Class Of Linear Hyperbolic Systems Research Articles

Sampled-data distributed control for homo-directional linear hyperbolic system with spatially sampled state measurements

This paper addresses the stability analysis problem for a class of linear hyperbolic systems with distributed controllers sampled in space and time. First, the considered system is recast in an equivalent form with a continuous time control loop and operators representing the discretization errors (spatio-temporal sampling errors). Then with the help of the Lyapunov–Razumikhin approach, the Rɛ-stability of the proposed linear hyperbolic systems of balance laws is verified via sufficient conditions. At last, the proposed method is illustrated numerically.

A Hybrid System Approach to Exponential Stability with Sampled-data Control for a Class of Linear Hyperbolic Systems

Abstract In this paper, we address the problem of exponential stability for a class of linear hyperbolic systems with distributed sampled-data control. First, we recast the original system into a hybrid model via an augmented system approach. Using this model, the link between the sampling interval, the system state and its sampled vector is characterized by an Integral Quadratic Constraint (IQC). The obtained IQC is used for deriving numerically tractable stability criteria. The method is illustrated by an open loop unstable numerical example.

Stability Analysis for A Class of Linear Hyperbolic System of Balance Laws with Sampled-data Control

Abstract The stability for a class of linear hyperbolic systems with distributed sampled-data controllers is discussed in this paper. The original sampled-data system is firstly transformed into a new equivalent system by modelling the sampling induced error as a reset integrator operator. Then we construct an appropriate Lyapunov function and obtain sufficient conditions for the Re - stability of the system based on linear matrix inequalities (LMIs). A numerical example illustrates our results: when the sampling interval is within the allowable range, the solution of the system converges from the domain of attraction to a positive invariant set.

Regional stability and stabilization of a class of linear hyperbolic systems with nonlinear quadratic dynamic boundary conditions

This paper addresses the boundary control problem of fluid transport in a Poiseuille flow taking the actuator dynamics into account. More precisely, sufficient stability conditions are derived to guarantee the exponential stability of a linear hyperbolic differential equation system subject to nonlinear quadratic dynamic boundary conditions by means of Lyapunov based techniques. Then, convex optimization problems in terms of linear matrix inequality constraints are derived to either estimate the closed-loop stability region or synthesize a robust control law ensuring the local closed-loop stability while estimating an admissible set of initial states. The proposed results are then applied to application-oriented examples to illustrate local stability and stabilization tools.

Singular Perturbation Approximation of Linear Hyperbolic Systems of Balance Laws

This technical note deals with a class of linear hyperbolic systems of balance laws with multiple time scales. The scale of time constants is modeled by a perturbation parameter. This parameter is introduced in both dynamics and boundary conditions. The solution of the full system is approximated by that of the reduced subsystem when the perturbation parameter is small enough. Lyapunov technique is used to prove it. The main result is illustrated by an academic example. Moreover, the boundary control synthesis to a gas flow transport model is shown based on singular perturbation approach.

Event-based stabilization of linear systems of conservation laws using a dynamic triggering condition

This paper deals with a new event-based stabilization strategy for a class of linear hyperbolic systems of conservation laws. It includes an internal dynamic which serves as a filter mechanism for the event-triggered condition previously introduced in Espitia et al. (2016). The well-posedness as well as the global exponential stability of the resulting closed-loop system is studied. Some numerical simulations are performed to validate the theoretical results.

Abstract State-Space Models for a Class of Linear Hyperbolic Systems of Balance Laws

The paper discusses and compares different abstract state-space representations for a class of linear hyperbolic systems defined on a one-dimensional spatial domain. It starts with their PDE representation in both weakly and strongly coupled forms. Next, the homogeneous state equation including the unbounded formal state operator is presented. Based on the semigroup approach, some results of well-posedness and internal stability are given. The boundary and observation operators are introduced, assuming a typical configuration of boundary inputs as well as pointwise observations of the state variables. Consequently, the homogeneous state equation is extended to the so-called boundary control state/signal form. Next, the classical additive statespace representation involving (A, B, C)-triple of state, input and output operators is considered. After short discussion on the appropriate Hilbert spaces, state-space equation in the so-called factor form is also presented. Finally, the resolvent of the system state operator A is discussed.

Exact representation of the asymptotic drift speed and diffusion matrix for a class of velocity-jump processes

This paper examines a class of linear hyperbolic systems which generalizes the Goldstein–Kac model to an arbitrary finite number of speeds vi with transition rates μij. Under the basic assumptions that the transition matrix is symmetric and irreducible, and the differences vi−vj generate all the space, the system exhibits a large-time behavior described by a parabolic advection–diffusion equation. The main contribution is to determine explicit formulas for the asymptotic drift speed and diffusion matrix in term of the kinetic parameters vi and μij, establishing a complete connection between microscopic and macroscopic coefficients. It is shown that the drift speed is the arithmetic mean of the velocities vi. The diffusion matrix has a more complicate representation, based on the graph with vertices the velocities vi and arcs weighted by the transition rates μij. The approach is based on an exhaustive analysis of the dispersion relation and on the application of a variant of the Kirchoff's matrix tree Theorem from graph theory.

Tikhonov theorem for linear hyperbolic systems

A class of linear hyperbolic systems of conservation laws with multiple time scales which are modeled by a perturbation parameter is considered in this paper. By setting the perturbation parameter to zero, two subsystems, the reduced subsystem standing for the slow dynamics and the boundary-layer subsystem representing the fast dynamics, are computed. It is first proved that the exponential stability of the full system implies the stability of both subsystems. Secondly, a counterexample is given to indicate that the stability of the two subsystems does not ensure the full system’s stability. Moreover a new Tikhonov theorem for this class of infinite dimensional systems is stated. The solution of the full system can be approximated by that of the reduced subsystem, and this is proved by Lyapunov techniques. An application to boundary feedback stabilization of gas transport model is used to illustrate the results.

Localized wave and vortical solutions to linear hyperbolic systems and their application to linear shallow water equations

The result of this paper is that any fast-decaying function can be represented as an integral over the canonical Maslov operator, on a special Lagrangian manifold, acting on a specific function. This representation enables one to construct effective explicit formulas for asymptotic solutions of a vast class of linear hyperbolic systems with variable coefficients.

Convex optimal control problem involving a class of linear hyperbolic systems with constraints

A convex optimal control problem with inequality constraints involving a class of linear hyperbolic partial differential equations is considered. A result on the existence of optimal control is proved and a necessary condition for optimality is given. An iterative algorithm is presented for solving the optimal control problem under consideration. The convergence properties are also studied.

A convex optimal control problem involving a class of linear hyperbolic systems

In this paper, we consider a convex optimal control problem involving a class of linear hyperbolic partial differential systems. A computational algorithm which generates minimizing sequences of controls is devised. The convergence properties of the algorithm are investigated.

Control Canonical Forms and Eigenvalue Assignment by Feedback for a Class of Linear Hyperbolic Systems

Canonical forms are developed for a class of linear hyperbolic systems. They are then applied to solve the problem of eigenvalue assignment by distributed feedback and boundary control. The duality of this problem is demonstrated to one of eigenvalue assignment by boundary feedback of an adjoint system subject to distributed control. For both systems it is shown that by feedback, the set $\{ \rho _j \} $, $j \in \mathbb{Z}$, can be assigned as eigenvalues of the closed loop system, subject to an asymptotic condition on the set $\{ \rho _j \} $. The feedback control is explicitly characterized. Analogous results are obtained for the problem of eigenvalue assignment by distributed feedback and distributed control.

Eigenvalue Assignment by Delayed Boundary Feedback for a Class of Linear Hyperbolic Systems

Eigenvalue Assignment by Delayed Boundary Feedback for a Class of Linear Hyperbolic Systems