Exponential Stability and Transfer Functions of Processes Governed by Symmetric Hyperbolic Systems

Abstract

Abstract. In this paper we study the frequency and time domain behaviour of a heat exchanger network system. The system is governed by hyperbolic partial differential equations. Both the control operator and the observation operator are unbounded but admissible. Using the theory of symmetric hyperbolic systems, we prove exponential stability of the underlying semigroup for the heat exchanger network. Applying the recent theory of well-posed infinite-dimensional linear systems, we prove that the system is regular and derive various properties of its transfer functions, which are potentially useful for controller design. Our results remain valid for a wide class of processes governed by symmetric hyperbolic systems.Mathematics Subject Classification. 93D09, 93D25, 80A20, 35L50.

Exponential stability of distributed parameter systems governed by symmetric hyperbolic partial differential equations using Lyapunov's second method

“…We have presented a new proof of some stability result in [31] by constructing Lyapunov functionals. The presented proof is a generalization of a previous one [40] in the sense that no symmetry condition is asked for the matrix b(x). Compared with the method of characteristics used in [31] the Lyapunov method is direct, simple and admits extension to cases of nonlinearities, multiple spatial variables and more general Banach spaces.…”

“…In our previous paper [40], using a Lyapunov functional we have proved the stability result of Rauch and Taylor by supposing that some matrix b(x) in the system is symmetric. Here we propose another Lyapunov functional for the proof without any restriction on b(x).…”

“…H2 means that the system is strictly hyperbolic. As illustrated in [40], H2 is not absolutely essential in applications. H3 says that the system is strictly dissipative on the boundary.…”

“…For the second example we construct a Lyapunov functional to prove its exponential stability. We show how the Lyapunov direct method is extended to PDE systems of higher spatial dimension (see also [37,40]). Through these examples we explain how our results are related to the reviewed literature.…”

An optimisation approach for stability analysis and controller synthesis of linear hyperbolic systems

“…Indeed, this class of functions allows to express conditions for stability as Matrix Inequalities (MI). We can cite also [5], [10], [15], and [17] for the linear case. LMI conditions derived by an operator approach is used in [11] for the H ∞ boundary control of parabolic and hyperbolic systems.…”