Backstepping Synthesis for Feedback Control of First-Order Hyperbolic PDEs with Spatial-Temporal Actuation

and Applied Analysis 3 which results in the following system: u t (x, t) = u x (x, t) + h (x) u (0, t)


Introduction
Motivated by the essence of spatial-temporal evolutions in nature, control of systems governed by partial differential equations (PDEs), or distributed parameter system, has been studied for decades (e.g., [1][2][3][4][5], just to name a few).As one important category of distributed parameter systems, hyperbolic PDEs are widely used to describe dynamics arising in traffic flows, chemical reactors, heat exchangers, and fluid transport pipelines and the control problems regarding hyperbolic PDEs have been widely studied [6][7][8][9].
We consider the stabilization problem of a hyperbolic PDE system with a spatial-temporal actuation over the full physical domain.Our approach in this work relies on the backstepping method, which was originally developed in the 1990s for adaptive and robust control of nonlinear lumped parameter systems governed by ordinary differential equations (ODEs) [10].In the last decade, the backstepping method for PDEs with boundary actuation has been widely developed [11][12][13][14][15].Besides successful applicability of the backstepping techniques to one-dimensional classical PDE systems of both hyperbolic and parabolic types, much progress has been made to establish the control design for higher-dimensional systems, including complex PDEs arising in applied physics, including magnetohydrodynamic (MHD) [16] and fluid flows [17,18] (governed by MHD and Navier-Stokes equations, resp.), pipeline dynamics for oil and gas transportation [19], and even 3-dimensional diffusion-reaction systems with varying parameters [20].In addition, the backstepping technique can be extended to handle PDE systems with nonlinear terms in the sense of local stabilization [21,22].

Abstract and Applied Analysis
Backstepping can be used to achieve the stabilization of unstable PDEs in a physically appealing way where the destabilizing terms are eliminated by means of an invertible integral transformation of the PDE together with the boundary feedback.In addition, unlike the linear quadratic regulator (LQR) approaches for boundary control [23,24], which require the solutions of operator Riccati equations, the backstepping technique takes advantage of the structure of systems and yields control gain formulas which can be evaluated using symbolic computation and, in some cases, can be given explicitly.For detailed information on the backstepping method, one can refer to the books [13,25] and the references therein.
The backstepping approach is so far a systematic method and gives a rather straightforward way for boundary feedback controller design for PDE systems.More recently, the authors in [26] have extended the backstepping method in order to deal with full domain control problems of parabolic PDEs.The authors first apply the backstepping transformation to map the original system into a simple target system.Then, an additional differential transformation is introduced to move the input to the boundary, and immediately an exponentially stabilizing state feedback controller is obtained.
The problem of boundary feedback stabilization of firstorder hyperbolic PDEs has been studied in [7] using the backstepping method.However, the backstepping technique has never been extended to deal with the full domainv control problem for the first-order hyperbolic PDE systems.In this paper, we assume that the interior actuator can be decomposed into a product of spatial and temporal components, and the spatial component satisfies a specific ODE such that the backstepping technique can be used to this problem.Following the backstepping procedure, in the first step, a simple system can be obtained from the original system by using the backstepping transformation.However, the input still remains in the obtained system for the interior actuation and we can not cancel the residual term using the classical backstepping technique, which causes the instability of the open-loop system.To solve this problem, a differential transformation is introduced to move the input from the interior of the domain to the boundary which enables removing the residual term [26].Then, a feedback control law is obtained such that the first-order hyperbolic PDE converges to zero in a finite time, which has been proved using the semigroup argument.The main contribution of the current work is to extend the strategy proposed in [26] to first-order hyperbolic PDE systems.
The rest of this paper is organized as follows.In Section 2, we state the problem formulation.In Section 3, we discuss the state transformation to restate the control problem; then a differential transformation is given to transform the problem into a classical boundary stabilization problem.In Section 4, the invertibility of the proposed transformation is investigated.In Section 5, a state feedback controller is designed and the stability result is proved.The numerical simulations are presented in Section 6.In Section 7, we close the paper by addressing the concluding remarks and future research topics.

Problem Formulation
We consider the following 1-dimensional hyperbolic PDE of the form   (, ) =   (, ) +  ()  (, ) + ℎ ()  (0, ) where (, ) ∈ R is the state; () ∈ R is the control input; , ℎ,  are continuous on [0, 1] and  is continuous on [0, 1] × [0, 1].To use the backstepping technique to deal with the stabilization problem of (1), we make the following assumption on the function  throughout of the paper, which represents the shape of actuation: (A)  satisfies the following integrodifferential equation: where ,  0 ∈ R are two design parameters.Since ( 2) is linear, for given ,  0 ∈ R, there exists a unique solution to (2).In the following, we assume that Otherwise, by the uniqueness theory of the solution for ODEs, () = 0 for any  ∈ [0, 1] if  0 = 0.The choosing methods of the parameters  and  0 will be specified in the following sections.The main objective of this paper is to stabilize the zero equilibrium of the system (1) with a state feedback controller.

Transformations
In the following, the design procedure is presented to the state feedback controller for the system (1).By using an infinitedimensional backstepping transformation, the system (1) can be converted into a simple system.Then, since the obtained system is not a boundary control system, an additional differential transformation is used to move the actuation function to the boundary.Based on the invertibility of these transformations, we construct a feedback control law such that the first-order hyperbolic PDE converges to zero in a finite time.
By [7], (9) has a unique solution  ∈ where we have exchanged the order of the integration in (10) and Here, we note that in fact  is the image of  under the backstepping transformation.Subtracting (11) from (10) and making use of (9), we have Moreover, by ( 7), (9), and ( 12 where the order of the integration in the above equation has been exchanged.Thus, it follows from ( 7) and ( 14) that  is the solution of the following ODE: Now, we will discuss the boundary condition that V should satisfy.Motivated by [7], the transformation (8) with the integral kernel  satisfying ( 9) is invertible and the inverse transformation is of the form where the integral kernel  ∈  1 ([0, 1] × [0.1]) is the solution of the following PDE: Since (1, ) = 0 for  ∈ (0, ∞), it follows from ( 16) that By combining ( 13) and ( 18), the backstepping transformation converts the system (5) into the following system: From ( 19), we see that the input  still remains on the internal domain and we can not eliminate the residual term at the boundary, which causes the instability of the open-loop system.

Differential Transformation.
We introduce the following differential transformation: Assume that the solution V of ( 19) is sufficiently smooth such that the continuous derivatives in the following reduction all exist.The regularity of the solution V will be discussed in the proof of Theorem 4.Then, under this assumption, by (19), we have Taking the derivative of ( 20) with respect to  and taking the derivative of ( 21) with respect to , respectively, we have V  (, ) =   (, ) + V  (, ) +   ()  () .
Substituting ( 21) and ( 23) into the right-hand side of ( 22) yields Thus, together with ( 15) and (20), this implies that On the other hand, taking the derivative of ( 18) with respect to , we have where we have used the following equation: Thus, we see from ( 25) and ( 26) that when the solution V of ( 19) is sufficiently smooth, the differential transformation (20) can map V to the solution  of the following system: (28)
Proof.To verify the existence of  −1 in  2 (0, 1), we need to show that, for any  ∈  2 (0, 1), there exists a unique function V ∈ D() such that  = V.For any given  ∈ R, the following ODE V  − V =  has a unique solution: which satisfies V(0) = .We can see easily V, V  ∈  2 (0, 1), that is, V ∈  1 (0, 1).According to (29) and (30), we remain to show that the function V satisfies the boundary condition By substituting (33) into (34), we have where Thus, if (1) ̸ = 0, we can choose such that (34) holds; that is, V satisfies the boundary condition.Thus, we have proved that the operator  has an inverse operator  −1 .
Next, we will prove (32 where Thus, we have proved the first inequality of (32).Moreover, since Thus the proof is complete.
Remark 7. If (47) has a classical solution , that is,  is continuously differentiable and satisfies (47) for every  and , then the classical solution is the same as the mild solution, which is mentioned in Remark 6.
Remark 9.For any given  0 ̸ = 0, to guarantee the existence of the feedback control law (46), we only need to design the parameter  in (7) such that (1) ̸ = 0. Equivalently, we can design the parameter  in (2) such that (1) ̸ = 0, which ensures the existence of the feedback control law (68).

Numerical Simulation
Now we consider the following 1-dimensional hyperbolic PDE (i.e., by taking () = , ℎ() = 0, (, ) =  (−) in ( 1 where  1 is the modified Bessel functions of order one.We choose  = 2,  = 2, and  = 0.5 in (69) and set the initial condition as (, 0) = sin(2).From Figure 1, we can observe that the system is unstable from the numerical result which is obtained using the finite difference method.The integral term in the PDE system (69) is the main reason to cause instability of the open-loop system.Thus, a feedback control law is necessary to achieve closed-loop stability.The interior actuation function includes two components, that is, the temporal function () and the spatial shape function ().The temporal component () is given by (68) and the spatial shape function () is given by the ODE (2) with the design parameter  = 4.5.By Remark 9, without loss of generality, we take in (2).For the closed-loop simulation, we solve the PDE system (69) with () and () given in Theorem 8 using the finite difference method.The response of the closed-loop system is shown in Figure 2. In addition, the corresponding temporal function () in the closed-loop simulation is shown in Figure 3.The result in Theorem 8 states that the closed-loop system should satisfy (, ) = 0 for any  ≥ 1 and  ∈ [0, 1].As shown in Figures 2 and 3, one may realize that the state trajectory  and the boundary trajectory () are close to but do not exactly reach zero, which is completely due to the unavoidable numerical errors.By improving the numerical accuracy, one may take the values closer to zero at  = 1, which implies that the numerical results validate the conclusion in Theorem 8.The shape function () is shown in Figure 4 which is a monotonically increasing function over  ∈ [0, 1].The interior actuation, that is, the multiplication term ()(), is shown in Figure 5.By Theorem 8, the constant  in (2) is a parameter to be designed such that (1) ̸ = 0.For every such , there exist  and  that can stabilize the unstable system.A natural problem is which the optimal parameter is in some proper sense.By varying  in the internal [−15, 5], we can solve the ODE (2) for (1) numerically and the result is shown in Figure 6.We find that, for  < 2.5, (1) is close to    zero (as shown in Figure 6).In order to avoid numerical singularities, we need to choose  properly such that (1) is obviously different from zero.For this reason, we will discuss the case for  ≥ 2.5.We compute () and its  2 -norm for  ∈ [2.5, 10].One can observe in Figure 7 that the  2norm is increasing dramatically as the value of  increases.Similarly, we also compute () and its  2 -norm for  ∈ [2.5, 10], shown in Figure 8.We note that the  2 -norm is a monotonically decreasing function of .It is shown in Figure 9 that the  2 -norm of ()() varies with respect to  and we find that there exists a point  = 5.9, such that the  2 -norm of the control function  is minimized (min ∈[2.5,10]‖‖ = 0.76).
We can realize that the spatial-temporal actuation ()() is not practically applicable in real world applications since it requires external forcing over the whole physical domain, that is,  ∈ [0,1].Motivated by the subdomain actuation mechanism in engineering practice, a more practical problem can be formulated as follows, associated with the simulation model in (69): Given () and () being synthesized based on Theorem 8, then we use only numerical simulation rather than complete theoretical analysis in the current work to validate the effectiveness of the feedback controller even for the subdomain actuation in practical applications.As shown in Figure 10, for  = 4.5, when the feedback control acts only on the subdomain [0.7, 1], the system (73) can be also stabilized by only using the controller which is synthesized for the spatial-temporal actuation.But the closed-loop system takes much longer to dissipate zero at  = 7 instead of  = 1 for the spatial-temporal actuation (see Figure 2).When we decrease the acting domain area further to [0.8, 1], then the system (73) can not be stabilized (see Figure 11).However, if we increase the value of  from  = 4.5 to  = 7, then the closed-loop system (73) appears to be stabilized again (see Figure 12).This is a quite interesting phenomenon and we try to give an explanation from Figure 13 where we solve () for different values of .One can readily find in the simulation that () increase dramatically as  does.For the characteristic function  [0.8,1] the controller is not capable of stabilizing the systems for  = 4.5.But, by increasing the value of , it is an equivalent approach to improve the actuation strength over the subdomain close to  = 1, where the spatial component () is much more significant than the values taken close to  = 0.Alternatively, one can conclude that subdomains, where () can achieve significant values, are better locations to deploy actuation.
A thorough theoretical analysis is definitely needed in the future.

Conclusions
In this paper, the stabilization problem using backstepping techniques for first-order hyperbolic PDEs with internal actuation has been investigated.For this, backstepping integral transformations together with a differential transformation are used to determine the state feedback controller.In future work, several directions can be pursued.First, just as said in Section 6, the theoretical study on the design of the parameter  is needed.Second, with this work done, we may expect to extend this paper to various PDE controlled systems arising in applied physics and engineering.Finally, the stabilization problem with internal actuation for nonlinear systems can be investigated.

Figure 1 : 3 Figure 2 :
Figure 1: The response of the open-loop system.
Assuming that (1) ̸ = 0, then the operator  has an inverse operator  −1 in  2 (0, 1).Moreover, there exist constants  1 and  2 such that       −1       ≤  1 , it is equivalent to show the invertibility of .Moreover, we can obtain that the inverse operator of  is continuous under certain condition.To this end, we define a continuous function  by () =   + ∫  0  (, )   .(31)Theorem 1.