Exact boundary controllability of coupled hyperbolic equations

Abstract

We study the exact boundary controllability of two coupled one dimensional wave equations with a control acting only in one equation. The problem is transformed into a moment problem. This framework has been used in control theory of distributed parameter systems since the classical works of A.G. Butkovsky, H.O. Fattorini and D.L. Russell in the late 1960s to the early 1970s. We use recent results on the Riesz basis property of exponential divided differences.

The Kalman condition for the boundary controllability of coupled 1-d wave equations

“…We recall that System (1) is exactly controllable in H at time T if, for every initial and final data (u 0 , u 1 ), (z 0 , z 1 ), both in H, there exists a control f ∈ L 2 (0, T ) such that the solution of System (1) corresponding to (u 0 , u 1 , f ) satisfies (2) u(x, T ) = z 0 (x), u t (x, T ) = z 1 (x).…”

“…Most of the known controllability results of (1) are in the case of two coupled equations: see [2,3,4], but the results are for a particular coupling matrix A. In the d-dimensional situation, that is, for a system of coupled wave equations in a domain Ω ⊂ R d , Alabau-Boussouria and collaborators have obtained several results in the case of two equations for particular coupling matrices (see e.g.…”

Uniform indirect boundary controllability of semi-discrete <inline-formula><tex-math id="M1">\begin{document}$ 1 $\end{document}</tex-math></inline-formula>-<inline-formula><tex-math id="M2">\begin{document}$ d $\end{document}</tex-math></inline-formula> coupled wave equations

“…Contrarly to the multidimensional case, the one dimensional case is well-understood now, the article [4] (in fact the problem considered in [4] is slightly more general than (1), see Section 5 for more details) answers totally the question and describes exactly controllable spaces for the coupled hyperbolic system (1) and gave explicit descriptions of non-controllable situations. More precisely, in [4] it was proved that if α = 0, (1) is controllable if, and only if…”

“…As mentioned in the introduction, lack of uniform controllability is caused by high frequency modes generated by the semi-discrete problem (4). Then, in order to obtain a bounded sequence of controls, one has to filter out these spurious frequency modes.…”

Controllability for a string with attached masses and Riesz bases for asymmetric spaces

“…An important feature that sometimes arises in such systems is that they are well posed with respect to asymmetric spaces. Examples of these systems include strings with attached masses and many other hybrid systems, such as the so-called cascade systems of the wave equations, where controls are applied directly to the first system and every next system is controlled through the previous one (see [5,15] and references therein).…”