A Well-posedness of the kernel equations

Local Exponential Stabilization of a Quasilinear Hyperbolic System using Backstepping 1

Abstract

In this work, we consider the problem of boundary stabilization for a quasilinear system of first-order hyperbolic PDEs. We design a new full-state feedback control law, with actuation on only one end of the domain, which achieves exponential stability of the closed-loop system. Our proof uses a backstepping transformation to find new variables for which a strict Lyapunov function can be constructed. The kernels of the transformation are found to verify a Goursat-type system of first-order hyperbolic PDEs, whose well-posedness is shown using the method of characteristics and successive approximations. Once the kernels are computed, the stabilizing feedback law can be explicitly constructed from them.

n

onlinear hyperbolic systems, boundary conditions, stability, Lyapunov function, backstepping, method of characteristics, integral equation, Goursat problem

{AMS}

TBD

1 Introduction

In this paper we are concerned with the problem of boundary stabilization for a system of first-order hyperbolic quasilinear PDEs, with actuation at only one of the boundaries. The quasilinear case is of interest since many relevant physical systems are described by systems of first-order hyperbolic quasilinear PDEs, such as open channels [11, 16, 17, 18], transmission lines [6], gas flow pipelines [14] or road traffic models [12].

This problem has been considered in the past for systems [13] and even systems [24], using the explicit evolution of the Riemann invariants along the characteristics. More recently, an approach using control Lyapunov functions has been developed, for systems [2] and systems [3]. These results use only static output feedback (the output being the value of the state on the boundaries). However, they do not deal with the same class of systems considered in this work (which includes an extra term in the equations); with this term, it has been shown in [1] that there are examples (even for linear system) for which there are no control Lyapunov functions of the “diagonal” form (see next section for notation) which would allow the computation of a static output feedback law to stabilize the system, even if feedback is allowed on both sides of the boundary.

Several other authors have also studied this problem. For instance, the linear case has been analyzed in [38] (using a Lyapunov approach) and in [25] (using a spectral approach). The nonlinear case has been considered by [8] and [15] using a Lyapunov approach, and in [26][27], and [11] using a Riemann invariants approach.

The basis of our design is the backstepping method [20]; initially developed for parabolic equations, it has been used for first-order hyperbolic equations [23], delay systems [21], second-order hyperbolic equations [31], fluid flows [34], nonlinear PDEs [35] and even PDE adaptive designs [32]. The method allows us to design a full-state feedback law (with actuation on only one end of the domain) making the closed-loop system locally exponentially stable in the sense. The gains of the feedback law are the solution of a 4 x 4 system of first-order hyperbolic linear PDEs, whose well-posedness is shown. The proof of stability is based on [3]; we construct a strict Lyapunov function, locally equivalent to the norm, and written in coordinates defined by the (invertible) backstepping transformation.

The paper is organized as follows. In Section 2 we formulate the problem. In Section 3 we consider the linear case and formulate a backstepping design that globally stabilizes the system in the sense. In Section 4 we present our main result, which shows that the linear design locally stabilizes the nonlinear system in the sense. The proof of this result is given in Section 5. We finish in Section 6 with some concluding remarks. We also include an appendix with the proof of well-posedness of the kernel equations, and some technical lemmas.

2 Problem Statement

Consider the system

(1)

where , , , with denoting the set of real matrices. We assume that is twice continuously differentiable with respect to and , and we assume that (possibly after an appropiate state transformation) is a diagonal matrix with nonzero eigenvalues and which are, respectively, positive and negative, i.e.,

(2)

where denotes the diagonal matrix with in the first position of the diagonal and in the second.

We also assume that , implying that there is an equilibrium at the origin, and that is twice continuously differentiable with respect to . Denote

(3)

and assume that .

Denoting , we study classical solutions of the system under the following boundary conditions

(4)

which are consistent (see [28]) with the signs of (2), at least for small values of . We assume that is twice differentiable and vanishes at the origin. In (4), is the actuation variable, and our task is to find a feedback law for to make the origin of system (1),(4) locally exponentially stable.

Remark 1

The case with in (1) was addressed in [2] and [3] by using control Lyapunov functions to design a static output feedback law; this approach has been shown to fail in [1] for some cases with , at least for a “diagonal” Lyapunov function of the form .

3 Stabilization of hyperbolic linear systems

Next, we present a new design, based on the backstepping method, to stabilize a hyperbolic linear system; this procedure will be used later to locally stabilize system (1), (4).

Consider the system

(5)

where , , where the matrices and are respectively diagonal and antidiagonal, as follows:

(6)

where are and are functions, verifying that , and with boundary conditions

(7)

where and the components of are . Our objective is to design a full-state feedback control law for to ensure that the closed-loop system is globally asymptotically stable in the norm, which is defined as . There are two cases, depending on whether in (7) is nonzero or . We first analyze the first case, thus assuming .

3.1 Target system and backstepping transformation

Our approach to designing , following the backstepping method, is to seek a mapping that transforms into a target variable with asymptotically stable dynamics as follows:

(8)

with boundary conditions

(9)

where the components of are denoted as

(10)

System (8), (9) verifies the properties expressed in the following proposition. {proposition} Consider system (8), (9) with initial condition . Then, for every , there exists such that

(11)

In fact, the equilibrium is reached in finite time , where is given by

(12)
{proof}

Define

(13)

where will be computed later. Select:

(14)

Notice that defines a norm equivalent to . Computing the derivative of and integrating by parts, we obtain

(15)

where we have used that and commute. Since

(16)

and, on the other hand,

(17)

choosing , , and , where , we get that , therefore:

(18)

where can be chosen as large as desired. This shows exponential stability of the origin for the system.

To show finite-time convergence to the origin, one can find the explicit solution of (8) as follows. Define first

(19)

noting that they are monotonically increasing functions of , and thus invertible. Note that the components of verify the differential equations

(20)
(21)

which can be rewritten as follows

(22)
(23)

The solution of these equations is and , where and are arbitrary functions. Now, if , are the initial condition for the states, one obtains (valid for ) and (valid for ). Using the boundary conditions (9) one finds the remaining values of and , and thus the solution of the system, as follows:

(24)
(25)

Thus, after , where

(26)

one has that .

3.2 Backstepping transformation and kernel equations

To map the original system (5) into the target system (8), we use the following transformation:

(27)

where

(28)

is a matrix of kernels. Defining

(29)

the original and target boundary conditions (respectively (7) and (9)) can be written compactly (omitting dependences in and ) as

(30)

Introducing (27) into (8), applying (5), integrating by parts and using the boundary conditions, we obtain that the original system (5) is mapped into the target system (8) if and only if one has the following three matrix equations:

(31)
(32)
(33)

Expanding (31), we get the following kernel equations:

(34)
(35)
(36)
(37)

with boundary conditions obtained from (32)–(33)

(38)
(39)
(40)
(41)

The equations evolve in the triangular domain . Notice that they can be written as two separate hyperbolic systems, one for and and another for and .

By Theorem A (see the Appendix), one finds that, for , under the assumption that are , are and that , there is a unique solution to (34)–(41), which is in .

3.3 The inverse transformation

To study the invertibility of transformation (27), we look for a transformation of the the target system (8) into the original system (5) as follows:

(42)

where

(43)

Introducing (42) into (5), applying (8), integrating by parts and using the boundary conditions, we obtain as before a set of kernel equations:

(44)
(45)
(46)
(47)

with boundary conditions

(48)
(49)
(50)
(51)

Again by Theorem A (see the Appendix), one finds that there is a unique solution to these equations, which is .

3.4 Control law and main result

From the transformation (27) evaluated at , one gets

(52)

With control law (52) the following result holds. {theorem} Consider system (5) with boundary conditions (7), control law (52), and initial condition . Then, for every , there exists such that

(53)

In fact, the equilibrium is reached in finite time , where is given by (12).

{proof}

Since the transformation (27) is invertible, when applying control law (52) the dynamical behavior of (5) is the same as the behavior of (8), which is well-posed from standard results and whose explicit solution and stability properties we know from Proposition 3.1. Thus, we obtain the explicit solutions of from the direct and inverse transformation, as follows:

(54)

where is the explicit solution of the , system, given by (24)–(25), with initial conditions:

(55)

In particular, we know that goes to zero in finite time , therefore also shares that property. Finally, since the origin of the system is exponentially stable with an arbitrary large exponential decay rate, we conclude, using the inverse transformation, that the origin of the system is also exponentially stable with an arbitrary large exponential decay rate. Equation (53) follows by using the inverse and direct transformations to relate the norms of and (using the fact that the kernels of the transformations are continuous, and thus bounded, functions).

3.5 The case

If the coefficient is zero in (7), the method presented in the paper is not valid since (38) would require the value of one of the control kernels to be infinity at the boundary of the domain . Similarly, if the coefficient is close to zero one still gets very large values for the kernels close to the boundary, resulting in potentially large control laws.

The method can be modified to accommodate zero or small values of by setting a slightly different target system (20)–(21), as follows:

(56)
(57)

where is to be obtained from the method; regardless of the value of , this is a cascade system which is still exponentially stable and converges in finite time by the same arguments of Proposition 3.1, since now, using the same Lyapunov function defined in (14), we obtain

(58)

The new term (which is the last one) can be controlled by slightly modifying the coefficients of in the proof of Proposition 3.1, obtaining the same result as before.

The kernel equations resulting from the transformation are still the same (34)–(37), with the same boundary conditions (39)–(41) for , , and (which reduces to when ), but one obtains an undetermined boundary conditions for :

(59)

where can be chosen as desired; by choosing at least a continuous function, one can apply Theorem A and thus the kernel equations are well-posed. After has been chosen and the kernels have been computed, one obtains the value of as

(60)

Invertibility of the transformation follows as before, thus one obtains the same result of Theorem 3.4. The non-uniqueness in (59) gives the designer some freedom in shaping the input function from to . Also note that this has no impact in the feedback law as the kernels and (which are the ones appearing in (52)) are uniquely defined and independent of the non-unique and .

4 Application of the linear backstepping controller to the nonlinear system

We wish to show that the linear controller (52) designed using backstepping works locally for the nonlinear system, in terms that will be made precise.

For that, we write our quasilinear system (1) in a form equivalent (up to linear terms) to (5). Define

(61)

We obtain a new state variable from using the following transformation:

(62)

so that

(63)

It follows that verifies the following equation:

(64)

where

(65)
(66)

It is evident that and that . Also,

(67)

Thus, it is possible to write (64) as a linear system with the same structure as (5) plus nonlinear terms:

(68)

where

(69)

and

(70)

Computing the boundary conditions of (68) by combining (4) with the transformation (62), and defining and , one obtains

(71)

where . In what follows we will consider the case ; the case is analogous (see Remark 2).

Notice that the linear parts of (68) and (71) are identical to (5) and (7), and that the coefficients and verify the assumptions of Section 3. Also, it is clear that the nonlinear terms verify , , and

Therefore, we consider using the feedback law:

(72)

which implies, in terms of the original variable:

(73)

where the kernels are computed from (34)–(41) using the coefficients and from (67) and (69).

Next, we show that the control law (73), which is computed for the linear part of the system, asymptotically stabilizes the nonlinear system, although locally. However, the right space to prove stability of the closed-loop system is , instead of the space that was used in Section 3 for the linear system.

Denoting:

(74)

the boundary conditions of the closed loop system would be written as:

(75)

A necessary condition for system (1) with boundary conditions (75) to be well-posed in the space is that the initial conditions verify the corresponding second-order compatibility condition. These are

(76)
(77)
(78)
(79)

While (76) and (78) are natural compatibility conditions, the conditions (77) and (79) are artificial (since they show up due to the feedback law that has been designed) and rather stringent, as they require very specific values of the initial conditions. Thus, we modify our control law in a way that, without losing its stabilizing character, does not require any specific values in the initial values beyond the natural conditions (76) and (78). The modification in the boundary conditions consists in adding a dynamic extension to the controller as follows:

(80)

where is one of the states of the following system:

(81)

where the constants and can be chosen as desired with the only conditions that and . It is evident that with positive values of these constants, (81) is always stable. The initial conditions of and are an additional degree of freedom that can be used to eliminate the compatibility conditions (77) and (79). With the modification of the control law, these compatibility conditions are now

(82)
(83)

Call

(84)
(85)

Selecting