Backstepping in Infinite Dimension for a Class of Parabolic Distributed Parameter Systems

Abstract

 In this paper a family of stabilizing boundary feedback control laws for a class of linear parabolic PDEs motivated by engineering
applications is presented. The design procedure presented here can handle systems with an arbitrary finite number of open-loop
unstable eigenvalues and is not restricted to a particular type of boundary actuation. Stabilization is achieved through the
design of coordinate transformations that have the form of recursive relationships. The fundamental difficulty of such transformations
is that the recursion has an infinite number of iterations. The problem of feedback gains growing unbounded as the grid becomes
infinitely fine is resolved by a proper choice of the target system to which the original system is transformed. We show how
to design coordinate transformations such that they are sufficiently regular (not continuous but L
∞). We then establish closed-loop stability, regularity of control, and regularity of solutions of the PDE. The result is accompanied
by a simulation study for a linearization of a tubular chemical reactor around an unstable steady state.

Full text

DOI: 10.1007/s00498-003-0128-6
Springer-Verlag London Ltd. (2003
Math. Control Signals Systems (2003) 16: 44–75
Mathematics of Control,
Signals, and Systems
Backstepping in Infinite Dimension for a Class of
Parabolic Distributed Parameter Systems*
Dejan M. Bos
ˇkovic
´,yAndras Balogh,yand Miroslav Krstic
´y
Abstract. In this paper a family of stabilizing boundary feedback control laws
for a class of linear parabolic PDEs motivated by engineering applications is pre-
sented. The design procedure presented here can handle systems with an arbitrary
finite number of open-loop unstable eigenvalues and is not restricted to a particu-
lar type of boundary actuation. Stabilization is achieved through the design of
coordinate transformations that have the form of recursive relationships. The
fundamental di‰culty of such transformations is that the recursion has an infinite
number of iterations. The problem of feedback gains growing unbounded as the
grid becomes infinitely fine is resolved by a proper choice of the target system to
which the original system is transformed. We show how to design coordinate
transformations such that they are su‰ciently regular (not continuous but Ly).
We then establish closed-loop stability, regularity of control, and regularity of
solutions of the PDE. The result is accompanied by a simulation study for a lin-
earization of a tubular chemical reactor around an unstable steady state.
Key words. Boundary control, Linear parabolic PDEs, Stabilization, Backstep-
ping, Coordinate transformations.
1. Introduction
Motivated by the model for the chemical tubular reactor, the model of unstable
burning in solid rocket propellants, and other PDE systems that appear in vari-
ous engineering applications, we present an algorithm for global stabilization of
a broader class of linear parabolic PDEs. The result presented here is a general-
ization of the ideas of Balogh and Krstic
´[BK1]. The goal is to obtain an Lyco-
ordinate transformation and a boundary control law that renders the closed-loop
system asymptotically stable, and additionally establish regularity of control and
regularity of solutions for the closed-loop system.
The key issue with arbitrarily unstable linear parabolic PDE systems is the
target system to which one is transforming the original system by coordinate
44
* Date received: June 22, 2001. Date revised: January 17, 2002. This work was supported by grants
from AFOSR, ONR, and NSF.
yDepartment of MAE, University of California at San Diego, La Jolla, California 92093-0411,
U.S.A. boskovic@mae.ucsd.edu. {abalogh, krstic}@ucsd.edu.

transformation. For example, if one takes the standard backstepping route lead-
ing to a tridiagonal form, the resulting transformations, if thought of as integral
transformations, end up with ‘‘kernels’’ that are not even finite. A proper selection
of the target system will result in a bounded kernel and the solutions correspond-
ing to the controlled problem are going to be at least continuous.
The class of parabolic PDEs considered in this paper is
utðt;xÞ¼euxxðt;xÞþBuxðt;xÞþlðxÞuðt;xÞþðx
0
fðx;xÞuðt;xÞdx;
xAð0;1Þ;t>0;ð1:1Þ
where e>0 and Bare constants, lðxÞALyð0;1Þand fðx;yÞALyð½0;1½0;1Þ,
with initial condition uð0;xÞ¼u0ðxÞ, for xA½0;1. The boundary condition at
x¼0 is either homogeneous Dirichlet,
uðt;0Þ¼0;t>0;ð1:2Þ
or homogeneous Neumann,
uxðt;0Þ¼0;t>0;ð1:3Þ
while the Dirichlet boundary condition (alternatively Neumann) at the other
end,
uðt;1Þ¼aðuðtÞÞ;1t>0;ð1:4Þ
is used as the control input, where the linear operator arepresents a control law
to be designed to achieve stabilization. It is assumed that the initial distribu-
tion is compatible with (1.2) (alternatively with (1.3)), i.e. u0ð0Þ¼0 (alternatively
u0
xð0Þ¼0).
Our interest in systems described by (1.1) is twofold. First, the physical moti-
vation for considering (1.1) is that it represents the linearization of the class of
reaction–di¤usion–convection equations that model many physical phenomena.
Examples are numerous and among others include the problem of compressor
rotating stall (the most recent model due to Mezic [HMBZ] is ut¼euxx þuu3),
whose linearization is (1.1) with lðxÞ11, B1fðx;yÞ10, the unstable heat equa-
tion [BKL] (e11, B1fðx;yÞ10, and lðxÞ1l¼constant), the linearization of
the unstable burning for solid rocket propellants [BK3] (e1B11, lðxÞ10, and
fðx;yÞ¼AexdðyÞ,A¼constant), and the linearization of an adiabatic chemi-
cal tubular reactor around either stable or unstable equilibrium [HH1] (e¼1=Pe,
B¼1, lðxÞis a spatially dependent function that corresponds to either stable or
unstable steady state profile, and fðx;yÞ10).
Second, from the perspective of control theory, systems described by (1.1) are
interesting since their discretization appears in the most general strict-feedback
form [KKK]. Therefore, developing backstepping control algorithms for such a
1Throughout the paper we use the simplified notation uðtÞ¼uðt;Þ.
Backstepping in Infinite Dimension 45

class of problems is of great importance as the first step in an attempt to extend
fully the existing backstepping techniques from the finite-dimensional setup to the
infinite-dimensional one.
For di¤erent combinations of the boundary condition at x¼0 (Dirichlet or
Neumann), and control applied at x¼1 (Dirichlet or Neumann), we use a back-
stepping method for the finite-di¤erence semi-discretized approximation of (1.1)
to derive a boundary feedback control law that makes the infinite-dimensional
closed-loop system stable with an arbitrary prescribed stability margin. We show
that the integral kernel in the control law resides in the function space Lyð0;1Þ
and that solutions corresponding to the controlled problem are classical.
We should stress that although we focus our attention in this paper on a class
of one-dimensional parabolic problems, the design procedures and results pre-
sented here can be easily extended to higher-dimensional problems. We have dem-
onstrated that fact in [BK2], where backstepping was successfully applied on a
two-dimensional nonlinear heat convection model from Burns et al. [BKR2]. Note
that a further extension to three dimensions would be conceptually the same
and the control would be applied via a planar array of wall actuators and the
coordinate transforms in the backstepping design would depend on three indices
ðaijk;bijk ;gijk Þ. As already mentioned, the main issue in our approach is not the
dimension of the system, but the choice of the target system that will result in a
bounded kernel as the grid becomes infinitely fine.
Prior work on stabilization of general parabolic equations includes, among
others, the results of Triggiani [T] and Lasiecka and Triggiani [LT1] who devel-
oped a general framework for the structural assignment of eigenvalues in para-
bolic problems through the use of semigroup theory. Separating the open-loop
system into a finite-dimensional unstable part and an infinite-dimensional stable
part, they apply feedback boundary control that stabilizes the unstable part while
leaving the stable part stable. An LQR approach in Lasiecka and Triggiani [LT2]
is also applicable to this problem. A unified treatment of both interior and
boundary observations/control generalized to semilinear problems can be found
in [A2]. Nambu [N] developed auxiliary functional observers to stabilize di¤usion
equations using boundary observation and feedback. Stabilizability by boundary
control in the optimal control setting is discussed by Bensoussan et al. [BDDM].
For the general Pritchard–Salamon class of state–space systems a number of
frequency-domain results has been established on stabilization during the last
decade (see, e.g. [C3] and [L2] for a survey). The placement of finitely many
eigenvalues were generalized to the case of moving infinitely many eigenvalues by
Russell [R1]. The stabilization problem can also be approached using the abstract
theory of boundary control systems developed by Fattorini [F1] that results in a
dynamical feedback controller (see remarks in Section 3.5 of [CZ]). Extensive sur-
veys on the controllability and stabilizability theory of linear PDEs can be found
in [R2] and [LT2].
The first result, to our knowledge, where backstepping was applied to a PDE is
the control design for a rotating beam by Coron and d’Andre
´a-Novel [CA]. They
designed a nonlinear feedback torque control law for a hyperbolic PDE model of
a rotating beam with no damping and no control on the free boundary. The scalar
46 D. M. Bos
ˇkovic
´, A. Balogh, and M. Krstic
´

control input, applied in a distributed fashion, is used to achieve global asymp-
totic stabilization of the system. In addition, the authors show regularity of con-
trol inputs.
Backstepping was successfully applied to parabolic PDEs in [LK] and [BK2]–
[BK4] in settings with only a finite number of steps.
Our work is also related to results of Burns et al. [BKR1]. Although their result
is quite di¤erent because of the di¤erent control objective (theirs is LQR optimal
control, ours is stabilization), and the fact that their plant is open-loop stable but
with the spatial domain of dimension higher than ours, the technical problem of
proving some regularity of the gain kernel ties the two results together.
In an attempt to generalize the backstepping techniques from finite dimensions
to linear parabolic infinite-dimensional systems, Boskovic et al. [BKL] considered
the unstable heat equation with parameters restricted so that the number of open-
loop unstable eigenvalues is no greater than one. In this limited case we derived a
closed-form and smooth coordinate transformation based on backstepping. In an
e¤ort to extend the results from [BKL] for an arbitrary level of instability, Balogh
and Krstic
´[BK1] obtained the first backstepping-type feedback control law for a
linear PDE that can accommodate an arbitrary level of instability, i.e. stabilize
the system that has an arbitrary number of unstable eigenvalues in an open loop.
By designing a su‰ciently regular (not continuous but Ly) coordinate transfor-
mation they were able to establish closed-loop stability, regularity of control, and
regularity of solutions of the PDE.
We emphasize that, in addition to being an important step in a generalization
of a finite-dimensional technique to infinite dimensions and with the ultimate goal
of potential applications to nonlinear problems, the backstepping control design
for linear parabolic PDEs presented here has advantages of its own. First, com-
pared with the pole placement type of designs that have been prevalent in the
control of parabolic PDEs, it has the standard advantage of a Lyapunov-based
approach that the designer does not have to look for the solution of the uncon-
trolled system to find the controller that stabilizes it. The problem of finding
modal data in the case of spatially dependent lðxÞand fðx;yÞbecomes nontrivial
and finding closed-form expressions for the system eigenvalues and eigenvectors
appears highly unlikely in the general case. In some instances, as is the case for
the tubular reactor example used in our simulation study, the only way to obtain
spatially dependent coe‰cients is numerically. In that case finding eigenvalues
and eigenvectors numerically becomes inevitable, which might be computationally
very expensive if a large number of grid points is necessary for simulating the sys-
tem. To obtain a backstepping controller that stabilizes the system, on the other
hand, the designer has to obtain a kernel given by a simple recursive expression
that is computationally inexpensive. Second, from an applications point of view,
numerical results both for the nonlinear [BK2]–[BK4] and linear (linearization of
the chemical tubular reactor presented here) parabolic PDEs suggest that reduced-
order backstepping control laws (designed on a much coarser grid) that use only a
few state measurements can successfully stabilize the system.
The main reason for choosing a model of a chemical tubular reactor in our
simulation study is because a large amount of research activity has been dedicated
Backstepping in Infinite Dimension 47

to the control designs based on PDE models of chemical reactors. Using a combi-
nation of Galerkin’s method with a procedure for the construction of approxi-
mate inertial manifolds, Christofides [C2] designed output feedback controllers for
nonisothermal tubular reactors that guarantee stability and enforce the output of
the closed-loop system to follow, up to a desired accuracy, a prescribed response
for almost all times. In a more recent paper by Orlov and Dochain [OD] a sliding
mode control developed for minimum phase semilinear infinite-dimensional sys-
tems was applied to stabilization of both plug flow (hyperbolic) and tubular (para-
bolic) chemical reactors. Both results use distributed control to stabilize the system
around prespecified temperature and concentration steady-state profiles. On the
other hand, we apply point actuation at x¼1 in our design.
The paper is organized as follows. In Section 2 we formulate our problem and
its discretization for two di¤erent cases of boundary conditions at x¼0 (either
homogeneous Dirichlet uðt;0Þ¼0, or homogeneous Neumann uxðt;0Þ¼0) and
we lay out our strategy for the solution of the stabilization problem. The pre-
cise formulations of our main theorems are contained in Section 3. In Lemmas 1
(homogeneous Dirichlet at x¼0) and 5 (homogeneous Neumann at x¼0) of
Section 4 we design coordinate transformations for semi-discretizations of the
system (for a less general case with no integral term on the right-hand side of the
system equation) which map them into exponentially stable systems. We show
in Lemmas 2 (homogeneous Dirichlet at x¼0) and 6 (homogeneous Neumann
at x¼0) that the discrete coordinate transformations remain uniformly bounded
as the grid gets refined and hence converge to coordinate transformations in the
infinite-dimensional case. The regularity Cwð½0;1;Lyð0;1ÞÞ of the transforma-
tion is established in Lemma 3. We complete the proofs of our main theorems
using Lemma 4 [BK1] that establishes the stability of the infinite-dimensional
controlled systems. The extension from Dirichlet to Neumann type of actuation
is presented in Section 5, followed by an extension of the result to the case when
the integral term is present on the right-hand side of the system equation in Sec-
tion 6. Finally, simulation study for a linearized model of an adiabatic chemical
tubular reactor presented in Section 7 shows, besides the e¤ectiveness of our
control, that reduced versions of the controller stabilize the infinite-dimensional
system as well.
2. Motivation
In this section we formulate our problem for a particular case of the system
(1.1) with no integral term on the right-hand side of the system equation, i.e.
for
utðt;xÞ¼euxxðt;xÞþBuxðt;xÞþlðxÞuðt;xÞ;xAð0;1Þ;t>0:ð2:5Þ
This particular case is the most interesting from the applications point of view
and we present results for all four combinations of di¤erent types of boundary
conditions at the uncontrolled end x¼0, and actuations at the control end
x¼1.
48 D. M. Bos
ˇkovic
´, A. Balogh, and M. Krstic
´

An extension of the result for the most general case of the system (1.1) (integral
term on the right-hand side of the system equation) with homogeneous Dirichlet
boundary condition at x¼0 and Dirichlet type of actuation at x¼1 is presented
in Section 6.
2.1. Case 1: Dirichlet Boundary Condition at x ¼0
In this subsection we analyze the case when the homogeneous Dirichlet bound-
ary condition is imposed at x¼0. We first introduce the case when actuation of
the Dirichlet type is applied at x¼1. The extension for the Neumann type of
actuation is presented is Section 5. The semi-discretized version of system (2.5)
with (1.2) and (1.4) using central di¤erencing in space is the finite-dimensional
system:
u0¼0;ð2:6Þ
_
uui¼euiþ12uiþui1
h2þBuiþ1ui
hþliui;i¼1;...;n;ð2:7Þ
unþ1¼anðu1;u2;...;unÞ;ð2:8Þ
where nAN,h¼1=ðnþ1Þand ui¼uðt;ihÞ,li¼lðihÞ, for i¼0;...;nþ1.
With unþ1as control, this system is in the strict-feedback form and hence it is
readily stabilizable by standard backstepping. However, the naive version of
backstepping would result in a control law with gains that grow unbounded as
n!y.
The problem with standard backstepping is that it would not only attempt to
stabilize the equation, but also place all of its poles, and thus, as n!y, change
its parabolic character. Indeed, an infinite-dimensional version of the tridiagonal
form in backstepping is not parabolic. Our approach will be to transform the sys-
tem, but keep its parabolic character, i.e. keep the second spatial derivative in the
transformed coordinates.
Towards this end, we start with a finite-dimensional backstepping-style coordi-
nate transformation
w0¼u0¼0;ð2:9Þ
wi¼uiai1ðu1;...;ui1Þ;i¼1;...;n;ð2:10Þ
wnþ1¼0;ð2:11Þ
for the discretized system (2.6)–(2.8), and seek the functions aisuch that the trans-
formed system has the form
w0¼0;ð2:12Þ
_
wwi¼ewiþ12wiþwi1
h2þBwiþ1wi
hcwi;i¼1;...;n;ð2:13Þ
wnþ1¼0:ð2:14Þ
The finite-dimensional system (2.12)–(2.14) is the semi-discretized version of the
Backstepping in Infinite Dimension 49

infinite-dimensional system
wtðt;xÞ¼ewxxðt;xÞþBwxðt;xÞcwðt;xÞ;xAð0;1Þ;t>0;ð2:15Þ
with boundary conditions
wðt;0Þ¼0;ð2:16Þ
wðt;1Þ¼0;ð2:17Þ
which is exponentially stable for c>ep2B2=ð4eÞ.
The backstepping coordinate transformation is obtained by combining (2.6)–
(2.8), (2.9)–(2.11), and (2.12)–(2.14) and solving the resulting system for the
ai’s. Namely, subtracting (2.13) from (2.7), expressing the obtained equation in
terms of ukwk,k¼i1;i;iþ1, and applying (2.10) we obtain the recursive
form
ai¼ðeþBhÞ1(ð2eþBh þch2Þai1eai2ðcþliÞh2ui
þX
i1
j¼1
qai1
quj
ððeþBhÞujþ1ð2eþBh ljh2Þujþeuj1Þ);ð2:18Þ
for i¼1;...;nwith initial values a0¼0 and2
a1¼ h2
eþBh ðcþl1Þu1:ð2:19Þ
Writing the ai’s in the linear form
ai¼X
i
j¼1
ki;juj;i¼1;...;n;ð2:20Þ
and performing simple calculations we obtain the general recursive relationship
ki;1¼h2
eþBh ðcþl1Þki1;1þe
eþBh ðki1;2ki2;1Þ;ð2:21Þ
ki;j¼h2
eþBh ðcþljÞki1;jþki1;j1þe
eþBh ðki1;jþ1ki2;jÞ;
j¼2;...;i2;ð2:22Þ
ki;i1¼h2
eþBh ðcþli1Þki1;i1þki1;i2;ð2:23Þ
ki;i¼ki1;i1h2
eþBh ðcþliÞ;ð2:24Þ
2From now on we assume that nis large enough to have the inequality eþBh >0 satisfied.
50 D. M. Bos
ˇkovic
´, A. Balogh, and M. Krstic
´

for i¼4;...;nwith initial conditions
k1;1¼ h2
eþBh ðcþl1Þ;ð2:25Þ
k2;1¼ h4
ðeþBhÞ2ðcþl1Þ2;ð2:26Þ
k2;2¼ h2
eþBh ðcþl1Þþ h2
eþBh ðcþl2Þ

;ð2:27Þ
k3;1¼ h6
ðeþBhÞ3ðcþl1Þ3e
ðeþBhÞ
h2
ðeþBhÞðcþl2Þ;ð2:28Þ
k3;2¼ h2
eþBh ðcþl2Þh2
eþBh ðcþl1Þþ h2
eþBh ðcþl2Þ

h4
ðeþBhÞ2ðcþl1Þ2;ð2:29Þ
k3;3¼ h2
eþBh ðcþl1Þþ h2
eþBh ðcþl2Þþ h2
eþBh ðcþl3Þ

:ð2:30Þ
For the simple case when lðxÞ1l¼constant, (2.21)–(2.30) can be solved explic-
itly to obtain
ki;ij¼ i
jþ1

Ljþ1
nðijÞX
½j=2
l¼1
1
l
jl
l1

il
j2l

Lj2lþ1
nMl
nð2:31Þ
for i¼1;...;n,j¼0;...;i1, where
Ln¼h2
eþBh ðcþlÞ;ð2:32Þ
Mn¼e
eþBh :ð2:33Þ
Regarding the infinite-dimensional system (2.5) with (1.2) and (1.4), the linearity
of the control law in (2.20) suggests a stabilizing boundary feedback control of the
form
aðuÞ¼ð1
0
kðxÞuðxÞdx;ð2:34Þ
where the function kðxÞis obtained as a limit of fðnþ1Þkn;jgn
j¼1as n!y. From
the complicated expression (2.31) it is not clear if such a limit exists. A quick
numerical simulation (see Fig. 1) shows that the coe‰cients fðnþ1Þkn;jgn
j¼1remain
bounded but it also shows their oscillation, and increasing nonly increases the
oscillation (see Fig. 2). A similar type of behavior was encountered in the related
work of Balogh and Krstic
´[BK1]. Clearly, there is no hope for pointwise conver-
gence to a continuous kernel kðxÞ. However, as we will see in the next sections,
there is weak* convergence in Lyas we go from the finite-dimensional case to the
Backstepping in Infinite Dimension 51

infinite-dimensional one. As a result, we obtain a solution to our stabilization
problem (2.5) with boundary conditions (1.2) and (1.4).
2.2. Case 2: Neumann Boundary Condition at x ¼0
If a homogeneous Neumann boundary condition is prescribed at x¼0, a slightly
di¤erent procedure has to be applied. Note that we may assume without loss
of generality that the boundary condition at x¼0 is homogeneous since the
boundary condition of the third kind can be easily converted into a homoge-
neous Neumann boundary condition without changing the qualitative structure
of the system equation. For example, the boundary condition uxð0Þ¼quð0Þ,q¼
constant, would be converted by a variable change zðt;xÞ¼uðt;xÞeqx. The sys-
tem equation for that case would be transformed into zt¼ezxx þðBþ2qeÞzxþ
ðlðxÞþBq þeq2Þz. The main idea for the case of a homogeneous Neumann
boundary condition is very similar to the case with homogeneous Dirichlet
boundary condition at x¼0, and we only outline the di¤erences.
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-
6
-
5
-
4
-
3
-
2
-
1
0
x
k 50(x)
Fig. 1. Oscillation of the approximating kernel for n¼50, l¼5, e¼1, B¼1, and c¼1.
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
-
6
-
5
-
4
-
3
-
2
-
1
0
x
k 100(x)
Fig. 2. Oscillation of the approximating kernel for n¼100, l¼5, e¼1, B¼1, and c¼1.
52 D. M. Bos
ˇkovic
´, A. Balogh, and M. Krstic
´

We start with a finite-dimensional backstepping-style coordinate transformation
w0¼u0;ð2:35Þ
w1¼u1;ð2:36Þ
wi¼uiai1ðu1;...;ui1Þ;i¼2;...;n;ð2:37Þ
wnþ1¼0;ð2:38Þ
that transforms the original system into the semi-discretized version of the infinite-
dimensional system
wtðt;xÞ¼ewxxðt;xÞþBwxðt;xÞcwðt;xÞ;xAð0;1Þ;t>0;ð2:39Þ
with boundary conditions
wxðt;0Þ¼0;ð2:40Þ
wðt;1Þ¼0;ð2:41Þ
which is exponentially stable for c>ep2B2=ð4eÞ. Note that the given bound
is not optimal. The optimal bound is c>eh2B2=ð4eÞ, where his the smallest
positive root of equation ð2=BÞh¼tanðhÞ.
Using the same approach as in Section 2.1 we obtain
ai¼ðeþBhÞ1(ð2eþBh þch2Þai1eai2ðcþliÞh2ui
þqai1
qu1
ððeþBhÞu2ðeþBh l1h2Þu1Þ
þX
i1
j¼2
qai1
quj
ððeþBhÞujþ1ð2eþBh ljh2Þujþeuj1Þ);ð2:42Þ
instead of (2.18), with a0¼0 and a1given by (2.19). Writing the ai’s in the linear
form (2.20) we obtain
ki;1¼h2
eþBh ðcþl1Þþ e
eþBh

ki1;1þe
eþBh ðki1;2ki2;1Þ;ð2:43Þ
and ki;j,ki;i1, and ki;igiven by (2.22)–(2.24). The initial conditions for the recur-
sion are given as
k2;1¼ h2
eþBh ðcþl1Þþ e
eþBh

h2
eþBh ðcþl1Þ;ð2:44Þ
k3;1¼ h2
eþBh ðcþl1Þþ e
eþBh

2h2
eþBh ðcþl1Þ
e
ðeþBhÞ
h2
ðeþBhÞðcþl2Þ;ð2:45Þ
k3;2¼ h2
eþBh ðcþl2Þh2
eþBh ðcþl1Þþ h2
eþBh ðcþl2Þ

h4
ðeþBhÞ2ðcþl1Þ2e
eþBh
h2
eþBh ðcþl1Þ;ð2:46Þ
Backstepping in Infinite Dimension 53

and k1;1,k2;2, and k3;3the same as for the Dirichlet case. For the simple case
when lðxÞ1l¼constant, (2.31) becomes
ki;ij¼ i
jþ1

Ljþ1
nðijÞX
½j=2
l¼1
1
l
jl
l1

il
j2l

Lj2lþ1
nMl
n
X
½ð j1Þ=2
l¼0X
j2l1
k¼0
lþk
l

il1
k

Mjlk
nLkþ1
n
þX
½ð j1Þ=2
l¼1X
j2l1
k¼1
lþk
l1

il1
k1

Mjlk
nLkþ1
n:ð2:47Þ
Same as for the Dirichlet case, the stabilizing boundary feedback control will be in
the form (2.34), where the function kðxÞis obtained as a limit of fðnþ1Þkn;jgn
j¼1
for kn;jfrom (2.47) as n!y.
3. Main Result
3.1. Case 1: Dirichlet Boundary Condition at x ¼0
As we stated earlier, we use a backstepping scheme for the semi-discretized finite-
di¤erence approximation of system (2.5), (1.2), (1.4), (2.34) to derive a linear
boundary feedback control law that makes the infinite-dimensional closed-loop
system stable with an arbitrary prescribed stability margin. The precise formula-
tion of our main result is given by the following theorem.
Theorem 1. For any lðxÞALyð0;1Þand e;c>0there exists a function k A
Lyð0;1Þsuch that for any u0ALyð0;1Þthe unique classical solution uðt;xÞA
C1ðð0;yÞ;C2ð0;1ÞÞ of system (2.5), (1.2), (1.4), (2.34) is exponentially stable in
the L2ð0;1Þand maximum norms with decay rate c.The precise statements of sta-
bility properties are the following: there exists a positive constant M3such that for
all t >0,
kuðtÞk2aMku0k2ect ð3:48Þ
and
max
xA½0;1juðt;xÞj aMsup
xA½0;1
ju0ðxÞject:ð3:49Þ
Remark 1. For a given integral kernel kALyð0;1Þthe existence and regularity
results for the corresponding solution uðt;xÞfollows from trivial modifications in
the proof of Theorem 4.1 of [L1]. See also [F2].
3.2. Case 2: Neumann Boundary Condition at x ¼0
Theorem 2. For any lðxÞALyð0;1Þand e;c>0there exists a function k A
Lyð0;1Þsuch that for any u0ALyð0;1Þthe unique classical solution uðt;xÞA
3Mgrows with c,l, and 1=e.
54 D. M. Bos
ˇkovic
´, A. Balogh, and M. Krstic
´

C1ðð0;yÞ;C2ð0;1ÞÞ of system (2.5), (1.3), (1.4), (2.34) is exponentially stable in
the L2ð0;1Þand maximum norms with decay rate c. The precise statements of sta-
bility properties are the following: there exists positive constant M 4such that for all
t>0,
kuðtÞk2aMku0k2ect ð3:50Þ
and
max
xA½0;1juðt;xÞj aMsup
xA½0;1
ju0ðxÞject:ð3:51Þ
4. Proof of Main Result
4.1. Case 1: Dirichlet Boundary Condition at x ¼0
As was already mentioned in the Introduction, the proof of Theorem 1 requires
four lemmas.
Lemma 1. The elements of the sequence fki;jgdefined in (2.21)–(2.30) satisfy
jki;ijjai
jþ1

Ljþ1
nþðijÞX
½j=2
l¼1
1
l
jl
l1

il
j2l

Lj2lþ1
nMl
n;ð4:52Þ
where l¼maxxA½0;1jlðxÞj.
Remark 2. There is equality in (4.52) when lðxÞ1l¼constant >0.
Proof. The right-hand side of (2.25)–(2.30) can be estimated to obtain estimates
for the initial values of k’s:
jk1;1jah2
eþBh ðcþlÞ¼Ln;ð4:53Þ
jk2;1jah4
ðeþBhÞ2ðcþlÞ2¼L2
n;ð4:54Þ
jk2;2ja2h2
eþBh ðcþlÞ¼2Ln;ð4:55Þ
jk3;1jah6
ðeþBhÞ3ðcþlÞ3þe
eþBh
h2
eþBh ðcþlÞ¼L3
nþMnLn;ð4:56Þ
jk3;2ja3h4
ðeþBhÞ2ðcþlÞ2¼3L2
n;ð4:57Þ
jk3;3ja3h2
eþBh ðcþlÞ¼3Ln:ð4:58Þ
4Mgrows with c,l, and 1=e.
Backstepping in Infinite Dimension 55

We then go from j¼ibackwards to obtain from (2.24) and (2.23)
jki;ijaih2
eþBh ðcþlÞ¼iLn;ð4;59Þ
jki;i1jaiði1Þ
2
h4
ðeþBhÞ2ðcþlÞ2¼iði1Þ
2L2
n:ð4:60Þ
Finally we obtain inequality (4.52) of Lemma 1 using the general identity (2.22)
and mathematical induction. 9
In order to prove that the finite-dimensional coordinate transformation (2.9),
(2.10), (2.20) converges to an infinite-dimensional one that is well defined, we
show the uniform boundedness of ðnþ1Þki;jwith respect to nANas i¼1;...;n,
j¼1;...;i. Note that the binomial coe‰cients in inequality (4.52) are monotone
increasing in iand hence it is enough to show the boundedness of terms ðnþ1Þkn;j,
or equivalently ðnþ1Þkn;nj. Also, we introduce notations
q¼j
nA½0;1;ð4:61Þ
and
E¼2lþc
e;ð4:62Þ
R¼2jBj
e;ð4:63Þ
so that we can write
jkn;njj¼jkn;nqnj
an
qn þ1

Lqnþ1
n
þðnqnÞX
½qn=2
l¼1
1
l
qn l
l1

nl
qn 2l

Lqn2lþ1
nMl
n;ð4:64Þ
Ln¼h2
eþBh ðcþlÞaE
ðnþ1Þ2;ð4:65Þ
and
Mn¼e
eþBh ¼1Bh
eþBh a1þjBjh
e=2¼1þR
nþ1;ð4:66Þ
for su‰ciently large n.
Lemma 2. The sequence fðnþ1Þkn;jgj¼1;...;n;nb1remains bounded uniformly in n
and j as n !y.
56 D. M. Bos
ˇkovic
´, A. Balogh, and M. Krstic
´

Proof. We can write, according to (4.64),
ðnþ1Þjkn;nqnj
aðnþ1Þn
qn þ1

E
ðnþ1Þ2
!
qnþ1
þðnþ1ÞðnqnÞX
½qn=2
l¼1
1
l
qn l
l1

nl
qn 2l

E
ðnþ1Þ2
!
qn2lþ1
Ml
n:
ð4:67Þ
The first term can be estimated as
ðnþ1Þn
qn þ1

E
ðnþ1Þ2
!
qnþ1
aðnþ1Þqnþ2E
nþ1

qn E
ðnþ1Þqnþ2
aEE
n

qn
aEeE=e;ð4:68Þ
where the last line shows that the bound is uniform in nand also in q.
In the following steps we will use the simple inequalities
ðnlÞ!
ðnqn þlÞ!an
nqn þ2l
n1
nqn þ2l1 nlþ1
nqn þlþ1
ðnlÞ!
ðnqn þlÞ!
¼n!
ðnqn þ2lÞ!ð4:69Þ
and
ðqn lÞ!
l!ðqn 2lþ1Þ!
1
nþ1

qn2l
aqð4:70Þ
to obtain
ðnþ1ÞðnnqÞX
½qn=2
l¼1
1
l
qn l
l1

nl
qn 2l

E
ðnþ1Þ2
!
qn2lþ1
Ml
n
aEðnþ1Þn
ðnþ1Þ2X
½qn=2
l¼1
ðqn lÞ!
l!ðqn 2lþ1Þ!
1
nþ1

qn2l
n!
ðqn 2lÞ!ðnqn þ2lÞ!
E
nþ1

qn2l
1þR
nþ1

l
aEq 1þR
nþ1

nqX
nq
s¼0
n
s
E
n

s
1ns
aEq 1þR
n

nq
1þE
n

nq
aEeRþE:
Here in the last step we used the fact that the convergence ð1þX=nÞn
!
n!yeXis
monotone increasing and qA½0;1. This proves the lemma. 9
Backstepping in Infinite Dimension 57

As a result of the above boundedness, we obtain a sequence of piecewise con-
stant functions
knðx;yÞ¼ðnþ1ÞX
n
i¼1X
i
j¼1
ki;jwIi;jðx;yÞ;ðx;yÞA½0;1½0;1;nb1;ð4:71Þ
where
Ii;j¼i
nþ1;iþ1
nþ1

j
nþ1;jþ1
nþ1

;j¼1;...;i;i¼1;...;n;nb1:
ð4:72Þ
The sequence (4.71) is bounded in Lyð½0;1½0;1Þ. The space Lyð½0;1½0;1Þ is
the dual space of L1ð½0;1½0;1Þ, hence, it has a corresponding weak*-topology.
Since the space L1ð½0;1½0;1Þ is separable, it follows now by Alaoglu’s theo-
rem, see, e.g. p. 140 of [K] or Theorem 6.62 of [RR], that (4.71) converges in
the weak*-topology to a function ~
kkðx;yÞALyð½0;1½0;1Þ. The uniform in
pANweak convergence in each Lpð½0;1½0;1Þ ILyð½0;1½0;1Þ, immedi-
ately follows.
Remark 3. Alternatively, using the Eberlein–Shmulyan theorem see, e.g. p. 141
of [Y], one finds that (4.71) has a weakly convergent subsequence in each
Lpð½0;1½0;1Þ space for 1 <p<ywith Lp-norms bounded uniformly in
p. Using a diagonal process we choose a subsequence mðnÞANsuch that
fkmðnÞðx;yÞgnb1converges weakly to the same function ~
kkðx;yÞin each of the
spaces Lpð½0;1½0;1Þ,pAN. The function ~
kkðx;yÞalong with fkmðnÞðx;yÞgnb1
is uniformly bounded in all these Lp-spaces with the same bound for all pAN.
Remark 4. In the case of constant lwe have equality in (4.52). The right-
hand side is strictly monotone increasing in i, which results in ~
kk ACð½0;1;
Lyð0;1ÞÞ.
Lemma 3. The map ~
kk:½0;1!Lyð0;1Þgiven by x 7! ~
kkðx;Þ is weakly con-
tinuous.
Proof. From the uniform boundedness in iof (4.52) we obtain that
X
½nx
j¼1
k½nx;juj¼X
½nx
j¼1
ððnþ1Þk½nx;jÞuj
1
nþ1!
n!yðx
0
~
kkðx;xÞuðxÞdx;
EuAL1ð0;1Þ;ExA½0;1:ð4:73Þ
Here ½nxdenotes the largest integer not larger than nx and the convergence is
uniform in x, meaning that for all e>0 there exists NðeÞANsuch that
ðx
0
~
kkðx;xÞuðxÞdxX
½nx
j¼1
k½nx;juj

<e;ExA½0;1;En>N:
58 D. M. Bos
ˇkovic
´, A. Balogh, and M. Krstic
´

For an arbitrary xA½0;1we now fix an n>Nðe=2Þand choose a d>0 such that
½nx¼½nðxþdÞ. We obtain
ð1
0
~
kkðx;xÞuðxÞdxð1
0
~
kkðxþd;xÞuðxÞdx

aðx
0
~
kkðx;xÞuðxÞdxX
½nx
j¼1
k½nx;juj

þX
½nx
j¼1
k½nx;jujX
½nðxþdÞ
j¼1
k½nðxþdÞ;juj

þX
½nðxþdÞ
j¼1
k½nðxþdÞ;jujðxþd
0
~
kkðxþd;xÞuðxÞdx

<e
2þ0þe
2¼eð4:74Þ
which proves weak continuity of ~
kk from the right. For an arbitrary xA½0;1we
now fix an n>Nðe=2Þsuch that ½nx0nx and choose a d<0 such that ½nx¼
½nðxþdÞ. Inequality (4.74) holds again, proving weak continuity from the left.
With this we obtain the statement of the lemma, i.e.
~
kk ACwð½0;1;Lyð0;1ÞÞ:9ð4:75Þ
The following lemma shows how norms change under the above transforma-
tion.
Lemma 4 [BK1]. Suppose that two functions wðxÞALyð0;1Þand uðxÞALyð0;1Þ
satisfy the relationship
wðxÞ¼uðxÞðx
0
~
kkðx;xÞuðxÞdx;ExA½0;1;ð4:76Þ
where ~
kk ACwð½0;1;Lyð0;1ÞÞ:ð4:77Þ
Then there exist positive constants m and M, whose sizes depend only on ~
kk, such
that
mkwkyakukyaMkwky
and
mkwk2akuk2aMkwk2:
Proof of Theorem 1. We now complete the proof of Theorem 1 by combining
the results of Lemmas 1–4. In Lemma 1 we derived a coordinate transformation
that transforms the finite-dimensional system (2.6)–(2.8) into the finite-dimensional
system (2.12)–(2.14). As a result of the uniform boundedness of the transformation
(shown in Lemma 2) we obtained the coordinate transformation (4.76) that trans-
forms the system (2.5), (1.2) into the asymptotically stable system (2.15)–(2.17).
Due to the weak continuity proven in Lemma 3 the infinite-dimensional coordinate
Backstepping in Infinite Dimension 59

transformation results in the specific boundary condition
uðt;1Þ¼aðuÞ¼ð1
0
kðxÞuðt;xÞdx;ð4:78Þ
where
kðxÞ¼~
kkð1;xÞ;xA½0;1ð4:79Þ
with kALyð0;1Þ.
Figures 1 and 2 suggest the existence of smooth upper and lower envelopes to
the strongly oscillating approximating kernel functions. This, in turn, could mean
that the kernel function kðxÞcoincides with the average of these smooth functions
and hence is smooth itself, at least in this simple case of constant coe‰cients.
However, a kernel function in Lyð0;1Þis su‰cient for us both in theory and in
practice.
The convergence in Sobolev spaces W2;1
2(see, e.g. [A1]) of the finite-di¤erence
approximations obtained from (2.6)–(2.8) and (2.12)–(2.14) to the solutions of
(1.1)–(1.4) and (2.15)–(2.17) respectively is obtained using interpolation techni-
ques (see, e.g. [BJ].) Using Green’s function and the fixed-point method as
was done in [L1], we see that solutions to (1.1)–(1.4), (4.78) are, in fact, classical
solutions.
Introducing a variable change
sðt;xÞ¼wðt;xÞeðB=ð2eÞÞxþðcþB2=ð4eÞÞtð4:80Þ
we transform the wsystem (2.15)–(2.17) into a heat equation
stðt;xÞ¼esxxðt;xÞð4:81Þ
with homogeneous Dirichlet boundary conditions. The well-known (see, e.g. [C1])
stability properties of the ssystem (4.81) along with Lemma 4 proves the stability
statements of Theorem 1. 9
4.2. Case 2: Neumann Boundary Condition at x ¼0
In this section we prove Theorem 2. The proof is completely analogous to the
proof of Theorem 1 and we only outline the di¤erences.
Lemma 5. The elements of the sequence fki;jgdefined in (2.43)–(2.46) satisfy
jki;ijj<i
jþ1

Ljþ1
nþðijÞX
½j=2
l¼1
1
l
jl
l1

il
j2l

Lj2lþ1
nMl
n
þ2X
½ð j1Þ=2
l¼0X
j2l1
k¼0
lþk
l

il1
k

Mjlk
nLkþ1
n;ð4:82Þ
where l¼maxxA½0;1jlðxÞj.
60 D. M. Bos
ˇkovic
´, A. Balogh, and M. Krstic
´

Proof. The proof goes along the same lines as the proof of the Lemma 1. We
first obtain estimates for the initial values of k’s as
jk1;1jah2
eþBh ðcþlÞ¼Ln;ð4:83Þ
jk2;1jah2
eþBh ðcþlÞþ e
eþBh

h2
eþBh ðcþlÞ¼L2
nþMnLn
aL2
nþ2MnLn;ð4:84Þ
jk2;2ja2h2
eþBh ðcþlÞ¼2Ln;ð4:85Þ
jk3;1jah2
eþBh ðcþlÞþ e
eþBh

2h2
eþBh ðcþlÞþ e
ðeþBhÞ
h2
ðeþBhÞðcþlÞ
¼ðLnþMnÞ2LnþMnLn¼L3
nþ2MnL2
nþM2
nLnþMnLn
aL3
nþ4MnL2
nþ2M2
nLnþMnLn;ð4:86Þ
jk3;2ja3h4
ðeþBhÞ2ðcþlÞ2þe
eþBh
h2
eþBh ðcþlÞ¼3L2
nþMnLn
a3L2
nþ2MnLn;ð4:87Þ
jk3;3ja3h2
eþBh ðcþlÞ¼3Ln;ð4:88Þ
and for ki;iand ki;i1as
jki;ijaih2
eþBh ðcþlÞ¼iLn;ð4:89Þ
jki;i1jaiði1Þ
2
h4
ðeþBhÞ2ðcþlÞ2þe
eþBh
h2
eþBh ðcþlÞ¼iði1Þ
2L2
nþMnLn
aiði1Þ
2L2
nþ2MnLn:ð4:90Þ
Finally we obtain inequality (4.82) of Lemma 5 using the general identity for ki;j
and mathematical induction. 9
The only thing left now is to prove the uniform boundedness of ðnþ1Þkn;nqn ,
with the bound on kn;nqn given by
jkn;njj¼jkn;nqnj
an
qn þ1

Lqnþ1
nþðnqnÞX
½qn=2
l¼1
1
l
qn l
l1

nl
qn 2l

Lqn2lþ1
nMl
n
þ2X
½ðqn1Þ=2
l¼0X
qn2l1
k¼0
lþk
l

nl1
k

Mqnlk
nLkþ1
n:ð4:91Þ
Backstepping in Infinite Dimension 61

Lemma 6. The sequence fðnþ1Þkn;jgj¼1;...;n;nb1remains bounded uniformly in n
and j as n !y.
Proof. We can write, according to (4.91),
ðnþ1Þjkn;nqnj
aðnþ1Þn
qn þ1

E
ðnþ1Þ2
!
qnþ1
þðnþ1ÞðnqnÞX
½qn=2
l¼1
1
l
qn l
l1

nl
qn 2l

E
ðnþ1Þ2
!
qn2lþ1
Ml
n
þ2ðnþ1ÞX
½ðqn1Þ=2
l¼0X
qn2l1
k¼0
lþk
l

nl1
k

E
ðnþ1Þ2
!
kþ1
Mqnlk
n:ð4:92Þ
Since the first two terms are identical to terms appearing in expression (4.67), we
only have to estimate the third term from (4.92). Using the simple inequalities
lþk
l

¼lþk
k

aðlþkÞk;ð4:93Þ
nl1
k

an
k

;ð4:94Þ
and
lþkalþkmax ¼lþqn 2l1¼qn l1aqn <nþ1;ð4:95Þ
we obtain
ðnþ1ÞX
½ðqn1Þ=2
l¼0X
qn2l1
k¼0
lþk
l

nl1
k

E
ðnþ1Þ2
!
kþ1
Mqnlk
n
aE
nþ1X
½ðqn1Þ=2
l¼0X
qn2l1
k¼0
lþk
nþ1

kn
k
 E
nþ1

k
1þR
nþ1

qnlk
aE
nþ11þR
n

nX
½ðqn1Þ=2
l¼0X
qn
s¼0
n
s
E
n

s
1ns
aE
nþ1eR1þE
n

nX
½ðqn1Þ=2
l¼0
1
aEeRþE:
This proves the lemma. 9
5. Extension of the Result to the Neumann Type of Actuation
In previous sections we derived control laws of the Dirichlet type to stabilize the
system. We show here briefly how to extend them to the Neumann case.
62 D. M. Bos
ˇkovic
´, A. Balogh, and M. Krstic
´

Dirichlet control uðt;1Þwas obtained by setting wðt;1Þ¼0 in the transformation
wðt;xÞ¼uðt;xÞðx
0
~
kkðx;xÞuðxÞdx;xA½0;1:ð5:96Þ
If one uses uxðt;1Þfor feedback, then the boundary condition of the target system
at x¼1 will be
wxðt;1Þ¼C1wðt;1Þ;ð5:97Þ
which can be shown to be exponentially stabilizing for both wðt;0Þ¼0 and
wxðt;0Þ¼0 for su‰ciently large c>0. We obtain the expression for the Neu-
mann actuation in the original ucoordinates by implementing the Neumann
boundary condition (5.97) as
uxðt;1Þ¼C1uðt;1Þþ~
kkð1;1Þuðt;1Þþð1
0
~
kkxð1;xÞuðxÞdxC1ð1
0
~
kkð1;xÞuðxÞdx;
xA½0;1;ð5:98Þ
where ~
kkxdenotes a partial derivative with respect to the first variable.
In terms of the discretized original and target systems, and discrete coordinate
transformation ai¼Pi
j¼1ki;juj,i¼1;...;n, the discretized equivalent udis
xðt;1Þof
uxðt;1Þis
udis
xðt;1Þ¼C1unþanan1
hC1an
¼C1unþkn;n
hunþX
n1
j¼1
kn;jkn1;j
hujC1X
n
j¼1
kn;juj:ð5:99Þ
Comparing (5.99) and (5.98) term by term, it is evident that uniform boundedness
of ki;j=hwill guarantee that
udis
xðt;1Þ!
n!yuxðt;1Þð5:100Þ
for all t>0.
6. Extension to the Case with Nonzero Integral Term on the Right-Hand
Side of the System Equation
In previous sections we showed how to stabilize a less general case of the system
(1.1) with no integral term on the right-hand side of the system equation. In this
section we present the extension to
utðt;xÞ¼euxxðt;xÞþBuxðt;xÞþlðxÞuðt;xÞþðx
0
fðx;xÞuðt;xÞdx;
xAð0;1Þ;t>0;ð6:101Þ
with a homogeneous Dirichlet boundary condition at x¼0,
uðt;0Þ¼0;t>0;ð6:102Þ
and Dirichlet boundary condition
uðt;1Þ¼aðuðtÞÞ;t>0;ð6:103Þ
used for actuation at the other end. One immediately notices the overlap between
the extension presented in this section and the results from Section 2.1 that are a
Backstepping in Infinite Dimension 63

special case. A natural question to ask would be why the problem was not given
in its most general form in Section 2.1. The reason is that as of now we have not
succeeded in extending the results from Section 2.2 to the most general case, and
it is easy to see why. By comparing Dirichlet and Neumann control results for the
case without the integral term one can see that the expression for the kernel in
the Neumann case (2.47) is much more complex than the Dirichlet one given by
expression (2.31). The level of complexity increases even more when the integral
term is present, as will be seen in this section, which prevented us from obtaining
a closed-form expression in the Neumann case with the integral term. The only
reason for presenting the less general Dirichlet result first was to draw the parallel
between the cases for uðt;0Þ¼0 and uxðt;0Þ¼0 and present only the di¤erences.
As in Section 2.1 we choose discretization of (2.15)–(2.17) for our target sys-
tem, obtain the recursive expression for the discretized backstepping-style coordi-
nate transformation as
ai¼ðeþBhÞ1(ð2eþBh þch2Þai1eai2ðliþcÞh2ui
h3X
i1
k¼1
fi;kukþqai1
qu1
ððeþBhÞu2ð2eþBh l1h2Þu1Þ
þX
i1
j¼2
qai1
quj ðeþBhÞujþ1ð2eþBh ljh2Þuj
þeuj1þh3X
j1
k¼1
fj;kuk!);ð6:104Þ
and then, assuming the linear form for ai’s (ai¼Pi
j¼1ki;juj), obtain the general
recursive relationship for the kernel as
ki;1¼h2
eþBh ðcþl1Þki1;1þe
eþBh ðki1;2ki2;1Þ
h3
eþBh fi;1þh3
eþBh X
i1
l¼2
fl;1ki1;l;ð6:105Þ
ki;j¼h2
eþBh ðcþljÞki1;jþki1;j1þe
eþBh ðki1;jþ1ki2;jÞ
h3
eþBh fi;jþh3
eþBh X
i1
l¼jþ1
fl;jki1;l;j¼2;...;i2;ð6:106Þ
ki;i1¼h2
eþBh ðcþli1Þki1;i1þki1;i2h3
eþBh fi;i1;ð6:107Þ
ki;i¼ki1;i1h2
eþBh ðcþliÞ;ð6:108Þ
64 D. M. Bos
ˇkovic
´, A. Balogh, and M. Krstic
´

for i¼4;...;nwith initial conditions
k1;1¼ h2
eþBh ðcþl1Þ;ð6:109Þ
k2;1¼ h4
ðeþBhÞ2ðcþl1Þ2h3
eþBh f2;1;ð6:110Þ
k2;2¼ h2
eþBh ðcþl1Þþ h2
eþBh ðcþl2Þ

;ð6:111Þ
k3;1¼ h6
ðeþBhÞ3ðcþl1Þ3h2
eþBh ðcþl1Þh3
eþBh f2;1e
eþBh
h2
eþBh ðcþl2Þ
h3
eþBh f3;1h3
eþBh
h2
eþBh ðcþl1Þþ h2
eþBh ðcþl2Þ

f2;1;ð6:112Þ
k3;2¼ h2
eþBh ðcþl2Þh2
eþBh ðcþl1Þþ h2
eþBh ðcþl2Þ

h4
ðeþBhÞ2ðcþl1Þ2h3
eþBh f2;1h3
eþBh f3;2;ð6:113Þ
k3;3¼ h2
eþBh ðcþl1Þþ h2
eþBh ðcþl2Þþ h2
eþBh ðcþl3Þ

:ð6:114Þ
For the simple case when lðxÞ1l¼constant and fðx;yÞ1f¼constant,
(6.105)–(6.114) can be solved explicitly to obtain
ki;ij¼ i
jþ1

Ljþ1
nðijÞX
½j=2
l¼1
1
l
jl
l1

il
j2l

Lj2lþ1
nMl
n
ðijÞX
½ð jþ1Þ=2
l¼1
1
lPl
nX
½ð jþ1Þ=2l
m¼0
lþm1
l1

Mm
n
X
jþ12l2m
k¼0
jl2mk
l1

kþlþm1
k

im
kþl1

Lk
nð6:115Þ
for i¼1;...;n,j¼0;...;i1, where
Pn¼h3
eþBh f:ð6:116Þ
The precise formulation of the main result in this section is summarized in the fol-
lowing theorem.
Theorem 3. For any lðxÞALyð0;1Þ,fðx;yÞALyð½0;1½0;1Þ, and e;c>0
there exists a function k ALyð0;1Þsuch that for any u0ALyð0;1Þthe unique
classical solution uðt;xÞAC1ðð0;yÞ;C2ð0;1ÞÞ of system (6.101)–(6.103), (2.34) is
exponentially stable in the L2ð0;1Þand maximum norms with decay rate c. The
precise statements of stability properties are the following: there exists positive con-
Backstepping in Infinite Dimension 65

stant M5such that for all t >0,
kuðtÞk2aMku0k2ect ð6:117Þ
and
max
xA½0;1juðt;xÞj aMsup
xA½0;1
ju0ðxÞject:ð6:118Þ
The proof of Theorem 3 is completely analogous to the proof of Theorem 1 and
we only outline the di¤erences.
Lemma 7. The elements of the sequence fki;jgdefined in (6.105)–(6.114) satisfy
jki;ijjai
jþ1

Ljþ1
nþðijÞX
½j=2
l¼1
1
l
jl
l1

il
j2l

Lj2lþ1
nMl
n
þðijÞX
½ð jþ1Þ=2
l¼1
1
lPl
nX
½ð jþ1Þ=2l
m¼0
lþm1
l1

Mm
n
X
jþ12l2m
k¼0
jl2mk
l1

kþlþm1
k

im
kþl1

Lk
n;
ð6:119Þ
where l¼maxxA½0;1jlðxÞj and f ¼supðx;yÞA½0;1½0;1jfðx;yÞj.
Remark 5. There is equality in (6.119) when lðxÞ1l¼constant >0 and
fðx;yÞ1f¼constant >0.
Proof. The right-hand side of (6.109)–(6.114) can be estimated to obtain esti-
mates for the initial values of k’s:
jk1;1jah2
eþBh ðcþlÞ¼Ln;ð6:120Þ
jk2;1jah4
ðeþBhÞ2ðcþlÞ2þh3
eþBh f¼L2
nþPn;ð6:121Þ
jk2;2ja2h2
eþBh ðcþlÞ¼2Ln;ð6:122Þ
jk3;1jah6
ðeþBhÞ3ðcþlÞ3þ3h2
eþBh ðcþlÞh3
eþBh fþe
eþBh
h2
eþBh ðcþlÞ
þh3
eþBh f¼L3
nþ3LnPnþMnLnþPn;ð6:123Þ
jk3;2ja3h4
ðeþBhÞ2ðcþlÞ2þ2h3
eþBh f¼3L2
nþ2Pn;ð6:124Þ
jk3;3ja3h2
eþBh ðcþlÞ¼3Ln:ð6:125Þ
5Mgrows with c,l, and 1=e.
66 D. M. Bos
ˇkovic
´, A. Balogh, and M. Krstic
´

We then go from j¼ibackwards and obtain
jki;ijaih2
eþBh ðcþlÞ¼iLn;ð6:126Þ
jki;i1jaiði1Þ
2
h4
ðeþBhÞ2ðcþlÞ2þði1Þh3
eþBh f
¼iði1Þ
2L2
nþði1ÞPn;ð6:127Þ
from (6.107) and (6.108), respectively. Finally we obtain inequality (6.119)
of Lemma 7 using the general identity (6.106) and mathematical induc-
tion. 9
To prove the uniform boundedness of ðnþ1Þkn;nqn we start by finding a bound
on kn;nqn as
jkn;njj¼jkn;nqnj
an
qn þ1

Lqnþ1
nþðnqnÞX
½qn=2
l¼1
1
l
qn l
l1

nl
qn 2l

Lqn2lþ1
nMl
n
þðnqnÞX
½ðqnþ1Þ=2
l¼1
1
lPl
nX
½ðqnþ1Þ=2l
m¼0
lþm1
l1

Mm
n
X
qnþ12l2m
k¼0
qn l2mk
l1

kþlþm1
k

nm
kþl1

Lk
n;ð6:128Þ
where
Pn¼h3
eþBh fajfjh3
e=2¼H
ðnþ1Þ3;ð6:129Þ
Hbeing defined as
H¼2jfj
e:ð6:130Þ
The uniform boundedness of the kernel is given by the following lemma.
Lemma 8. The sequence fðnþ1Þkn;jgj¼1;...;n;nb1remains bounded uniformly in n
and j as n !y.
Backstepping in Infinite Dimension 67

Proof. We can write, according to (6.128),
ðnþ1Þjkn;nqnj
aðnþ1Þn
qn þ1

E
ðnþ1Þ2
!
qnþ1
þðnþ1ÞðnqnÞX
½qn=2
l¼1
1
l
qn l
l1

nl
qn 2l

E
ðnþ1Þ2
!
qn2lþ1
Ml
n
þðnþ1ÞðnqnÞX
½ðqnþ1Þ=2
l¼1
1
lPl
nX
½ðqnþ1Þ=2l
m¼0
lþm1
l1

Mm
n
X
qnþ12l2m
k¼0
qn l2mk
l1

kþlþm1
k

nm
kþl1

E
ðnþ1Þ2
!
k
:ð6:131Þ
Since the first two terms are identical to terms appearing in expression (4.67), we
only have to estimate the third term in (6.131). We start with simple inequalities
nm
kþl1

an
k

;ð6:132Þ
qn l2mk
l1

ðnþ1Þl1a1;ð6:133Þ
kþlþm1
k

ðnþ1Þka1;ð6:134Þ
and obtain
ðnþ1ÞðnqnÞX
½ðqnþ1Þ=2
l¼1
1
lPl
nX
½ðqnþ1Þ=2l
m¼0
lþm1
l1

Mm
n
X
qnþ12l2m
k¼0
qn l2mk
l1

kþlþm1
k

nm
kþl1

E
ðnþ1Þ2
!
k
aðnþ1Þ2X
½ðqnþ1Þ=2
l¼1
1
l
Hl
ðnþ1Þ2lþ1X
½ðqnþ1Þ=2l
m¼0
1þR
n

mlþm1
l1

68 D. M. Bos
ˇkovic
´, A. Balogh, and M. Krstic
´

X
qnþ12l2m
k¼0
qn l2mk
l1

ðnþ1Þl1
kþlþm1
k

ðnþ1Þk
nm
kþl1

E
nþ1

k
aHX
½ðqnþ1Þ=2
l¼1
Hl1
ðnþ1Þ2l1X
½ðqnþ1Þ=2l
m¼0
1þR
n

mlþm1
l1

X
qnþ12l2m
k¼0
n
k
 E
nþ1

k
1nk
aH1þR
n

nX
½ðqnþ1Þ=2
l¼1
Hl1
ðnþ1Þ2l10
B
@
qn þ1
2

1
l11
C
A
X
½ðqnþ1Þ=2l
m¼0
1X
n
k¼0
n
k
E
n

k
1nk
aH1þR
n

n
1þE
n

nX
½ðqnþ1Þ=2
l¼10
B
@
qn þ1
2

1
l11
C
A
H
n

l1½ðqn þ1Þ=2lþ1
ðnþ1Þl
aH1þR
n

n
1þE
n

nX
n
l¼1
n1
l1

H
n

l1n
ðnþ1Þl
aH1þR
n

n
1þE
n

nX
n
l¼0
n1
l

H
n

l1
1nl1
aH1þR
n

n
1þE
n

n
1þH
n

n
aHeðRþEþHÞ:
This proves the lemma. 9
7. Simulation Study
In this section we present the simulation results for a linearization of an adiabatic
chemical tubular reactor. For the case when Peclet numbers for heat and mass
transfer are equal (Lewis number of unity) the two equations for the temperature
and concentration can be reduced to one equation [HH1]:
yt¼1
Pe yxx yxþDaðbyÞey=ð1þmyÞ;xAð0;1Þ;t>0;ð7:135Þ
yxðt;0Þ¼Peyðt;0Þ;ð7:136Þ
yxðt;1Þ¼0;ð7:137Þ
Backstepping in Infinite Dimension 69

where Pe stands for the Peclet number, Da for the Damko
¨hler number, mfor the
dimensionless activation energy, and bfor the dimensionless adiabatic temperature
rise. For a particular choice of system parameters (Pe ¼6, Da ¼0:05, m¼0:05,
and b¼10) system (7.135)–(7.137) has three equilibria [HH2]. As shown in [HH2],
the middle profile is unstable while the outer two profiles are stable. The equilib-
rium profiles for this case are shown in Fig. 3. Linearizing the system around the
unstable equilibrium profile yðxÞwe obtain
yt¼1
Pe yxx yxþDaGðyðxÞÞy;ð7:138Þ
yxðt;0Þ¼Peyðt;0Þ;ð7:139Þ
yxðt;1Þ¼0;ð7:140Þ
where ynow stands for the deviation variable from the steady state yðxÞ, and Gis
a spatially dependent coe‰cient defined as
GðyÞ¼ by
ð1þmyÞ21
"#
ey=ð1þmyÞ:ð7:141Þ
Although not obvious from (7.138)–(7.140), it is physically justifiable to apply
feedback boundary control at the 0-end only. In real applications control would
be implemented through small variations of the inlet temperature and the inlet
reactant concentration (see [VA] and [HH1]). Since our control algorithm assumes
actuation at the 1-end we transform the original system (7.138)–(7.140) by intro-
00.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1
0
1
2
3
4
5
6
7
8
9
10
¯
θ(ξ)
ξ
Fig. 3. Steady-state profiles for the adiabatic chemical tubular reactor with Pe ¼6, Da ¼0:05,
m¼0:05, and b¼10.
70 D. M. Bos
ˇkovic
´, A. Balogh, and M. Krstic
´

ducing a variable change
uðt;xÞ¼yð1xÞ:ð7:142Þ
In the new set of variables the system (7.138)–(7.140) becomes
utðt;xÞ¼ 1
Pe uxxðt;xÞþuxðt;xÞþDagðxÞuðt;xÞ;ð7:143Þ
uxðt;0Þ¼0;ð7:144Þ
uxðt;1Þ¼Peuðt;1ÞþDuxðt;1Þ;ð7:145Þ
where gðxÞis defined as
gðxÞ¼Gðyð1xÞÞ;ð7:146Þ
and Duxðt;1Þstands for the control law to be designed. All simulations pre-
sented in this study were done using the BTCS finite-di¤erence method for
n¼200 and a time step equal to 0.001 s. Although we have tested the con-
troller for several di¤erent combinations of initial distributions and target sys-
tems, we only present results for c¼0:1 and uð0;xÞ ¼ ððo=ðPeÞÞ cosðoxÞþ
sinðoxÞÞ,o¼1:48396. This particular initial distribution has been constructed
to satisfy the imposed boundary conditions on both ends in the open-loop case
exactly.
As expected, since the system (7.143)–(7.145) represents a linearization around
the unstable steady state, the open-loop system (Duxðt;1Þ¼0) is unstable, and
the state grows exponentially as shown in Fig. 4. We now apply the approach
outlined in Section 2.2 and obtain a coordinate transformation that transforms
0
0.5
1
1.5
2
0
0.2
0.4
0.6
0.8
1
-
14
-
12
-
10
-
8
-
6
-
4
-
2
0
u(t,x)
t
x
Fig. 4. Open-loop response of the system.
Backstepping in Infinite Dimension 71

the discretization of (7.143)–(7.145) into discretization of the asymptotically
stable system
wtðt;xÞ¼ 1
Pe wxxðt;xÞþwxðt;xÞcwðt;xÞ;ð7:147Þ
wxðt;0Þ¼0;ð7:148Þ
wxðt;1Þ¼Pewðt;1Þ:ð7:149Þ
The control is implemented as
Duxðt;1Þ¼anðu1;...;unÞan1ðu1;...;un1Þ
hþPeanðu1;...;unÞ;ð7:150Þ
where hstands for the discretization step in the controller design. The closed-loop
response of the system with a controller designed for n¼200 and c¼0:1 and the
corresponding control e¤ort Duxðt;1Þare shown in Fig. 5.
From an applications point of view it would of interest to see whether the sys-
tem (7.143)–(7.145) could be stabilized with a reduced version of the control law
(7.150). By a reduced-order controller we assume a controller designed on a much
coarser grid than the one used for simulating the response of the system. The
expectation that the system might be rendered stable with a low-order back-
stepping controller is based on our past experience in designing nonlinear low-
order backstepping controllers for the heat convection loop [BK2], stabilization of
unstable burning in solid propellant rockets [BK3], and stabilization of chemical
tubular reactors [BK4]. The idea of using controllers designed using only a small
00.5 11.5 2
0
0.2
0.4
0.6
0.8
1
-
2
0
2
00.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2
0
10
20
30
40
xt
u(t,x)
Dux(t,1)
t
Fig. 5. Closed-loop response of the system with a controller that uses full state information. (First
row: uðt;xÞ; second row: the control e¤ort Duxðt;1Þ.)
72 D. M. Bos
ˇkovic
´, A. Balogh, and M. Krstic
´

number of steps of backstepping to stabilize the system for a certain range of the
open-loop instability is based on the fact that in most real-life systems only a finite
number of open-loop eigenvalues is unstable. The conjecture is then to apply a
low-order backstepping controller (controller that uses only a small number of
state measurements) that is capable of detecting the occurrence of instability from
a limited number of measurements, and stabilize the system. Indeed, simulation
results show that we can successfully stabilize the unstable equilibrium using a
kernel obtained with only two steps of backstepping (using only two state mea-
surements ut;1
3

and ut;2
3

) with the same c¼0:1. By a controller designed using
only two steps of backstepping we assume a controller designed on a very coarse
grid, namely, on a grid with just three points. In this case control is implemented
by substituting a1and a2in expression (7.150) for Duxðt;1Þ, where a1and a2are
obtained from expressions (2.25), (2.44), and (2.27) with h¼1
3,e¼1=ðPeÞ,B¼1,
l1¼Dag 1
3
,l2¼Dag 2
3
,u1¼ut;1
3

, and u2¼ut;2
3

. The closed-loop response
of the system with a reduced-order controller and corresponding control e¤ort
Duxðt;1Þare shown in Fig. 6.
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0
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Backstepping in Infinite Dimension 75