# Encyclopedia of Systems and Control

Living Edition
| Editors: John Baillieul, Tariq Samad

# Backstepping for PDEs

• Rafael Vazquez
• Miroslav Krstic
Living reference work entry
DOI: https://doi.org/10.1007/978-1-4471-5102-9_100023-1

## 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

## 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

Consider a reaction-diffusion equation
\displaystyle \begin{aligned} \begin{array}{rcl} u_t&\displaystyle =&\displaystyle \epsilon u_{xx}+\lambda(x) u,{} \end{array} \end{aligned}
(1)
where 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
\displaystyle \begin{aligned} \begin{array}{rcl} u(t,0)&\displaystyle =&\displaystyle 0,\quad u(t,1)=U(t),{} \end{array} \end{aligned}
(2)
where U(t) is the actuation variable. These are the so-called Dirichlet-type boundary conditions, but Neumann- (ux(t, 0) = 0) or Robin-type (ux(t, 0) = qu(t, 0)) boundary conditions are also frequently considered. Denote the initial condition u(0, x) as u0(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)

Since the last term in (1) is a potential source of instability, the most natural objective of feedback would be to eliminate it, thus reaching the following target system
\displaystyle \begin{aligned} \begin{array}{rcl} w_t&\displaystyle =&\displaystyle \epsilon w_{xx},{} \end{array} \end{aligned}
(3)
that is, a heat equation, with homogeneous Dirichlet boundary conditions
\displaystyle \begin{aligned} \begin{array}{rcl} w(t,0)&\displaystyle =&\displaystyle 0,\quad w(t,1)=0.{} \end{array} \end{aligned}
(4)
To study the target system stability, consider the L2([0, 1]) space of functions (whose square is integrable in the interval [0, 1]), endowed with the norm $$\Vert f \Vert ^2 = \int _0^1 f^2(x) dx$$.
Denoting by w0 the initial condition of (3)–(4), it is well-known that if w0 ∈ L2([0, 1]), then w(t, ⋅) ∈ L2([0, 1]) for each t > 0 and, using ∥w(t, ⋅)∥2 as a Lyapunov function, the following exponential stability result is verified:
\displaystyle \begin{aligned} \Vert w(t,\cdot) \Vert ^2 \leq \mathrm{e}^{-\alpha t} \Vert w_0 \Vert ^2, {} \end{aligned}
(5)
where α is a positive number.

### Backstepping Transformation (and Its Inverse)

Now, pose the following coordinate transformation
\displaystyle \begin{aligned} w(t,x)=u(t,x)-\int_0^x k(x,\xi) u(t,\xi) d\xi,{} \end{aligned}
(6)
to map the system (1) into (3). The transformation (6), whose second term is a Volterra-type integral transformation (because of its lower-triangular structure), is the backstepping transformation, and its kernel 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
\displaystyle \begin{aligned} u(t,x)=w(t,x)+\int_0^x l(x,\xi)w(t,\xi)d\xi, {} \end{aligned}
(7)
where 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 L2([0, 1]) functions into L2([0, 1]) functions. In particular, if u and w are related by the transformations as in (6) and (7), it follows that ∥u2 ≤ C1w2 and ∥w2 ≤ C2u2 for some C1, C2 > 0.

### Kernel Equations

The kernel k(x, ξ) verifies the following equation
\displaystyle \begin{aligned} \begin{array}{rcl}\epsilon k_{xx}(x,\xi)=\epsilon k_{\xi\xi}(x,\xi)+\lambda(\xi)k(x,\xi),{} \end{array} \end{aligned}
(8)
with boundary conditions
\displaystyle \begin{aligned} \epsilon k (x,0)=-0,\quad k(x,x)=-\frac{1}{2\epsilon} \int_0^x \lambda(\xi) d\xi.\quad {} \end{aligned}
(9)
Equation (8) is a hyperbolic equation of Goursat type. The way to derive (8), along with (9), is by substituting the transformation (6) into the target system (3) and eliminating w. Then, integrating by parts, one gets a weak formulation of (8)–(9).
The PDE (8)–(9) is well-posed and can be solved numerically fast and efficiently. It can be also reduced to an integral equation, by defining 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:
\displaystyle \begin{aligned} \begin{array}{rcl} 4\epsilon G_{\delta \eta}&\displaystyle =&\displaystyle \lambda \left(\frac{\delta-\eta}{2}\right)G(\delta,\eta), {} \end{array} \end{aligned}
(10)
with boundary conditions
\displaystyle \begin{aligned} \begin{array}{rcl} G(\delta,\delta)&\displaystyle =&\displaystyle 0, \\ G(\delta,0)&\displaystyle =&\displaystyle -\frac{1}{4\epsilon} \int_0^\delta \lambda \left(\frac{\tau}{2} \right) d\tau.\qquad \end{array} \end{aligned}
(11)
Equation (10) for G can be integrated, yielding an integral equation for G
\displaystyle \begin{aligned} G(\delta,\eta)&= -\frac{1}{4\epsilon} \int_\eta^\delta \lambda \left(\frac{\tau}{2} \right) d\tau\\ &\quad +\frac{1}{4\epsilon} \int_\eta^\delta \int_0^\eta \lambda\left(\frac{\tau-s}{2}\right) G(\tau,s) ds d\tau.{} \end{aligned}
(12)
The method of successive approximations can be used to solve (12). Define $$G_0(\delta ,\eta ) = -\frac {1}{4\epsilon } \int _\eta ^\delta \lambda \left (\frac {\tau }{2} \right ) d\tau$$ and, for n ≥ 1, Then it can be shown that G(δ, η) =limnGn(δ, η). The function Gn 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
\displaystyle \begin{aligned} k(x,\xi)=-\frac{\lambda}{\epsilon} \xi \frac{I_{1}\left(\sqrt{\frac{\lambda}{\epsilon}(x^{2}-\xi^{2})}\right)} {\sqrt{\frac{\lambda}{\epsilon}(x^{2}-\xi^{2})}}, \end{aligned}
where I1 is the first-order modified Bessel function of the first kind.

### Feedback Law and Closed-Loop System Stability

The feedback 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
\displaystyle \begin{aligned} U(t)=\int_0^1 k(1,\xi) u(t,\xi) d\xi.{} \end{aligned}
(14)
Finally, using (5) and the properties of the transformations (6) and (7), one can obtain
\displaystyle \begin{aligned} \Vert u(t) \Vert^2 &\leq C_2 \Vert w(t) \Vert^2 \leq C_2 \mathrm{e}^{-\alpha t} \Vert w_0 \Vert ^2\\ &\leq C_1 C_2 \mathrm{e}^{-\alpha t} \Vert u_0 \Vert ^2, \end{aligned}
(15)
thus showing exponential stability for the origin of the system (1)–(2).

## 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).

## Bibliography

1. Anfinsen H, Aamo OM (2019) Adaptive control of hyperbolic PDEs. Springer, Cham
2. Krstic M (2009) Delay compensation for nonlinear, adaptive, and PDE systems. Birkhauser, Basel
3. Krstic M, Smyshlyaev A (2008) Boundary control of PDEs: a course on backstepping designs. SIAM, Philadelphia
4. Smyshlyaev A, Krstic M (2010) Adaptive control of parabolic PDEs. Princeton University Press, Princeton
5. Vazquez R, Krstic M (2008) Control of turbulent and magnetohydrodynamic channel flow. Birkhauser, Basel