A constructive method to controllability and observability for quasilinear hyperbolic systems

Abstract

Abstract. A simple constructive method is presented to get the exact boundary controllability, the exact boundary controllability of nodal profile and the exact boundary observability for 1-D quasilinear hyperbolic systems.Key words. Exact boundary controllability, Exact boundary controllability of nodal profile, Exact boundary observability, Quasilinear hyperbolic system, Contructive method.AMS subject classifications. 35B37, 93B05, 93B07, 35L501. Introduction and preliminaries. The controllability and observability are of great importance in both theory and applications. A complete theory has been established for linear hyperbolic systems, in particular, for linear wave equations [25][26][27]. There have also been some results for semilinear wave equations [4,11,[29][30]. For quasilinear hyperbolic systems that have numerous applications in mechanics, physics and other applied sciences, however, very few results are available even in one-space-dimensional case [2][3].In this paper we will present a simple and efficient constructive method to the exact boundary controllability, the exact boundary controllability of nodal profile and the exact boundary observability for general 1-D quasilinear hyperbolic systems with general nonlinear boundary conditions [12,[14][15][16][17][19][20][21].Noting that for the weak solution to quasilinear hyperbolic systems, which includes shock waves and corresponds to an irreversible process, generically speaking, it is impossible to have the exact boundary controllability for any arbitrarily given initial and final states [1]. In order to give a general and systematic presentation, we consider only the classical solution to quasilinear hyperbolic systems, which corresponds to a reversible process.Since this method is given in the framework of classical solutions, only the local exact boundary controllability and the local exact boundary observability can be obtained generically, however, in the special case that the problem is linear, this method will directly lead to the global exact boundary controllability and the global exact boundary observability.In what follows, we consider only the case of first order quasilinear hyperbolic systems, for higher order quasilinear hyperbolic systems the corresponding problem can be discussed in a similar way [13,[22][23]28].Consider the following 1-D first order quasilinear hyperbolic system

PDE Observer Design for Counter-Current Heat Flows in a Heat-Exchanger

“…Using the fact that controllability and observability of the class of hyperbolic transport systems to which (4) belongs has been proved by (Li, 2010), we can build a boundary observer to estimate the distributed temperatures. The measurements from the sensors at the boundaries of the PHE are used by the observer to correct the initial conditions of the PDE system and thus estimate the distributed values.…”

“…Alinhac (2009) has, for example, dedicated an entire volume to the study of this class of PDE. Li (2010), on the other hand, proposes a thorough investigation of the controllability and observability of quasilinear hyperbolic systems. The issue of boundary control of such class of systems is also present in the literature.…”

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