{\infty}}\|u\|\leq\sqrt{2V}\leq\left(1+\|\hat{k}\|_{\infty}\right)\|u\|.$
(52)

In conclusion,

$\|u(t)\|\leq\left(1+\|\hat{l}\|_{\infty}\right)\left(1+\|\hat{k}\|_{\infty}\right){\rm
e}^{-(t-t_{0})/2}\|u_{0}\|.$ (53)

With (40), (41), (42), the proof is completed. ∎

##  5 Observer Design

State estimators (observers) with boundary measurements can be formulated with
four measurement choices on the interval $[0,1]$: the measured quantities can
be $u(0,t),u_{x}(0,t),u(1,t),u_{x}(1,t)$. That leads to many possible problem
formulations. The possibilities multiply once we note that, on the opposite
boundary from the one at which measurement is conducted, one can have either a
Dirichlet or Neumann (or even Robin) boundary condition. Our objective in this
paper is not to solve all the possible problems. We are concerned only with
illustrating how NOs can be combined with PDE observers. Hence, our choice
among the many possibilities is the simplest choice, of the highest
pedagogical value.

Since our goals with observers are twofold—to estimate the unmeasured state
but also to use it in output-feedback control for stabilization—our choice of
measurement needs to be consistent with the actuation choice we have already
pursued in this note, namely, Dirichlet actuation of $u(1,t)=U(t)$. So, we
cannot use $u(1,t)$ for measurement but we can use
$u(0,t),u_{x}(0,t),u_{x}(1,t)$. We make the last among these three choices. We
let the output $u_{x}(1,t)$ be measured.

Figure 4: The PDE backstepping observer (54), (55), (56) uses boundary
measurement of the flux $u_{x}(1,t)$. The gain $\hat{k}(1,t)$ is produced with
the DeepONet $\hat{\mathcal{K}}$.

Our choice of $u_{x}(1,t)$ for measurement, as indicated in the observer
diagram in Figure 4, is motivated by the fact that, with this measurement, an
observer can be built using the same kernel $k(x,y)$ as for the control law.
In other words, with a single training of a neural operator
$\hat{\mathcal{K}}$, we obtain gains for both a controller and an observer—we
kill two birds (pedagogically speaking) with one stone. We don’t have to
expend an undue amount of the reader’s effort on the verification of the
conditions of the DeepONet approximation theorem. It is enough for the reader
to see once how this is done. The rest of the effort is better spent on
illustrating the uses of this approximation capability in observers and
output-feedback controllers.

######  Theorem 6.

Let $B_{\lambda},B_{\lambda^{\prime}}>0$ be arbitrarily large and consider the
system (1), (2), (3) with any $\lambda\in C^{1}([0,1])$ whose derivative
$\lambda^{\prime}$ is Lipschitz and which satisfies $\|\lambda\|_{\infty}\leq
B_{\lambda}$ and $\|\lambda^{\prime}\|_{\infty}\leq B_{\lambda^{\prime}}$.
There exists a sufficiently small
$\epsilon^{*}(B_{\lambda},B_{\lambda^{\prime}})>0$ such that the observer

$\displaystyle\hat{u}_{t}(x,t)$ $\displaystyle=$
$\displaystyle\hat{u}_{xx}(x,t)+\lambda(x)\hat{u}(x,t)$ (54)
$\displaystyle-\hat{k}(1,x)\left[u_{x}(1,t)-\hat{u}_{x}(1,t)\right]$
$\displaystyle\hat{u}(0,t)$ $\displaystyle=$ $\displaystyle 0$ (55)
$\displaystyle\hat{u}(1,t)$ $\displaystyle=$ $\displaystyle U(t),$ (56)

with all NO gain kernels $\hat{k}=\hat{\mathcal{K}}(\lambda)$ of approximation
accuracy $\epsilon\in(0,\epsilon^{*})$ in relation to the exact backstepping
kernel $k={\mathcal{K}}(\lambda)$ ensure that the observer error system, for
all $u_{0},\hat{u}_{0}\in L^{2}[0,1]$, satisfies the exponential stability
bound

$\|u(t)-\hat{u}(t)\|\leq M{\rm
e}^{-(t-t_{0})/2}\|u_{0}-\hat{u}_{0}\|,\quad\forall t\geq t_{0},$ (57)

where $M(\epsilon,\bar{\lambda})$ is defined in (27).

We start by postulating a PDE backstepping observer in the form

$\displaystyle\hat{u}_{t}(x,t)$ $\displaystyle=$
$\displaystyle\hat{u}_{xx}(x,t)+\lambda(x)\hat{u}(x,t)$ (58)
$\displaystyle+p_{1}(x)\left[u_{x}(1,t)-\hat{u}_{x}(1,t)\right]$
$\displaystyle\hat{u}(0,t)$ $\displaystyle=$ $\displaystyle 0$ (59)
$\displaystyle\hat{u}(1,t)$ $\displaystyle=$ $\displaystyle U(t).$ (60)

The observer error $\tilde{u}(x,t)=u(x,t)-\hat{u}(x,t)$ is governed by the
system

$\displaystyle\tilde{u}_{t}(x,t)$ $\displaystyle=$
$\displaystyle\tilde{u}_{xx}(x,t)+\lambda(x)\tilde{u}(x,t)$ (61)
$\displaystyle-p_{1}(x)\tilde{u}_{x}(1,t)$ $\displaystyle\tilde{u}(0,t)$
$\displaystyle=$ $\displaystyle 0$ (62) $\displaystyle\tilde{u}(1,t)$
$\displaystyle=$ $\displaystyle 0.$ (63)

The backstepping transformation

$\tilde{u}(x,t)=\tilde{w}(x,t)-\int_{x}^{1}p(x,y)\tilde{w}(y,t)dy$ (64)

converts (61), (62), (63) into

$\displaystyle\tilde{w}_{t}(x,t)$ $\displaystyle=$
$\displaystyle\tilde{w}_{xx}(x,t)$ (65) $\displaystyle\tilde{w}(0,t)$
$\displaystyle=$ $\displaystyle 0$ (66) $\displaystyle\tilde{w}(1,t)$
$\displaystyle=$ $\displaystyle 0$ (67)

provided $p(x,y)$ satisfies

$p(x,y)=k(y,x)$ (68)

with $k$ that is governed by (11), (12), (13), and with the observer gain

$p_{1}(x)=-k(1,x).$ (69)

It is crucial to note in (68) that the arguments $x$ and $y$ have been
commuted in $k(\cdot,\cdot)$. The commuting of the spatial arguments of the
backstepping kernel is akin to transposing matrices in going between designs
for controllers and observers in finite-dimensional LTI systems. The commuted
order of the arguments $x$ and $y$ continues in the rest of the proof. The
observer (58), (59), (60) is next rewritten as

$\displaystyle\hat{u}_{t}(x,t)$ $\displaystyle=$
$\displaystyle\hat{u}_{xx}(x,t)+\lambda(x)\hat{u}(x,t)$ (70)
$\displaystyle-k(1,x)\left[u_{x}(1,t)-\hat{u}_{x}(1,t)\right]$
$\displaystyle\hat{u}(0,t)$ $\displaystyle=$ $\displaystyle 0$ (71)
$\displaystyle\hat{u}(1,t)$ $\displaystyle=$ $\displaystyle U(t)$ (72)

and the transformation (64) as

$\tilde{u}(x,t)=\tilde{w}(x,t)-\int_{x}^{1}k(y,x)\tilde{w}(y,t)dy.$ (73)

Theorem 4 applies to the kernel $k(y,x)$ of the observer backstepping
transformation and the observer gains $-k(1,x)$. The observer (70), (71), (72)
is henceforth implemented with the approximate kernel $\hat{k}$ as in (54),
(55), (56) whereas the backstepping transformation (73) is applied with
$\hat{k}$ as

$\tilde{u}(x,t)=\omega(x,t)-\int_{x}^{1}\hat{k}(y,x)\omega(y,t)dy.$ (74)

The target system under the approximate kernel $\hat{k}$ becomes

$\displaystyle\omega_{t}(x,t)$ $\displaystyle=$
$\displaystyle\omega_{xx}(x,t)+\Omega_{0}(x,t)+\Omega_{1}(x,t)$ (75)
$\displaystyle\omega(0,t)$ $\displaystyle=$ $\displaystyle 0$ (76)
$\displaystyle\omega(1,t)$ $\displaystyle=$ $\displaystyle 0,$ (77)

where

$\displaystyle\Omega_{0}(x,t)$ $\displaystyle=$
$\displaystyle\delta_{k0}(x)\omega(x,t)+\int_{x}^{1}\hat{l}(y,x)\delta_{k0}(y)\omega(y,t)dy$
(78) $\displaystyle\Omega_{1}(x,t)$ $\displaystyle=$
$\displaystyle\int_{x}^{1}\left(\delta_{k1}(y,x)\omega(y,t)\right.$ (79)
$\displaystyle\left.+\hat{l}(y,x)\int_{y}^{1}\delta_{k1}(s,y)\omega(s,t)ds\right)dy$

and $\delta_{k0},\delta_{k1}$ are defined in (32), (33), with bounds (35),
(36). Note that the arguments in $\delta_{k1}$ have been commuted in the
integral in (75). Similar as in the proof of Theorem 5, the Lyapunov
functional

$V={1\over 2}\|\omega\|^{2}$ (80)

has a derivative

$\dot{V}\leq-{1\over 4}V,$ (81)

namely, $V(t)\leq V_{0}{\rm e}^{-(t-t_{0})/4}$, provided
$\epsilon\in[0,\epsilon^{*}]$, with $\epsilon^{*}$ obtained from (50). The
result (54) follows from (80), (74), (40), and the inverse backstepping
transformation

$\omega(x,t)=\tilde{u}(x,t)+\int_{x}^{1}\hat{l}(y,x)\tilde{u}(y,t)dy$ (82)

whose kernel $\hat{l}$ satisfies the bound (42). ∎

##  6 Collocated Output-Feedback Stabilization

In this section we put together the observer (54), (55), (56), along with the
observer-based controller

$U(t)=\int_{0}^{1}\hat{k}(1,x)\hat{u}(x,t)dx$ (83)

to stabilize the system (1), (2), (3) by output feedback.

The backstepping transformations

$\displaystyle{\check{w}}(x,t)$ $\displaystyle=$
$\displaystyle\hat{u}(x,t)-\int_{0}^{x}\hat{k}(x,y)\hat{u}(y,t)dy$ (84)
$\displaystyle\tilde{u}(x,t)$ $\displaystyle=$
$\displaystyle\omega(x,t)-\int_{x}^{1}\hat{k}(y,x)\omega(y,t)dy.$ (85)

transform the overall system into the cascade

$\displaystyle\check{w}_{t}(x,t)$ $\displaystyle=$
$\displaystyle\check{w}_{xx}(x,t)+\delta_{k0}(x)\check{w}(x,t)dy$
$\displaystyle+\delta_{k0}(x,y)\int_{0}^{x}\hat{l}(x,y)\check{w}(y,t)dy$
$\displaystyle+\int_{0}^{x}\delta_{k1}(x,y)\check{w}(y,t)dy$
$\displaystyle+\int_{0}^{x}\delta_{k1}(x,y)\int_{0}^{y}\hat{l}(y,\eta)\check{w}(\eta,t)d\eta
dy$
$\displaystyle-\left(\hat{k}(1,x)-\int_{0}^{x}\hat{k}(x,y)\hat{k}(1,y)dy\right)\omega_{x}(1,t)$
$\displaystyle\check{w}(0,t)$ $\displaystyle=$ $\displaystyle 0$ (87)
$\displaystyle\check{w}(1,t)$ $\displaystyle=$ $\displaystyle 0$ (88)
$\displaystyle\omega_{t}(x,t)$ $\displaystyle=$
$\displaystyle\omega_{xx}(x,t)+\delta_{k0}(x)\omega(x,t)$ (89)
$\displaystyle+\int_{x}^{1}\hat{l}(y,x)\delta_{k0}(y)\omega(y,t)dy$
$\displaystyle+\int_{x}^{1}\left(\delta_{k1}(y,x)\omega(y,t)\right.$
$\displaystyle\left.+\hat{l}(y,x)\int_{y}^{1}\delta_{k1}(s,y)\omega(s,t)ds\right)dy$
$\displaystyle\omega(0,t)$ $\displaystyle=$ $\displaystyle 0$ (90)
$\displaystyle\omega(1,t)$ $\displaystyle=$ $\displaystyle 0.$ (91)

Both the $\omega$-subsystem (89)–(91), which is autonomous, and the
$\check{w}$-subsystem (6)–(88), which is driven by the output
$\omega_{x}(1,t)$ of the $\omega$-subsystem, are exponentially stable in
$L^{2}[0,1]$ and higher norms for sufficiently small $\epsilon$. However,
because the trace term $\omega_{x}(1,t)$ in the last line of (6) cannot be
easily bounded even by an $H^{2}$ norm of $\omega$, we do not pursue a
stability analysis of the composite system, i.e., we leave the “separation
principle” unproven for the observer-based feedback (54), (55), (56), (83)
acting on the system (1), (2), (3). The technical challenge has nothing to do
with the NO implementation of the kernel $\hat{k}$, as the challenge does not
arise due to the perturbation kernels $\delta_{k0},\delta_{k1}$. The challenge
is due to the unbounded nature of the output mapping
$\omega(t)\mapsto\omega_{x}(1,t)$, a challenge not encountered in ODEs but
only in PDE control with boundary sensing or actuation.

The result given next, which is of a slightly more complicated form, is
provable but we give it without a proof because the calculations are very,
very lengthy and partly duplicate the calculations in the previous sections.
The actuation-sensing setup is from the last row of Table 1, namely, a
collocated Neumann actuation and Dirichlet sensing. Stability established is
in the $H^{1}$ norm $\|u(t)\|_{H^{1}}+\|\hat{u}(t)\|_{H^{1}}$.

######  Theorem 7.

Consider the system

$\displaystyle u_{t}(x,t)$ $\displaystyle=$ $\displaystyle
u_{xx}(x,t)+\lambda(x)u(x,t),\qquad\mbox{$x\in[0,1)$}$ (92) $\displaystyle
u(0,t)$ $\displaystyle=$ $\displaystyle 0$ (93) $\displaystyle u_{x}(1,t)$
$\displaystyle=$ $\displaystyle U(t)$ (94)

with a measured Dirichlet output $u(1,t)$, along with the collocated observer-
based Neumann-actuated controller

$\displaystyle\hat{u}_{t}(x,t)$ $\displaystyle=$
$\displaystyle\hat{u}_{xx}(x,t)+\lambda(x)\hat{u}(x,t)$ (95)
$\displaystyle+\kappa(x)\left[u(1,t)-\hat{u}(1,t)\right]$
$\displaystyle\hat{u}(0,t)$ $\displaystyle=$ $\displaystyle 0$ (96)
$\displaystyle\hat{u}_{x}(1,t)$ $\displaystyle=$ $\displaystyle
U(t)-\hat{k}(1,1)\left(u(1,t)-\hat{u}(1,t)\right)$ (97) $\displaystyle U(t)$
$\displaystyle=$
$\displaystyle\hat{k}(1,1)u(1,t)+\int_{0}^{1}\kappa(x)\hat{u}(x,t)dx,$ (98)

where the gain function of both the controller and the observer is given by

$\kappa(x):=\left.\hat{k}_{\xi}(\xi,x)\right|_{\xi=1}.$ (99)

For all $B_{\lambda},B_{\lambda^{\prime}}>0$ and for all $\lambda\in
C^{1}([0,1])$ whose derivative is Lipschitz and which satisfies
$\|\lambda\|_{\infty}\leq B_{\lambda}$ and $\|\lambda^{\prime}\|_{\infty}\leq
B_{\lambda^{\prime}}$, there exists a sufficiently small
$\epsilon^{*}(B_{\lambda},B_{\lambda^{\prime}})>0$ such that for all NO gain
kernels $\hat{k}=\hat{\mathcal{K}}(\lambda)$ of approximation accuracy
$\epsilon\in(0,\epsilon^{*})$ in relation to the exact backstepping kernel
$k={\mathcal{K}}(\lambda)$ there exists sufficiently large
$M(\epsilon,\bar{\lambda})>0$ such that the above observer-based feedback
ensures that the closed-loop system, for all $u_{0},\hat{u}_{0}\in
H^{1}[0,1]$, satisfies the exponential stability bound

$\|u(t)\|_{H^{1}}+\|\hat{u}(t)\|_{H^{1}}\leq M{\rm
e}^{-(t-t_{0})/4}\left(\|u_{0}\|_{H^{1}}+\|\hat{u}_{0}\|_{H^{1}}\right)$ (100)

for all $t\geq t_{0}$.

The proof is based on the backstepping transformations (84), (85) into a
perturbed version of the target system

$\displaystyle\check{w}_{t}(x,t)$ $\displaystyle=$
$\displaystyle\check{w}_{xx}(x,t)$ (101)
$\displaystyle+\left(\kappa(x)-\int_{0}^{x}\kappa(y)\hat{k}(x,y)dy\right)\omega(1,t)$
$\displaystyle\check{w}(0,t)$ $\displaystyle=$ $\displaystyle 0$ (102)
$\displaystyle\check{w}(1,t)$ $\displaystyle=$ $\displaystyle 0$ (103)
$\displaystyle\omega_{t}(x,t)$ $\displaystyle=$
$\displaystyle\omega_{xx}(x,t)$ (104) $\displaystyle\omega(0,t)$
$\displaystyle=$ $\displaystyle 0$ (105) $\displaystyle\omega(1,t)$
$\displaystyle=$ $\displaystyle 0.$ (106)

The perturbation terms are as in (6) and (89), employing the functions
$\delta_{k0}$ and $\delta_{k1}$ (which are uniformly bounded by $\epsilon$).
The Lyapunov analysis employs the $H^{1}$ norms of $\check{w}$ and $\omega$,
along with Agmon’s inequality to bound the perturbation term $\omega(1,t)$ in
the $\check{w}$-system using the norm $\|\omega_{x}\|$.

Figure 5: Open-loop instability (top) for the two respective reaction
coefficients $\lambda(x)$ shown on the bottom row. Figure 6: Examples of the
kernel $k(x,y)$ (top row), learned kernel $\hat{k}(x,y)$ (middle row), and the
kernel error $k(x,y)-\hat{k}(x,y)$ (bottom row). The two respective
$\lambda(x)$ values correspond to the same respective values as in Fig 5.
Figure 7: For the two respective $\lambda(x)$ values as in Fig 5, the top row
showcases closed-loop solutions with the learned kernel $\hat{k}(x,y)$,
whereas the bottom row shows the closed-loop PDE error between applying the
original kernel $k(x,y)$ and the learned kernel $\hat{k}(x,y)$.  Figure 8:
Left: PDE solution with $\lambda(x)=20\cos(5\cos^{-1}(x))$ and
$U(t)=7\sin(16\pi t)+10\cos(2\pi t)$. Right: the observer with the neural
operator-learned kernel. Note the difference between the plant initial
condition $u(x,0)=10$ and the twice as large initial condition of the exact
and neural operator observers, $\hat{u}_{\rm NO}(x,0)=\hat{u}(0,x)=20$. The
peak error between the analytical observer (not shown) and the neural operator
observer is around 0.3.

##  7 Numerical Results: Full-State Feedback, Observer, and Output Feedback

In Figure 5, we show that the system is open-loop unstable for the reaction
term $\lambda(x)=50\cos(\gamma\cos^{-1}(x))$ for $\gamma=5,8$. The increased
oscillation in larger $\gamma$ yields a lower rate of instability as shown on
the right. We simulate the PDE and its control using the finite difference
scheme in Appendix B.

In Figure 6, we demonstrate both the analytical and learned DeepONet kernels
for the two $\gamma$ values corresponding to Figure 5. To learn the mapping
$\mathcal{K}:\lambda(x)\mapsto k(x,y)$, we construct a dataset of 900
different $\lambda(x)$ as the Chebyshev polynomials define above with
$\gamma\sim$ uniform $(4,9)$. We choose $\lambda$ of this form due to the rich
set of kernel functions generated by varying only a single parameter. To
effectively utilize the DeepONet without modifying the grid, we stack
$\lambda(x)$ repeatedly $n_{y}$ times over the $y$ axis to make a 2D input to
the network. Then, we capitalize on the 2D mapping by implementing a CNN for
the DeepONet branch network. In the future, one can explore neural operators
on irregular girds along the direction of [52]. For training, the relative
$L_{2}$ error is $3.5e-2$ and the testing error is $3.6e-2$. With the learned
neural operator, we achieve speedups on the magnitude of $10^{3}$ compared to
an efficient finite difference implementation. In Figure 7, we demonstrate
closed-loop stability with the neural operator approximated gain function for
the control feedback law. Additionally, we see the error is largest at the
beginning achieving a maximum in both cases of approximately $10\%$.

In Figure 8, we test the observer (55), (55), (56) with a DeepONet-
approximated kernel trained as above using $\lambda(x)=20\cos(5\cos^{-1}(x))$
with $\gamma\sim$ uniform $(4,9)$. Additionally, we apply a boundary signal of
$U(t)=7\sin(16\pi t)+10\cos(2\pi t)$ to generate a challenging and rich PDE
motion for estimation. The true system state begins with initial conditions
$u(x,0)=10$ while the DeepONet observer has initial conditions of
$\hat{u}_{NO}(x,0)=20$. Despite this, the observer approximates the PDE well
with a peak error of less than $5\%$ compared to the analytical observer while
maintaining the same $10^{3}$x speedup over the finite difference scheme.

##  8 Conclusions

In this paper, we build on the framework introduced in [10], and depicted in
Figures 1 and 2, and extend the neural operator-supported PDE backstepping
methodology from the hyperbolic to the harder parabolic case. We limit
ourselves to the reaction-diffusion parabolic class for the clarity of
exposition.

With the foundation laid for hyperbolic and parabolic PDE backstepping
designs, which free the user from having to solve kernel PDEs in real time and
result in a thousandfold speedup, the road is open to developing this
methodology for two important control domains in which the backstepping
kernels constantly evolve in the course of implementation: (1) gain scheduling
for nonlinear PDEs, where the kernel depends on the current state of the PDE;
and (2) adaptive control of PDEs whose functional coefficients are unknown,
have to be adaptively estimated online, and the kernel has to be continuously
updated. In both applications, the solving of the $k$-PDE online is eliminated
with the aid of the pre-determined neural operator $\hat{\mathcal{K}}$.

## Appendix

##  Appendix A Neural networks notation

An $n$-layer neural network (NN)
$f^{\mathcal{N}}:\mathbb{R}^{d_{1}}\rightarrow\mathbb{R}^{d_{n}}$ is given by

$\displaystyle f^{\mathcal{N}}(x,\theta):=(l_{n}\circ l_{n-1}\circ...\circ
l_{2}\circ l_{1})(x,\theta)$ (107)

where layers $l_{i}$ start with $l_{0}=x\in\mathbb{R}^{d_{1}}$ and continue as

$l_{i+1}(l_{i},\theta_{i+1}):=\sigma(W_{i+1}l_{i}+b_{i+1}),\quad
i=1,\ldots,n-1$ (108)

$\sigma$ is a nonlinear activation function, and weights
$W_{i+1}\in\mathbb{R}^{d_{i+1}\times d_{i}}$ and biases
$b_{i+1}\in\mathbb{R}^{d_{i+1}}$ are parameters to be learned, collected into
$\theta_{i}\in\mathbb{R}^{d_{i+1}(d_{i}+1)}$, and then into
$\theta=[\theta_{1}^{\rm T},\ldots,\theta_{n}^{\rm T}]^{\rm
T}\in\mathbb{R}^{\sum_{i=1}^{n-1}d_{i+1}(d_{i}+1)}$. Let
$\vartheta^{(k)},\theta^{(k)}\in\mathbb{R}^{\sum_{i=1}^{k-1}d_{k,(i+1)}(d_{k,i}+1)}$
denote a sequence of NN weights.

A neural operator (NO) for approximating a nonlinear operator
$\mathcal{G}:{\mathcal{U}}\mapsto{\mathcal{V}}$ is defined as

$\displaystyle{\mathcal{G}}_{\mathbb{N}}(\mathbf{u}_{m})(y)=\sum_{k=1}^{p}g^{\mathcal{N}}(\mathbf{u}_{m};\vartheta^{(k)})f^{\mathcal{N}}(y;\theta^{(k)})$
(109)

where ${\mathcal{U}},{\mathcal{V}}$ are function spaces of continuous
functions $u\in{\mathcal{U}},v\in{\mathcal{V}}$. $\mathbf{u}_{m}$ is the
evaluation of function $u$ at points $x_{i}=x_{1},...,x_{m}$, $p$ is the
number of chosen basis components in the target space, $y\in Y$ is the
location of the output function $v(y)$ evaluations, and $g^{\mathcal{N}}$,
$f^{\mathcal{N}}$ are NNs termed branch and trunk networks. Note,
$g^{\mathcal{N}}$ and $f^{\mathcal{N}}$ are not limited to feedforward NNs
(107), but can also be of convolutional or recurrent.

##  Appendix B FD Scheme for Goursat-Form Kernel PDE

For the PDE in (1), (2), (3), we utilize the following finite difference
scheme adapted from [81]:

$\displaystyle k_{j}^{i+1}$ $\displaystyle=$ $\displaystyle-
k_{j}^{i-1}+k_{j+1}^{i}+k_{j-1}^{i}+h^{2}\lambda_{j}\frac{k_{j+1}^{i}+k_{j-1}^{i}}{2}$
(110) $\displaystyle k^{i+1}_{i}$ $\displaystyle=$ $\displaystyle
k_{i}^{i}+\frac{h}{2}\lambda_{i}$ (111) $\displaystyle k_{i+1}^{i+1}$
$\displaystyle=$ $\displaystyle
k_{i}^{i}-\frac{h}{4}(\lambda_{i}+\lambda_{i+1}),\qquad k_{1}^{j+1}=0$ (112)

with
$k_{i}^{j}=k((i-1)h,(j-1)h),i=2,...,N,j=2,...,i-1,\lambda_{i}=\bar{\lambda}((i-1)h),h=1/N$
where $N$ is the number of spatial steps.

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