Exponential Stability for Degenerate/Singular Hyperbolic Equations with Delayed Boundary Feedback

Stabilization of a 2 × 2 system of hyperbolic PDEs with recirculation in the unactuated channel

This paper solves the problem of boundary feedback stabilization of a class of coupled ordinary differential equations–hyperbolic equations with boundary, trace, and integral nonlocal terms. Using the backstepping approach, the controller is designed by formulating an integral operator, whose kernel is required to satisfy a coupled hyperbolic partial integral differential equation. By applying the method of successive approximations, the kernel's well-posedness is given. We prove the exponential stability of the origin of the system in a suitable Hilbert space. Moreover, a wave system with nonlocal terms is stabilized by applying the above result.

Boundary PI controllers for a star-shaped network of 2 × 2 systems governed by hyperbolic partial differential equations

In this paper we study exponential stability of a heat exchanger system with diffusion and without diffusion in the context of Banach spaces. The heat exchanger system is governed by hyperbolic partial differential equations (PDE) and parabolic PDEs, respectively, according to the diffusion impact ignored or not in the heat exchange. The exponential stability of the model with diffusion in the Banach space (C[0, 1])4 is deduced by establishing the exponential Lp stability of the considered system, and using the sectorial operator theory. The exponential decay rate of stability is also computed for the model with diffusion. Using the perturbation theory, we establish the exponential stability of the model without diffusion in the Banach space (C[0, 1])4 with the uniform topology. However the exponential decay rate of stability without diffusion is not exactly computed, since its associated semigroup is non analytic. Indeed the purpose of our paper is to investigate the exponential stability of a heat exchanger system with diffusion and without diffusion in the real Banach space X1 = (C[0, 1])4 with the uniform norm. The exponential stability of these two models in the Hilbert space X2 = (L2(0, 1))4 has been proved in [31] by using Lyapunov’s direct method. The first step consists to study the stability problem in the real Banach space Xp = (Lp(0, 1))4 equipped with the usual Lp norm, p > 1. By passing to the limit (p ! 1) we can extend some results of exponential stability from Xp = (Lp(0, 1))4 to the space X1 = (C[0, 1])4. In particular the dissipativity of the system in all the Xp spaces implies its dissipativity in X1 (see Lemma 3). The section 1 is dedicated to recall the heat exchanger models. The process with diffusion is governed by a system of parabolic PDEs, and the process without diffusion is described by degenerate hyperbolic PDEs of first order. The section 2 deals with exponential stability of the parabolic system in the Lebesgue spaces Lp(0, 1) , 1 < p < 1. Certain results can be extended to the X1 space. Unfortunately this study doesn’t allow us to deduce the expected stability of the system in X1. In the section 3, the sectorial operator theory is made use of to get exponential stability results on the model with diffusion in Xp. Specifically the theory enables us to determine the exponential decay rate in (C[0, 1])4 by computing the spectrum bound. In the section 4, using a perturbation technique we show the exponential stability for the model without diffusion in all Xp spaces, 1 < p < 1. We then take the limit, as p goes to 1, to deduce the exponential stability of the system in the Banach space X1. We call the diffusion model the heat exchanger model with diffusion taken into account and the convection model the heat exchanger without diffusion, respectively. We use the analyticity property of the semigroup associated to the diffusion model in order to determine its exponential decay rate. However the semigroup associated to the convection model is not analytic. In the latter case we have not yet found an efficient method to compute exactly the exponential decay rate. The main tools we use for our investigations are the notion of dissipativity in the Banach spaces, specifically in the Lp spaces, and the sectorial operator theory. As the reader will see our work presents some extensions of the Lyapunov’s direct method to a context of Banach spaces. We will denote the system operator associated to the diffusion model by Ad,p, and that of the convection model by Ac,p, respectively. The index p indicates the Lp( ) space in which the system evolves and the operator Ad,p or Ac,p is considered. Thus Ad,p (resp. Ac,p) indicates the diffusive (resp. convective) operator in the Xp space.

Lyapunov functions for switched linear hyperbolic systems

On boundary feedback stabilization of non-uniform linear [formula omitted] hyperbolic systems over a bounded interval

Further results on boundary feedback stabilisation of 2 × 2 hyperbolic systems over a bounded interval

We summarize recent theoretical results as well as numerical results on the feedback stabilization of first order quasilinear hyperbolic systems (on networks). For the stabilization linear feedback controls are applied at the nodes of the network. This yields the existence and uniqueness of a C 1-solution of the hyperbolic system with small C 1-norm. For this solution an appropriate L 2-Lyapunov function decays exponentially in time. This implies the exponential stability of the system. A numerical discretization of the Lyapunov function is presented and a numerical analysis shows the expected exponential decay for a class of first-order discretization schemes. As an application for the theoretical results the stabilization of the gas flow in fan-shaped pipe networks with compressors is considered.

Simultaneous downstream and upstream output-feedback stabilization of cascaded freeway traffic

This paper presents the output feedback stabilization of a first‐order hyperbolic equation with multi‐point nonlocal term by boundary control. First of all, we utilize the backstepping method to design a boundary state feedback controller. On the basis of equivalence of original and target system, we guarantee the exponential stability of closed‐loop system. Hereafter, the output feedback control in the light of an infinite‐dimensional observer is constructed. We demonstrate that the resulting closed‐loop system is exponentially stable via operator semigroup theory. Finally, we display the validity of the proposed controller by some numerical simulations.

This paper is motivated by, and ultimately directed to, boundary feedback partial differential equations of both parabolic and hyperbolic type, defined on a bounded domain. It is written, however, in abstract form. It centers on the (feedback) operator AF=A+P; A the infinitesimal generator of a s.c. semigroup on H; P an Abounded, one dimensional range operator (typically nondissipative), so that P=(A·, a)b, for a, b ∈ H. While Part I studied the question of generation of a s.c. semigroup on H by AF and lack thereof, the present Part II focuses on the following topics: (i) spectrum assignment of AF, given A and a ∈ H, via a suitable vector b ∈ H; alternatively, given A, via a suitable pair of vectors a, b ∈ H; (ii) spectrality of AF—and lack thereof—when A is assumed spectral (constructive counterexamples include the case where P is bounded but the eigenvalues of A have zero gap, as well as the case where P is genuinely Abounded). The main result gives a set of sufficient conditions on the eigenvalues {λn} of A, the given vector a ∈ H and a given suitable sequence {en} of nonzero complex numbers, which guarantee the existence of a suitable vector b ∈ H such that AF possesses the following two desirable properties: (i) the eigenvalues of AF are precisely equal to λn+en; (ii) the corresponding eigenvectors of AF form a Riesz basis (a fortiori, AF is spectral). While finitely many en′s can be preassigned arbitrarily, it must be however that en → 0 « sufficiently fast ». Applications include various types of boundary feedback stabilization problems for both parabolic and hyperbolic partial differential equations. An illustration to the damped wave equation is also included.

The stabilization problems for parabolic and hyperbolic partial differential equations with Dirichlet boundary condition are considered. The systems are stabilized by a boundary feedback in (1) The operator equation, (2) The boundary condition, (3) Both the operator equation and the boundary condition; the existence of feedback semigroups in these cases is also proved. The main tool in the investigation is a pseudodifferential transformation that transforms the domains of the feedback semigroup generators into classical operator domains, where a direct resolvent analysis can be employed. The transformation turns out to be a shortcut to some of the stabilization results of Lasiecka and Triggiani in [ J. Differential Equations, 47 (1983), pp. 245–272], [SIAM J. Control Optim., 21 (1983), pp. 766–802], and [Appl. Math. Optim., 8 (1981), pp. 1–37], and it illuminates to some extent how a change of boundary condition influences the systems.

This paper is motivated by, and ultimately directed to, boundary feedback partial differential equations of both parabolic and hyperbolic type, defined on a bounded domain. It is written, however, in abstract form. It centers on the (feedback) operator AF=A+P; A the infinitesimal generator of a s.c. semigroup on H; P an Abounded, one dimensional range operator (typically nondissipative), so that P=(A·, a)b, for a, b ∈ H. While Part I studied the question of generation of a s.c. semigroup on H by AF and lack thereof, the present Part II focuses on the following topics: (i) spectrum assignment of AF, given A and a ∈ H, via a suitable vector b ∈ H; alternatively, given A, via a suitable pair of vectors a, b ∈ H; (ii) spectrality of AF—and lack thereof—when A is assumed spectral (constructive counterexamples include the case where P is bounded but the eigenvalues of A have zero gap, as well as the case where P is genuinely Abounded). The main result gives a set of sufficient conditions on the eigenvalues {λn} of A, the given vector a ∈ H and a given suitable sequence {εn} of nonzero complex numbers, which guarantee the existence of a suitable vector b ∈ H such that AF possesses the following two desirable properties: (i) the eigenvalues of AF are precisely equal to λn+εn; (ii) the corresponding eigenvectors of AF form a Riesz basis (a fortiori, AF is spectral). While finitely many εn′s can be preassigned arbitrarily, it must be however that εn → 0 « sufficiently fast ». Applications include various types of boundary feedback stabilization problems for both parabolic and hyperbolic partial differential equations. An illustration to the damped wave equation is also included.

In this paper, 1-D mathematical model of the coagulation process of the polyacrylonitrile (PAN) carbon fiber is established using Fick diffusion law. Boundary stabilization for a linear parabolic diffusion-reaction partial differential equation (PDE) is considered. We use the method of backstepping to implement the boundary control of the concentration diffusion in the forming process of carbon fiber. By using the coordinate transformation, we transform the original system to a standard static system. The transformation depends on a so called gain kernel function, and we can design the boundary feedback controller using the kernel function. For the model in this paper, the kernel function itself is a hyperbolic PDE, and there is no explicit formation. Therefore, we use numerical methods to obtain the kernel function, and give the simulation results for the closed-loop control response. The simulation results show that the open-loop unstable system is stabilized by a boundary feedback.

The last chapter of the book is devoted to the study of parabolic–hyperbolic PDE loops by means of the small-gain methodology. Since there are many possible interconnections that can be considered, we focus on two particular cases, which are analyzed in detail. The first case considered in the chapter is the feedback interconnection of a parabolic PDE with a special first-order hyperbolic PDE: a zero-speed hyperbolic PDE. The study of this particular loop is of special interest because it arises in an important application: the movement of chemicals underground. Moreover, the study of this loop can be used for the analysis of wave equations with Kelvin–Voigt damping. The second case considered in the chapter is the feedback interconnection of a parabolic PDE with a first-order hyperbolic PDE by means of a combination of boundary and in-domain terms. The interconnection is effected by linear, non-local terms. For both cases, results for existence/uniqueness of solutions as well as sufficient conditions for ISS or exponential stability in the spatial sup-norm are provided.