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Boundary stabilization in finite time of one-dimensional linear hyperbolic balance laws with coefficients depending on time and space
Abstract
In this article we are interested in the boundary stabilization in finite time of one-dimensional linear hyperbolic balance laws with coefficients depending on time and space. We extend the so called “backstepping method” by introducing appropriate time-dependent integral transformations in order to map our initial system to a new one which has desired stability properties. The kernels of the integral transformations involved are solutions to non standard multi-dimensional hyperbolic PDEs, where the time dependence introduces several new difficulties in the treatment of their well-posedness. This work generalizes previous results of the literature, where only time-independent systems were considered.
... PDE backstepping design has also been utilized to control various PDE-ODE cascaded systems [12]- [15]. Boundary stabilization of one-dimensional linear hyperbolic PDEs with time and space varying parameters is presented, the well-posedness of time and space varying kernel PDEs has been solved in [16]. Global stabilization of a class of switched nonlinear systems is investigated in [17], in which a state feedback sampled-data controller is constructed by backstepping design. ...
... and B(t) are given by (19). Based on the successive approximation method, Coron, et al. solved the existence and uniqueness of time-varying kernel PDEs in Theorem 2.6 in [16], but the boundary conditions of the kernel PDE (15)- (18) are different from those of the kernel PDEs in Theorem 2.6 in [16]. So the result of [16] cannot be directly applied to the kernel PDEs (15)- (18). ...
... and B(t) are given by (19). Based on the successive approximation method, Coron, et al. solved the existence and uniqueness of time-varying kernel PDEs in Theorem 2.6 in [16], but the boundary conditions of the kernel PDE (15)- (18) are different from those of the kernel PDEs in Theorem 2.6 in [16]. So the result of [16] cannot be directly applied to the kernel PDEs (15)- (18). ...
We investigate stabilization of a class of cascaded systems of nonlinear ordinary differential equation (ODE)/wave partial differential equation (PDE) with time-varying propagation speed based on a two-step PDE backstepping transformation. A time-varying propagation velocity of wave PDE leads to two difficulties. One is how to prove the well-posedness and uniqueness of the time-varying kernel PDEs in the first-step backstepping transformation, the other is how to construct a backstepping transform to map the original system into a suitable target system during the second-step transformation. We prove that there exists a unique continuous
matrix-valued solution to the time-varying kernel PDEs, and design a predictor control for the original cascaded system. An example is provided to illustrate the feasibility of the proposed design.
... In addition to leaving the classical framework of Cauchy theory, the kernels resulting from the Volterra transformation present boundary conditions on the diagonal k(x, x) that proves to be very difficult to handle. Despite these difficulties, several methods have been developed to solve the PDE on the kernel of the Volterra transformation (successive approximations [27], explicit representations [27] or method of characteristics [15]) leading to a rich literature, the invertibility of the Volterra transformation being guaranteed. ...
... Conclude on the rapid stabilization using the operator equality. Aside of the seemingly different approach of hyperbolic systems [13,14,15,41,42], the proof Step 1 and 2 relied heavily in the literature on the quadratically close criterion. Roughly speaking it amounts to show after some computations that n∈N p∈N\{n} 1 |λ n − λ p + λ| 2 < +∞ which holds if the eigenvalues λ n of the operator A scales as n α with α > 3/2 but fails as soon as α ≤ 3/2. ...
... We have so far excluded from our discussion the case α = 1 as it seems to be a very specific case with techniques on its own. Indeed, the rapid stabilisation for hyperbolic systems was established in [14,15] through direct methods or by identifying the isomorphism applied to the eigenbasis leading to the Riesz basis [13,41,42]. The other results found in the literature were concerned with operators such that α ≥ 2, and in these case the Riesz basis properties was proved through the quadratically close criterion, thanks to the sufficient growth of the eigenvalues. ...
Fredholm-type backstepping transformation, introduced by Coron and L\"u, has become a powerful tool for rapid stabilization with fast development over the last decade. Its strength lies in its systematic approach, allowing to deduce rapid stabilization from approximate controllability. But limitations with the current approach exist for operators of the form for . We present here a new compactness/duality method which hinges on Fredholm's alternative to overcome the threshold. More precisely, the compactness/duality method allows to prove the existence of a Riesz basis for the backstepping transformation for skew-adjoint operator verifying , a key step in the construction of the Fredholm backstepping transformation, where the usual methods only work for . The illustration of this new method is shown on the rapid stabilization of the linearized capillary-gravity water wave equation exhibiting an operator of critical order .
... The Volterra transformation having the advantage of always being invertible, only the existence remains to prove, which is equivalent to solving a PDE of the kernel on a triangular domain. These PDEs usually do not enter in the classical Cauchy problem framework, but different techniques are now known to solve the kernel equation: successive approximations [33], explicit representations [33] or method of characteristics [19]. There exists now a vast literature on the backstepping method with Volterra transformations: let us cite for the heat/parabolic equation [3,9,22], for hyperbolic systems [6] and for the viscous Burgers equation [24]. ...
... The idea of using transformations remains the same, but proving the existence and invertibility of the transformation is generally more involved. There are mainly two ways to prove the existence of the transformation, either by direct methods [18,19] or, more commonly, by proving the existence of a Riesz basis. For the latter, we again distinguish two cases: either the Riesz basis is deduced directly by an isomorphism applied on an eigenbasis [17,50,51] or the existence of a Riesz basis follows by controllability assumptions and sufficient growth of the eigenvalues of the spatial operator allowing in particular to prove that the family is quadratically close to the eigenfunctions [16,20,21,27] (see Section 2.2 and Section 4 for a definition). ...
... define B T := {y ∈ C 0 ([0, T ]; L 2 1 ) ∩ L 2 ([0, T ]; M > 0, we also define B T (M ) as B T (M ) := {y ∈ B T | y B T ≤ M }. (5.18)Suppose that y 0 L 2 1 = R. Then, we consider the map L : z ∈ B T (3R) → y ∈ B T (5.19) ...
We study the rapid stabilization of the heat equation on the 1-dimensional torus using the backstepping method with a Fredholm transformation. We prove that, under some assumption on the control operator, two scalar controls are necessary and sufficient to get controllability and rapid stabilization. This classical framework allows us to present the backstepping method with Fredholm transformations on Laplace operators in a sharp functional setting, which is the main objective of this work. Finally, we prove that the same Fredholm transformation also leads to the local rapid stability of the viscous Burgers equation.
... By solution we mean "solution along the characteristics" or "broad solution" (see e.g. Appendix A of [6]). The same statement remains true if, in the boundary condition at x = 1, u is replaced by u(t) = 1 0 (f 1 (ξ)y 1 (t, ξ) + f 2 (ξ)y 2 (t, ξ)) dξ, (1.3) for any f 1 , f 2 ∈ L ∞ (0, 1). ...
... The transformation used is usually a Volterra transformation of the second kind. One can refer to the tutorial book [21] to design boundary feedback laws stabilizing systems modeled by various PDEs and to the introduction of [6] for a complementary state of the art on this method. This technique turned out to be a powerful tool to stabilize general coupled hyperbolic systems, moreover in finite time. ...
... However, the control time obtained in these works was larger than the one in [29] and it was only shown in [1,5] that we can stabilize with the same time as the one of [29]. These works have recently been generalized to time-dependent systems in [6]. Finally, let us also mention the two recent works [8,9] concerning the finite-time stabilization of homogeneous quasilinear systems, with the same control time as in [7,10]. ...
The goal of this article is to present the minimal time needed for the null controllability and finite-time stabilization of one-dimensional first-order 2×2 linear hyperbolic systems. The main technical point is to show that we cannot obtain a better time. The proof combines the backstepping method with the Titchmarsh convolution theorem.
... Controllability and finite-time boundary control have also been studied in e.g. [30,31] and [32,33]. Boundary output feedback stabilization for (9)- (11) has been investigated in [34] in the case of measurement errors, leading to the study of input-to-state stability for those systems. ...
... Consider z ∈ D(A). Expanding the left-hand side of (33) yields that ⟨Az, Zz⟩ + ⟨Zz, Az⟩ + ⟨Cz, Cz⟩ ...
We derive an explicit solution to the operator Riccati equation solving the Linear-Quadratic (LQ) optimal control problem for a class of boundary controlled hyperbolic partial differential equations (PDEs). Different descriptions of the system are used to obtain different representations of the operator Riccati equation. By means of an example, we illustrate the importance of considering an extended operator Riccati equation to solve the LQ-optimal control problem for our class of systems.
... Moreover, an explicit solution can be obtained using the Marcum-Q functions [38]. sing the control law (11), the closed-loop system (7)-(10) is well-posed and exponentially stable in L 2 norm [15], [16]. We have the following theorem. ...
... The NO-approximated backstepping kernels have the same functional form with the nominal kernels. Therefore, the initial-value problem of system (1)-(4) under the control law (24) has one, and only one solution using the results in [15,Theorem A.4] and [16,Appendix. A] as the system in this paper is a particular case of them. Then iterating the process for each time interval on the whole time domain, we can get the stochastic system has a unique solution for any t ≥ 0 that satisfies (25). ...
In this paper, we address the problem of robust stabilization for linear hyperbolic Partial Differential Equations (PDEs) with Markov-jumping parameter uncertainty. We consider a 2 x 2 heterogeneous hyperbolic PDE and propose a control law using operator learning and backstepping method. Specifically, the backstepping kernels used to construct the control law are approximated with neural operators (NOs) in order to improve computational efficiency. The key challenge lies in deriving the stability condition with respect to the Markov-jumping parameter uncertainty and NO approximation errors. The mean-square exponential stability of the stochastic system is achieved through Lyapunov analysis, indicating that the system can be stabilized if the random parameters are sufficiently close to the nominal parameters on average, and NO approximation errors are small enough. The theoretical results are applied to freeway traffic control under stochastic upstream demands and then validated through numerical simulations.
... Another strategy that has been studied is the Backstepping method, which aims to design a control law that achieves stabilization. The Backstepping method has been applied in [1,22] and typically requires full-state feedback control. However, it is possible to achieve boundary state feedback through backstepping control by designing an appropriate observer, as demonstrated in [23,24]. ...
... Therefore, an alternative control approach is worth exploring in this case. Building upon the works of Hu et al. [1,22] and Holta et al. [23], we develop a Backstepping control combined with observer design that stabilizes the system even when L ≥ L c , and without the need to observe the full state. Notably, the proposed control law drives the system to its zero equilibrium in finite time. ...
This paper is devoted to discuss the stabilizability of a class of non-homogeneous hyperbolic systems. Motivated by the example in \cite[Page 197]{CB2016}, we analyze the influence of the interval length L on stabilizability of the system. By spectral analysis, we prove that either the system is stabilizable for all or it possesses the dichotomy property: there exists a critical length such that the system is stabilizable for but unstabilizable for . In addition, for , we obtain that the system can reach equilibrium state in finite time by backstepping control combined with observer. Finally, we also provide some numerical simulations to confirm our developed analytical criteria.
... Another strategy that has been studied is the Backstepping method, which aims to design a control law that achieves stabilization. The Backstepping method has been applied in [13,21] and typically requires full-state feedback control. However, it is possible to achieve boundary state feedback through backstepping control by designing an appropriate observer, as demonstrated in [2,3]. ...
... Therefore, an alternative control approach is worth exploring in this case. Building upon the works of Hu et al. [13,21] and Holta et al. [2], we develop a Backstepping control combined with observer design that stabilizes the system even when L ≥ L c , and without the need to observe the full state. Notably, the proposed control law drives the system to its zero equilibrium in finite time. ...
... Another strategy that has been studied is the Backstepping method, which aims to design a control law that achieves stabilization. The Backstepping method has been applied in [13,21] and typically requires full-state feedback control. However, it is possible to achieve boundary state feedback through backstepping control by designing an appropriate observer, as demonstrated in [2,3]. ...
... Therefore, an alternative control approach is worth exploring in this case. Building upon the works of Hu et al. [13,21] and Holta et al. [2], we develop a Backstepping control combined with observer design that stabilizes the system even when L ≥ L c , and without the need to observe the full state. Notably, the proposed control law drives the system to its zero equilibrium in finite time. ...
This paper is devoted to discuss the stabilizability of a class of non-homogeneous hyperbolic systems. Motivated by the example in \cite[Page 197]{CB2016}, we analyze the influence of the interval length L on stabilizability of the system. By spectral analysis, we prove that either the system is stabilizable for all or it possesses the dichotomy property: there exists a critical length such that the system is stabilizable for but unstabilizable for . In addition, for , we obtain that the system can reach equilibrium state in finite time by backstepping control combined with observer. Finally, we also provide some numerical simulations to confirm our developed analytical criteria.
... The transformation used is usually a Volterra transformation of the second kind. One can refer to the tutorial book [KS08] to design boundary feedback laws stabilizing systems modeled by various PDEs and to the introduction of [CHOS21] for a complementary state of the art on this method. This technique turned out to be a powerful tool to stabilize general coupled hyperbolic systems, moreover in finite time. ...
... However, the control time obtained in these works was larger than the one in [Rus78b] and it was only shown in [ADM16,CHO17] that we can stabilize with the same time as the one of [Rus78b]. These works have recently been generalized to time-dependent systems in [CHOS21]. Finally, let us also mention the two recent works [CN20a, CN20b] concerning the finite-time stabilization of homogeneous quasilinear systems, with the same control time as in [CN19b,CN19a]. ...
The goal of this article is to present the minimal time needed for the null controllability and finite-time stabilization of one-dimensional first-order linear hyperbolic systems. The main technical point is to show that we cannot obtain a better time. The proof combines the backstepping method with the Titchmarsh convolution theorem.
... Controllability and finite-time boundary control have also been studied in e.g. [7,8] and [3,10]. Boundary output feedback stabilization for (1.1)-(1.4) ...
A solution to the suboptimal H-control problem is given for a class of hyperbolic partial differential equations (PDEs). The first result of this manuscript shows that the considered class of PDEs admits an equivalent representation as an infinite-dimensional discrete-time system. Taking advantage of this, this manuscript shows that it is equivalent to solve the suboptimal H-control problem for a finite-dimensional discrete-time system whose matrices are derived from the PDEs. After computing the solution to this much simpler problem, the solution to the original problem can be deduced easily. In particular, the optimal compensator solution to the suboptimal H-control problem is governed by a set of hyperbolic PDEs, actuated and observed at the boundary. We illustrate our results with a boundary controlled and boundary observed vibrating string.
... It continues to be an important research focus today because its application in many important engineering systems is natural (see e.g., [3]). Such a problem has been studied in [9], [16], [44], [45] in the controllability context, in [7], [8], [23], [47], [51], [58] in terms of stabilization. ...
This paper deals with the stabilization of a class of linear infinite-dimensional systems with unbounded control operators and subject to a boundary disturbance. We assume that there exists a linear feedback law that makes the origin of the closed-loop system globally asymptotically stable in the absence of disturbance. To achieve our objective, we follow a sliding mode strategy and we add another term to this controller in order to reject the disturbance. We prove the existence of solutions to the closed-loop system and its global asymptotic stability, while making sure the disturbance is rejected.
... 13 For additional insights, refer to the following literature. [14][15][16][17][18][19] When investigating evolutionary phenomena like temperature conduction, 20 fluid dynamics, 21 and neuroscience, 22 it is imperative to consider both time and spatial diffusion distribution. Reaction-diffusion systems 23,24 aptly describe those types of phenomena. ...
This paper examines fixed-time synchronization (FxTS) for two-dimensional coupled reaction–diffusion complex networks (CRDCNs) with impulses and delay. Utilizing the Lyapunov method, a FxTS criterion is established for impulsive delayed CRDCNs. Herein, impulses encompass both synchronizing and desynchronizing variants. Subsequently, by employing a Lyapunov–Krasovskii functional, two FxTS boundary controllers are formulated for CRDCNs with Neumann and mixed boundary condition, respectively. It is observed that vanishing Dirichlet boundary contributes to the synchronization of the CRDCNs. Furthermore, this study calculates the optimal constant for the Poincaré inequality in the square domain, which is instrumental in analyzing FxTS conditions for boundary controllers. Conclusive numerical examples underscore the efficacy of the proposed theoretical findings.
... In these references, the authors considered quite often systems driven by the Saint-Venant-Exner equations as a particular application for illustrating the results. Questions like controllability and finite-time boundary control have also been studied in [10,13] and [5,12] to cite a few. Boundary output feedback stabilization for the class (1.1)-(1.3) ...
Linear-Quadratic optimal controls are computed for a class of boundary controlled, boundary observed hyperbolic infinite-dimensional systems, which may be viewed as networks of waves. The main results of this manuscript consist in converting the infinite-dimensional continuous-time systems into infinite-dimensional discrete-time systems for which the operators dynamics are matrices, in solving the LQ-optimal control problem in discrete-time and then in interpreting the solution in the continuous-time variables, giving rise to the optimal boundary control input. The results are applied to two examples, a small network of three vibrating strings and a co-current heat-exchanger, for which boundary sensors and actuators are considered.
... The extension of [33] to n + m systems in [34] was followed by major progress on adaptive control design [35]. The recent results in [36,37] allow for finite-time stabilization of linear and coupled hyperbolic systems with space and time dependent parameters. In light of the current context, it's worth noting the emergence of nonlinear controllers for nonlinear infinite-dimension systems of conservation laws [38,39]. ...
Deep neural network approximation of nonlinear operators, commonly referred to as DeepONet, has proven capable of approximating PDE backstepping designs in which a single Goursat-form PDE governs a single feedback gain function. In boundary control of coupled PDEs, coupled Goursat-form PDEs govern two or more gain kernels --- a PDE structure unaddressed thus far with DeepONet. In this note, we open the subject of approximating systems of gain kernel PDEs for hyperbolic PDE plants by considering a simple counter-convecting coupled system in whose control a kernel PDE systems in Goursat form arises.
Applications include oil drilling, Saint-Venant model of shallow water waves, and Aw-Rascle-Zhang model of stop-and-go instability in congested traffic flow. In this paper we establish the continuity of the mapping from (a total of five) plant PDE functional coefficients to the kernel PDE solutions, prove the existence of an arbitrarily close DeepONet approximation to the kernel PDEs, and establish that the DeepONet-approximated gains guarantee stabilization when replacing the exact backstepping gain kernels. Taking into account anti-collocated boundary actuation and sensing, our \emph{-Globally-exponentially} stabilizing (GES) approximate gain kernel-based output feedback design implies the deep learning of both the controller's and the observer's gains. Moreover, the encoding of the output-feedback law into DeepONet ensures \emph{semi-global practical exponential stability (SG-PES).} The DeepONet operator speeds up the computation of the controller gains by multiple orders of magnitude. Its theoretically proven stabilizing capability is demonstrated through simulations.
... Some attempt to introduce a generalized backstepping approach which does not necessarily rely on Volterra transforms have also been introduced in [20,17,27,49,50,18,26]. The Volterra approach has been used in many areas and for many systems in the last decades including parabolic equations (see for instance [5,21,24]), hyperbolic system (see for instance [39,46,3,2,29,30,19]), etc. However, no result exists on diffusion system of the form (1.4) where the domain extends with time (in a way that is not compensated in the dynamics). ...
We study the boundary stabilization of one-dimensional cross-diffusion systems in a moving domain. We show first exponential stabilization and then finite-time stabilization in arbitrary small-time of the linearized system around uniform equilibria, provided the system has an entropic structure with a symmetric mobility matrix. One example of such systems are the equations describing a Physical Vapor Deposition (PVD) process. This stabilization is achieved with respect to both the volume fractions and the thickness of the domain. The feedback control is derived using the backstepping technique, adapted to the context of a time-dependent domain. In particular, the norm of the backward backstepping transform is carefully estimated with respect to time.
... • The closed-loop feedback stabilization is stable under perturbation in many circumstances; for example, let us mention [21,39,40,48]. ...
We construct explicit time-varying feedback laws that locally stabilize the two-dimensional internal controlled incompressible Navier–Stokes equations in arbitrarily small time. We also obtain quantitative rapid stabilization via stationary feedback laws, as well as quantitative null-controllability with explicit controls having e^{C/T} costs.
... • The backstepping approach: this method, very powerful, was introduced in [31,99,144] for finite dimensional systems. It was then adapted in [42] and modified in [9,28,136,103] for parabolic partial differential equations and used for linear and then non-linear 1-D hyperbolic systems in [102,146,6,5,96,97,50]. It has also been used for other more complicated operator (e.g. ...
Hyperbolic systems model the phenomena of propagations at finite speeds. They are present in many fields of science and, consequently, in many human applications. For these applications, the question of stability or stabilization of their stationary state is a major issue. In this paper we present state-of-the-art tools to stabilize 1-D nonlinear hyperbolic systems using boundary controls. We review the power and limits of energy-like Lyapunov functions; the particular case of density–velocity systems; a method to stabilize shock steady-states; an extraction method allowing to use the spectral information of the linearized system in order to stabilize the nonlinear system; and some results on proportional-integral boundary control. We also review open questions and perspectives for this field, which is still largely open.
... x,t (\Pi ) according to the solution concept given by Definition 1.1, or even to b j \in C(\Pi ) if one uses the solution concept as in [7,Definition A.1]. The assumptions (1.5) are imposed to simplify the presentation (in particular, they are assumed in the relevant result in [33] that we cite as Theorem 1.2 below). ...
... In literature there are some stabilizability related results available for coupled system with variable coefficients. For example, feedback stabilization by backstepping has been established for the coupled parabolic system with space and time dependent coefficients in [64], [52], [20], [32], [61] and for coupled hyperbolic system with space and time dependent coefficients in [16], [17], [22], [31]. In these cases to apply backstepping, some bound for the coefficients has been assumed. ...
... The first result of the paper reveals that the optimal time for the null-controllability of system (1.11), (1.5), and (1.6) might be significantly larger than the one for the time-independent setting even when Σ is constant and C is indefinitely differentiable. More precisely, we have In a recent work, Coron et al. [14] establish the null-controllability of (1.11), (1.5), and (1.6) for time τ k + τ k+1 for all k × m matrices B. They also obtain stabilizing feedbacks and derive similar results when Σ depends on t. Combining Theorem 1.1 and their results, one obtains the optimality for the time τ k + τ k+1 when m ≥ 2 and k ≥ 1, and for a large class of B. ...
The optimal time for the controllability of linear hyperbolic systems in one dimensional space with one-side controls has been obtained recently for time-independent coefficients in our previous works. In this paper, we consider linear hyperbolic systems with time-varying zero-order terms. We show the possibility that the optimal time for the null-controllability becomes significantly larger than the one of the time-invariant setting even when the zero-order term is indefinitely differentiable. When the analyticity with respect to time is imposed for the zero-order term, we also establish that the optimal time is the same as in the time-independent setting.
In this paper we introduce a method to find the minimal control time for the null controllability of 1D first-order linear hyperbolic systems by one-sided boundary controls when the coefficients are regular enough.
Control of mixed-autonomy traffic where Human-driven Vehicles (HVs) and Autonomous Vehicles (AVs) coexist on the road has gained increasing attention over the recent decades. This paper addresses the boundary stabilization problem for mixed traffic on freeways. The traffic dynamics are described by uncertain coupled hyperbolic partial differential equations (PDEs) with Markov jumping parameters, which aim to address the distinctive driving strategies between AVs and HVs. Considering that the spacing policies of AVs vary in mixed traffic, the stochastic impact area of AVs is governed by a continuous Markov chain. The interactions between HVs and AVs such as overtaking or lane changing are mainly induced by impact areas. Using backstepping design, we develop a full-state feedback boundary control law to stabilize the deterministic system (nominal system). Applying Lyapunov analysis, we demonstrate that the nominal backstepping control law is able to stabilize the traffic system with Markov jumping parameters, provided the nominal parameters are sufficiently close to the stochastic ones on average. The mean-square exponential stability conditions are derived, and the results are validated by numerical simulations.
The Input-to-State Stability of the non-horizontal cascade channels with different arbitrary cross section, slope and friction modeled by Saint-Venant equations is addressed in this paper. The control input and measured output are both on the collocated boundary. The PI control is proposed to study both the exponential stability and the output regulation of the closed-loop systems with the aid of Lyapunov approach. An explicit quadratic Lyapunov function as a weighted function of a small perturbation of the non-uniform steady-states of different channels is constructed. We show that by a suitable choice of the boundary feedback controls, the local exponential stability and the Input-to-State Stability of the nonlinear Saint-Venant equations for the
norm are guaranteed, then validated with numerical simulations. Meanwhile, the output regulation and the rejection of constant disturbances are realized as well.
This paper investigates the stabilization problem of stop–and–go waves in vehicle traffic flow with bilateral boundary feedback control. The stop–and–go waves induce the discontinuities of vehicle speed and density, which in turn lead to different traffic states on the front and back sides of the shock front. According to the Rankine–Hugoniot condition, a propagation equation of the shock front is proposed depending on the characteristic velocities of the Aw–Rascle–Zhang (ARZ) traffic flow model. Then the complete dynamics of the stop–and–go waves is formulated as a coupled hyperbolic PDE–PDE system with a common moving boundary. The well–posedness of the coupled system with the moving boundary is established via the fixed–domain method. To stabilize the discontinuous traffic state and the location of shock front simultaneously, the bilateral boundary feedback control is formulated for the stop–and–go waves of traffic flow. Some sufficient conditions in terms of matrix inequalities are derived for ensuring the local exponential stability of the closed–loop system in the
–norm. Finally, the theoretical results are illustrated with numerical simulations.
The relevance of the stated subject is conditioned upon the presence of a real possibility to simulate vibrations of elastic membranes in space according to the Hamilton principle using degenerate three-dimensional hyperbolic equations, which is of particular practical importance from the standpoint of the prospects for mathematical modelling of the heat propagation process in oscillating elastic membranes. The purpose of this paper is to study the sequence of the procedure for mathematical modelling of heat propagation in oscillating elastic membranes which is leading to degenerate three-dimensional hyperbolic equations. The methodological approach of this study is based on a combination of theoretical study of the possibilities of constructing mathematical models of heat propagation in oscillating elastic membranes with the practical application of methods for constructing three-dimensional hyperbolic equations with type and order degeneracy to find a single solution to a mixed problem. In the course of this study, the results were presented in the form of a mathematical proof of the possibility of obtaining a single solution to a mixed problem for three-dimensional hyperbolic equations with type and order degeneracy. The results obtained in this study and the conclusions formulated on their basis are of significant practical importance for developers of methods of mathematical modelling of heat propagation processes in oscillating artificial membranes, which is of key importance from the standpoint of prospects for improving methods of mathematical modelling of processes occurring in technical devices used in various fields of modern industries.
We study the rapid stabilization of the heat equation on the 1-dimensional torus using the backstepping method with a Fredholm transformation. This classical framework allows us to present the backstepping method with Fredholm transformations for the Laplace operator in a sharp functional setting, which is the main objective of this work. We first prove that, under some assumptions on the control operator, two scalar controls are necessary and sufficient to get controllability and rapid stabilization. Then, we prove that the Fredholm transformation constructed for the Laplacian also leads to the local rapid stability of the viscous Burgers equation.
In this paper, we are interested in the minimal null control time of one-dimensional first-order linear hyperbolic systems by one-sided boundary controls. Our main result is an explicit characterization of the smallest and largest values that this minimal null control time can take with respect to the internal coupling matrix. In particular, we obtain a complete description of the situations where the minimal null control time is invariant with respect to all the possible choices of internal coupling matrices. The proof relies on the notion of equivalent systems, in particular the backstepping method, a canonical LU-decomposition for boundary coupling matrices and a compactness-uniqueness method adapted to the null controllability property.
In this paper, we consider the stabilization issues of a reaction-diffusion equation with variable coefficients and boundary input delay. At first, we design an observer based on the system output to estimate the state of the system. Due to the present of time delay in control, we design a dynamic feedback controller based on the state information of observer, that is called the integral-type controller. By selecting appropriate kernel functions, we prove that the closed-loop system is exponentially stable. Herein, our approach mainly is based on the idea of ‘feedback equivalence’. By some equivalence transformations, we establish connection between the closed-loop system and a stable system.
In this paper, the well-posedness and stability of a 1-d 2 × 2 hyperbolic system under proportional feedback control is addressed. The semigroup approach and norm equivalence theorem is adopted to obtain the explicit dissipative condition for control parameters. The asymptotic spectral analysis of the system is presented. It is shown that the real part of the eigenvalues goes to some negative constant as the modulus of eigenvalues goes to +∞. Moreover, we find that there is a group of generalized eigenfunctions which forms a Riesz basis of the Hilbert state space. Hence, the spectrum determined growth condition holds, and then the exponential stability is established.
This article studies a composite disturbance rejection control strategy containing disturbance-observer-based control (DOBC) and H∞ control for a Korteweg-de Vries-Burgers (KdVB) equation under the point measurements. Here two types of disturbances are considered: one is described by an exogenous system, and the other is an external disturbance in the L²-sense. To significantly reduce the transmitted measurements, an event-triggering mechanism (ETM) is utilized. In the framework of networked control system, quantization and communication delay of the measured signals are also taken into account. Sufficient regional exponential stability conditions are established by linear matrix inequalities (LMIs). The effectiveness of the proposed control law is verified by simulation results.
This paper concerns the problem of boundary time-varying feedback controller for fixed-time stabilization of a linear parabolic distributed parameter system with spatially and temporally varying reactivity. By utilizing the continuous backstepping approach, the invertible Volterra integral transformation with the time-dependent gain kernel is introduced to convert the closed-loop system into a target system with a time-dependent coefficient. Meanwhile, the convergence of the target system within the prescribed time is guaranteed via the Lyapunov method. The well-posedness of the resulting kernel partial differential equations is also proven by exploiting the method of successive approximation. In addition, the growth-in-time of the kernel functions is estimated by applying the generalized Laguerre polynomials and the modified Bessel functions. Subsequently, the fixed-time stability of the closed-loop system under state feedback control within the prescribed time is proven by using the fixed-time stability of the target system and the time-varying kernel functions. Finally, a numerical example is provided to illustrate the effectiveness of the proposed control method.
In this paper, we are interested in the minimal null control time of one-dimensional first-order linear hyperbolic systems by one-sided boundary controls. Our main result is an explicit characterization of the smallest and largest values that this minimal null control time can take with respect to the internal coupling matrix. In particular, we obtain a complete description of the situations where the minimal null control time is invariant with respect to all the possible choices of internal coupling matrices. The proof relies on the notion of equivalent systems, in particular the backstepping method, a canonical LU-decomposition for boundary coupling matrices and a compactness-uniqueness method adapted to the null controllability property.
In this paper, the backstepping design of stabilising state feedback controllers for coupled linear parabolic PDEs with spatially varying distinct diffusion coefficients as well as space and time dependent reaction is presented. The selected target system is a cascade of exponentially stable, time-invariant systems, with time dependent couplings, that is uniformly exponentially stable with a prescribed rate of convergence. To determine the state feedback controller, the kernel equations are derived, which results in a set of coupled PDEs for a time dependent and spatially varying kernel. For this, the method of successive approximations is extended from the time-invariant case to the present problem. The applicability of the method is demonstrated by the stabilization of two coupled unstable parabolic PDEs with space and time dependent coefficients.
We are concerned about the controllability of a general linear hyperbolic system of the form dtw(t,x) = Σ(x)∂ x w(t,x) + γC(x)w(t, x) (γ ∈ ℝ) in one space dimension using boundary controls on one side. More precisely, we establish the optimal time for the null and exact controllability of the hyperbolic system for generic γ. We also present examples which yield that the generic requirement is necessary. In the case of constant σ and of two positive directions, we prove that the null-controllability is attained for any time greater than the optimal time for all γ ∈ R and for all C which is analytic if the slowest negative direction can be alerted by both positive directions. We also show that the null-controllability is attained at the optimal time by a feedback law when C = 0. Our approach is based on the backstepping method paying a special attention on the construction of the kernel and the selection of controls.
Using the backstepping approach we recover the null controllability for the heat equations with variable coefficients in space in one dimension and prove that these equations can be stabilized in finite time by means of periodic time-varying feedback laws. To this end, on the one hand, we provide a new proof of the well-posedness and the “optimal” bound with respect to damping constants for the solutions of the kernel equations; this allows one to deal with variable coefficients, even with a weak regularity of these coefficients. On the other hand, we establish the well-posedness and estimates for the heat equations with a nonlocal boundary condition at one side.
We consider the one dimensional Schr\"odinger equation with a bilinear control and prove the rapid stabilization of the linearized equation around the ground state. The feedback law ensuring the rapid stabilization is obtained using a transformation mapping the solution to the linearized equation on the solution to an exponentially stable target linear equation. A suitable condition is imposed on the transformation in order to cancel the non-local terms arising in the kernel system. This conditions also insures the uniqueness of the transformation. The continuity and invertibility of the transformation follows from exact controllability of the linearized system.
An integral transform is introduced which allows the construction of boundary controllers and observers for a class of first-order hyperbolic PIDEs with Fredholm integrals. These systems do not have a strict-feedback structure and thus the standard backstepping approach cannot be applied. Sufficient conditions for the existence of the backstepping-forwarding transform are given in terms of spectral properties of some integral operators and, more conservatively but easily verifiable, in terms of the norms of the coefficients in the equations. An explicit transform is given for particular coefficient structures. In the case of strict-feedback systems, the procedure detailed in this paper reduces to the well-known backstepping design. The results are illustrated with numerical simulations.
Abstract We consider feedback transformations of the backstepping/feedback linearization type that have been prevalent in nite dimensional nonlinear stabilization, and, with the objective of ulti- mately addressing nonlinear PDE’s, generate the rst such transformations for a linear PDE that can have an arbitrary nite number,of open-loop unstable eigenvalues. These transformations have the form of recursive relationships and the fundamental,difculty is that the recursion has an innite number,of iterations. Naive versions of backstepping lead to unbounded,coefcients in those transformations. We show how to design them such that they are sufciently,regu- lar (not continuous but L∞). We then establish closedñloop stability, regularity of control, and regularity of solutions of the PDE.
By means of the theory on the semi-global C
1 solution to the mixed initial-boundary value problem (IBVP) for first order quasilinear hyperbolic systems, we establish
the exact controllability for general nonautonomous first order quasilinear hyperbolic systems with general nonlinear boundary
conditions.
We consider a problem of stabilization of the Euler-Bernoulli beam. The beam is controlled at one end (using position and moment actuators) and has the ldquoslidingrdquo boundary condition at the opposite end. We design the controllers that achieve any prescribed decay rate of the closed loop system, removing a long-standing limitation of classical ldquoboundary damperrdquo controllers. The idea of the control design is to use the well-known representation of the Euler-Bernoulli beam model through the Schrodinger equation, and then adapt recently developed backstepping designs for the latter in order to stabilize the beam. We derive the explicit integral transformation (and its inverse) of the closed-loop system into an exponentially stable target system. The transformation is of a novel Volterra/Fredholm type. The design is illustrated with simulations.
This paper deals with the stabilization of a rotating body-beam
system with torque control. The system we consider is the one studied by
Baillieul and Levi (1987). Xu and Baillieul proved (1993) that, for any
constant angular velocity smaller than a critical one, this system can
be stabilized by means of a feedback torque control law if there is
damping. We prove that this result also holds if there is no damping
This monograph explores the modeling of conservation and balance laws of one-dimensional hyperbolic systems using partial differential equations. It presents typical examples of hyperbolic systems for a wide range of physical engineering applications, allowing readers to understand the concepts in whichever setting is most familiar to them. With these examples, it also illustrates how control boundary conditions may be defined for the most commonly used control devices.
The authors begin with the simple case of systems of two linear conservation laws and then consider the stability of systems under more general boundary conditions that may be differential, nonlinear, or switching. They then extend their discussion to the case of nonlinear conservation laws and demonstrate the use of Lyapunov functions in this type of analysis. Systems of balance laws are considered next, starting with the linear variety before they move on to more general cases of nonlinear ones. They go on to show how the problem of boundary stabilization of systems of two balance laws by both full-state and dynamic output feedback in observer-controller form is solved by using a “backstepping” method, in which the gains of the feedback laws are solutions of an associated system of linear hyperbolic PDEs. The final chapter presents a case study on the control of navigable rivers to emphasize the main technological features that may occur in real live applications of boundary feedback control.
Stability and Boundary Stabilization of 1-D Hyperbolic Systems will be of interest to graduate students and researchers in applied mathematics and control engineering. The wide range of applications it discusses will help it to have as broad an appeal within these groups as possible.
This paper is devoted to a simple and new proof on the optimal finite control time for general linear coupled hyperbolic system by using boundary feedback on one side. The feedback control law is designed by first using a Volterra transformation of the second kind and then using an invertible Fredholm transformation. Both existence and invertibility of the transformations are easily obtained.
We consider the problem of boundary stabilization of 3 × 3 linear first-order hyperbolic systems with one positive and two negative transport speeds by using backstepping. The main result of the paper is to supplement the previous works on how to choose multi-boundary feedback inputs applied on the states corresponding to the negative velocities to obtain finite-time stabilization of the original system in the spatial L 2 sense. Our method is still valid for boundary stabilization of general n × n hyperbolic system with arbitrary numbers of states traveling in either directions. 2010 Mathematics Subject Classification. 93D15, 35L04.
In this paper, a Luenberger-type boundary observer is presented for a class of distributed-parameter systems described by time-varying linear hyperbolic partial integro-differential equations. First, known limitations due to the minimum observation time for simple transport equations are restated for the considered class of systems. Then, the backstepping method is applied to determine the unknown observer gain term. By avoiding the framework of Gevrey-functions, which is typically used for the time-varying case, it is shown that the backstepping method can be employed without severe limitations on the regularity of the time-varying terms. A modification of the underlying Volterra transformation ensures that the observer error dynamics is equivalent to the behavior of a predefined exponentially stable target system. The magnitude of the observer gain term can be traded for lower decay rates of the observer error. After the theoretic results have been proven, the effectiveness of the proposed design is demonstrated by simulation examples.
This paper deals with the problem of boundary stabilization of first-order
n\times n inhomogeneous quasilinear hyperbolic systems. A backstepping method
is developed. The main result supplements the previous works on how to design
multi-boundary feedback controllers to realize exponential stability of the
original nonlinear system in the spatial H^2 sense.
In this work, we consider the problem of boundary stabilization for a quasilinear 2×2 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 H2 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 4×4 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.
In the present article we study the stabilization of first-order linear
integro-differential hyperbolic equations. For such equations we prove that the
stabilization in finite time is equivalent to the exact controllability
property. The proof relies on a Fredholm transformation that maps the original
system into a finite-time stable target system. The controllability assumption
is used to prove the invertibility of such a transformation. Finally, using the
method of moments, we show in a particular case that the controllability is
reduced to the criterion of Fattorini.
We solve the problem of stabilization of a class of linear first-order
hyperbolic systems featuring n rightward convecting transport PDEs and m
leftward convecting transport PDEs. Using the backstepping approach yields
solutions to stabilization in minimal time and observer based output feedback.
Research on stabilization of coupled hyperbolic PDEs has been dominated by
the focus on pairs of counter-convecting ("heterodirectional") transport PDEs
with distributed local coupling and with controls at one or both boundaries. A
recent extension allows stabilization using only one control for a system
containing an arbitrary number of coupled transport PDEs that convect at
different speeds against the direction of the PDE whose boundary is actuated.
In this paper we present a solution to the fully general case, in which the
number of PDEs in either direction is arbitrary, and where actuation is applied
on only one boundary (to all the PDEs that convect downstream from that
boundary). To solve this general problem, we solve, as a special case, the
problem of control of coupled "homodirectional" hyperbolic linear PDEs, where
multiple transport PDEs convect in the same direction with arbitrary local
coupling. Our approach is based on PDE backstepping and yields solutions to
stabilization, by both full-state and observer-based output feedback,
trajectory planning, and trajectory tracking problems.
This paper is devoted to the study of the local rapid exponential
stabilization problem for a controlled Kuramoto-Sivashinsky equation on a
bounded interval. We build a feedback control law to force the solution of the
closed-loop system to decay exponentially to zero with arbitrarily prescribed
decay rates, provided that the initial datum is small enough. Our approach uses
a method we introduced for the rapid stabilization of a Korteweg-de Vries
equation. It relies on the construction of a suitable integral transform and
can be applied to many other equations.
We propose a backstepping boundary control law for Burgers’ equation with actuator dynamics. While the control law without actuator dynamics depends only on the signals u(0,t) and u(1,t), the backstepping control also depends on u x (0,t),u x (1,t),u xx (0,t) and u xx (1,t), making the regularity of the control inputs the key technical issue of the paper. With elaborate Lyapunov analysis, we prove that all these signals are sufficiently regular and the closed-loop system, including the boundary dynamics, is globally H 3 stable and well posed.
This paper is devoted to the study of the rapid exponential stabilization
problem for a controlled Korteweg-de Vries equation on a bounded interval with
homogeneous Dirichlet boundary conditions and Neumann boundary control at the
right endpoint of the interval. For every noncritical length, we build a
feedback control law to force the solution of the closed-loop system to decay
exponentially to zero with arbitrarily prescribed decay rates, provided that
the initial datum is small enough. Our approach relies on the construction of a
suitable integral transform.
Using a result on the existence and uniqueness of the semiglobal C1 solution to the mixed initial-boundary value problem for first order quasi-linear hyperbolic systems with general nonlinear boundary conditions, we establish the exact boundary controllability for quasi-linear hyperbolic systems if the C1 norm of initial and final states is small enough.
The boundary value problemc
t=c
xx−c
yy+q(t,x)c with {fx349-1} was solved by Colton [1] forq analytic int. The solution may be used for mapping solutions of the heat equation into solutions ofu
t=u
xx+q(t,x)u. Solutions (of the boundary value problem) no longer exist ifq is not analytic int.
(1) The known nullcontrollability result for boundary control ofu
t = uxx + q(x)u is generalized to consider a time-dependent coefficientq. (2) For boundary control ofu
t =Au (where it is known thatC
T
: (initial data) (optimal nullcontrol for timeT) exists for allT>0) it is shown that logC
T
=145-1 asT 0.
In this paper the recently introduced backstepping method for boundary control of linear partial differential equations (PDEs) is extended to plants with non-constant diffusivity/thermal conductivity and time-varying coefficients. The boundary stabilization problem is converted to a problem of solving a specific Klein–Gordon-type linear hyperbolic PDE. This PDE is then solved for a family of system parameters resulting in closed-form boundary controllers. The results of a numerical simulation are presented for the case when an explicit solution is not available.
In this article we examine the effect of linear feedback control in the hyperbolic distributed parameter control system By means of a reduction to canonical form similar to the one already familiar for finite-dimensional systems we show this system to be equivalent to the controlled difference-delay system The theory of nonharmonic Fourier series is then employed to investigate the placement of eigenvalues in the closed loop system. Boundary value control and canonical form for observed systems are also studied.
In this paper we study the problem of boundary feedback stabilization for the unstable heat equation u t(x, t) = u xx(x, t) + a(x)u(x, t). This equation can be viewed as a model of a heat conducting rod in which not only is the heat being diffused (mathematically due to the diffusive term u xx] but also the destabilizing heat is generating (mathematically due to the term au with a > 0). We show that for any given continuously differentiable function a and any given positive constant λ we can explicitly construct a boundary feedback control law such that the solution of the equation with the control law converges to zero exponentially at the rate of λ. This is a continuation of the recent work of Boskovic, Krstic, and Liu [IEEE Trans. Automat. Control, 46 (2001), pp. 2022-2028] and Balogh and Krstic [European J. Control, 8 (2002), pp. 165-176].
This SIAM edition is an unabridged, corrected republication of th Incluye bibliografía e índice
In this note, a feedback boundary controller for an unstable heat
equation is designed. The equation can be viewed as a model of a thin
rod with not only the heat loss to a surrounding medium (stabilizing)
but also the heat generation inside the rod (destabilizing). The heat
generation adds a destabilizing linear term on the right-hand side of
the equation. The boundary control law designed is in the form of an
integral operator with a known, continuous kernel function but can be
interpreted as a backstepping control law. This interpretation provides
a Lyapunov function for proving stability of the system. The control is
applied by insulating one end of the rod and applying either Dirichlet
or Neumann boundary actuation on the other
A class of bounded continuous time-invariant finite-time
stabilizing feedback laws is given for the double integrator. Lyapunov
theory is used to prove finite-time convergence. For the rotational
double integrator, these controllers are modified to obtain
finite-time-stabilizing feedback that avoid
“unwinding”