4 The Three String Network

4 The Three String Network

This chapter is devoted to study the control problem for the simplest non trivial network of strings that cannot be reduced to a single string: the three string network. Most of the results presented here will be generalized later in Chapter 5 to the case of general networks supported by tree-shaped graphs. However, the generality of the problem in that case involves complex notations. It is therefore convenient to first address the simple case of the three string network, for which the main ideas involved in our analysis, that will allow us to address the case of general networks, can be described more transparently. We first consider the case when two of the three external nodes of the network are controlled. In this case, standard methods based on the d’Alembert formula and energy arguments allow showing that the observability and controllability properties hold in the optimal energy space. We then address the case when a single control acts on one exterior node. In this case the problem is much more complex since the space in which observability and controllability hold depend on the irrationallity properties of the ratios of the lengths of the strings entering in the network. The methods to analyze it also vary significantly and are based on results from Number Theory.

4.1 The Three String Network with Two Controlled Nodes 4.1.1 Equations of Motion of the Network Let T ,
0 ,
1 ,
2 be positive numbers. We consider the following nonhomogeneous system

54

4 The Three String Network

⎧ i utt − uixx = 0 in R×[0,
i ], i = 0, 1, 2, ⎪ ⎪ ⎪ ⎪ 0 1 2 ⎪ (t, 0) = u (t, 0) = u (t, 0) t ∈ R, u ⎪ ⎨ 0 1 2 ux (t, 0) + ux (t, 0) + ux (t, 0) = 0 t ∈ R, ⎪ ⎪ i i 2 ⎪ u (t,
i ) = v (t), u (t,
2 ) = 0 t ∈ R, i = 0, 1, ⎪ ⎪ ⎪ ⎩ i u (0, x) = ui0 (x), uit (0, x) = ui1 (x) x ∈ [0,
i ], i = 0, 1, 2

(4.1)

which models the vibrations of a network formed by three elastic strings e0 , e1 , e2 with lengths
0 ,
1 ,
2 coupled at one of their extremes. The functions ui = ui (t, x) : [0,
i ] → R, i = 0, 1, 2, represent the transversal displacements of the strings. On the free nodes of the strings e0 and e1 some external controls v 0 and v 1 act regulating their motion. Let us observe that in (4.1), the parametrization of the strings has been chosen so that x = 0 corresponds to the common node, while x =
i correspond to the exterior nodes of the strings ei , i = 1, 2.

• (uncontrolled node)
2

•
0

•

v0


1

•

v1

Fig. 4.1. The three string network with two controlled nodes

Let T > 0. According to the general results described in Chapter 2, the homogeneous system resulting in the absence of control in (4.1) (v 0 = v 1 = 0) ⎧ i φtt − φixx = 0 in R×[0,
i ], i = 0, 1, 2, ⎪ ⎪ ⎪ ⎪ 0 1 2 ⎪ t ∈ R, ⎪ ⎨ φ (t, 0) = φ (t, 0) = φ (t, 0) 0 1 2 (4.2) φx (t, 0) + φx (t, 0) + φx (t, 0) = 0 t ∈ R, ⎪ ⎪ i ⎪ ⎪ t ∈ R, i = 0, 1, 2, φ (t,
i ) = 0 ⎪ ⎪ ⎩ i i i i φ (0, x) = φ0 (x), φt (0, x) = φ1 (x) x ∈ [0,
i ], i = 0, 1, 2, ¯ with initial state (φ ¯ ,φ ¯ ) ∈ V × H satisfying has a unique solution φ 0 1  ¯ ∈ C([0, T ] : V ) C 1 ([0, T ] : H). φ

(4.3)

Recall that the spaces V and H are those defined in Chapter 2, Section 2.2, that is