# Backstepping for PDEs

**DOI:**https://doi.org/10.1007/978-1-4471-5102-9_100023-1

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## Abstract

Backstepping is an elegant constructive method, with roots in finite-dimensional control theory, that allows the solution of numerous boundary control and estimation problems for partial differential equations (PDEs). This entry reviews the main ingredients of the method, namely, the concepts of a backstepping invertible transformation, a target system, and the kernel equations. As a basic example, stabilization of a reaction-diffusion equation is explained.

## Keywords

Backstepping Boundary control Distributed parameter systems Lyapunov function Partial differential equations (PDEs) Stabilization## Synonyms

## Introduction to PDE Backstepping

In the context of partial differential equations (PDEs), backstepping is a constructive method, originated in the 2000s, mainly used to design boundary controllers and observers, both adaptive and nonadaptive, for numerous classes of systems. Its name comes from the use of the so-called backstepping transformation (a Volterra-type integral transform) as a tool to find feedback control laws and observer gains.

The method has three main ingredients. First, one needs to select a *target system* which verifies the desired properties (most often stability, proven with a Lyapunov function), but still closely resembles the original system. Next, an integral transformation (the backstepping transformation) is posed to map the original plant into the target system in the appropriate functional spaces. The invertibility of the transformation needs to be shown. Finally, using the original and target systems and the transformation, the *kernel equations* are found. Their solution is the kernel of the integral transformation, which in turn determines the control law. These equations are typically of Goursat type, that is, hyperbolic boundary problems on a triangular domain (with boundary values on two sides and the third side determining the control law), and can usually be proven solvable by transforming the boundary problems to integral equations and then using the method of successive approximations. Tying up all these elements, stability of the closed-loop system is then obtained, based on the stability of the target system which is inherited by the closed-loop system as a result of the structure and well-posedness of the transformation.

These ingredients are always present when using PDE backstepping and are closely connected among themselves. Judiciously choosing the target system will result in solvable kernel equations and an invertible transformation. On the other hand, an ill-chosen target system typically results in kernel equations that cannot be solved or even properly formulated, or a non-invertible transformation.

Next, the rationale of the method is illustrated for a basic case, namely, stabilization of a reaction-diffusion equation.

## Reaction-Diffusion Equation

*u*(

*t*,

*x*) is the state, for

*x*∈ [0, 1] and

*t*> 0, with

*𝜖*> 0 and

*λ*(

*x*) a continuous function, with boundary conditions

*U*(

*t*) is the actuation variable. These are the so-called Dirichlet-type boundary conditions, but Neumann- (

*u*

_{x}(

*t*, 0) = 0) or Robin-type (

*u*

_{x}(

*t*, 0) =

*qu*(

*t*, 0)) boundary conditions are also frequently considered. Denote the initial condition

*u*(0,

*x*) as

*u*

_{0}(

*x*).

The system (1)–(2) has an equilibrium, namely, *u* ≡ 0, which is unstable if *λ*(*x*) is sufficiently large. Thus, the control problem is to design a feedback *U*(*t*) such that the origin becomes stable (stability will be made precise by using a Lyapunov approach).

### Target System (and Its Stability)

*L*

^{2}([0, 1]) space of functions (whose square is integrable in the interval [0, 1]), endowed with the norm \(\Vert f \Vert ^2 = \int _0^1 f^2(x) dx\).

*w*

_{0}the initial condition of (3)–(4), it is well-known that if

*w*

_{0}∈

*L*

^{2}([0, 1]), then

*w*(

*t*, ⋅) ∈

*L*

^{2}([0, 1]) for each

*t*> 0 and, using ∥

*w*(

*t*, ⋅)∥

^{2}as a Lyapunov function, the following exponential stability result is verified:

*α*is a positive number.

### Backstepping Transformation (and Its Inverse)

*k*(

*x*,

*ξ*) is known as the backstepping kernel. One of the properties of a Volterra-type transformation is that it is invertible under very mild conditions on the kernel

*k*(

*x*,

*ξ*), for instance, if it is bounded. Assuming for the moment that the kernel verifies this property, then an inverse transformation can be posed as

*l*(

*x*,

*y*) is known as the inverse kernel, and it is also bounded.

It is easy to see that both transformations (6) and (7) map *L*^{2}([0, 1]) functions into *L*^{2}([0, 1]) functions. In particular, if *u* and *w* are related by the transformations as in (6) and (7), it follows that ∥*u*∥^{2} ≤ *C*_{1}∥*w*∥^{2} and ∥*w*∥^{2} ≤ *C*_{2}∥*u*∥^{2} for some *C*_{1}, *C*_{2} > 0.

### Kernel Equations

*k*(

*x*,

*ξ*) verifies the following equation

*w*. Then, integrating by parts, one gets a weak formulation of (8)–(9).

*δ*=

*x*+

*ξ*and

*η*=

*x*−

*ξ*, and denoting \(G(\delta ,\eta )=k(x,\xi )=k\left (\frac {\delta +\eta }{2},\frac {\delta -\eta }{2}\right )\). Then,

*G*(

*δ*,

*η*) verifies the following equation:

*G*can be integrated, yielding an integral equation for

*G*

*n*≥ 1, Then it can be shown that

*G*(

*δ*,

*η*) =lim

_{n→∞}

*G*

_{n}(

*δ*,

*η*). The function

*G*

_{n}can be computed recursively and used to approximate symbolically

*G*and thus

*k*. This procedure shows that (8)–(9) is well-posed (the limit always exists) and its solution

*k*is continuous (and thus invertible). In fact, for constant

*λ*, the exact solution is

*I*

_{1}is the first-order modified Bessel function of the first kind.

### Feedback Law and Closed-Loop System Stability

*U*in (2) is what makes the original system behave like the target system and can be found by setting

*x*= 1 in the backstepping transformation (6) and using the boundary conditions (2) and (4), which yields the feedback law

## Summary and Future Directions

Backstepping as a method for control of PDEs is a fairly recent development, and the area of control of PDEs is in itself an emerging and active area of research with great potential in many engineering applications that cannot be modeled solely by finite-dimensional dynamics. To learn more about the basics, we recommend the book (Anfinsen and Aamo 2019) which contains a didactic exposition of the initial backstepping results, providing details to design controllers and observers for parabolic equations and others. While we only dealt with the basic theoretical developments, the subject on the other hand has been steadily growing in the last decade. The technical literature now contains a wealth of results in application areas as diverse as oil pipe flows, battery health estimation, thermoacoustics, multi-phase flows, 3-D printing, or traffic control, among others, with hundreds of additional references. To name a few, we would like to mention several books written on the subject which include interesting recent and forth coming developments in topics such as flow control (Krstic 2009), adaptive control of parabolic PDEs (Krstic and Smyshlyaev 2008), adaptive control of hyperbolic PDEs (Smyshlyaev and Krstic 2010), or delays (Vazquez and Krstic 2008).

## Cross-References

## Bibliography

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- Smyshlyaev A, Krstic M (2010) Adaptive control of parabolic PDEs. Princeton University Press, PrincetonCrossRefGoogle Scholar
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