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Simultaneous Observability of Infinitely Many Strings and Beams NETWORKS AND HETEROGENEOUS MEDIA doi:10.3934/nhm.2020017 c American Institute of Mathematical Sciences Volume 15, Number 4, December 2020 pp. 633{652 SIMULTANEOUS OBSERVABILITY OF INFINITELY MANY STRINGS AND BEAMS Vilmos Komornik College of Mathematics and Computational Science Shenzhen University Shenzhen 518060, Peoples Republic of China Dpartement de mathmatique Universit de Strasbourg 7 rue Ren Descartes 67084 Strasbourg Cedex, France Anna Chiara Lai and Paola Loreti∗ Sapienza Universit di Roma Dipartimento di Scienze di Base e Applicate per l'Ingegneria Via A. Scarpa, 16 00161, Roma, Italy (Communicated by Enrique Zuazua) Abstract. We investigate the simultaneous observability of infinite systems of vibrating strings or beams having a common endpoint where the observation is taking place. Our results are new even for finite systems because we allow the vibrations to take place in independent directions. Our main tool is a vectorial generalization of some classical theorems of Ingham, Beurling and Kahane in nonharmonic analysis. 1. Introduction. In this paper we are investigating finite and infinite systems of strings or beams having a common endpoint, whose transversal vibrations may take place in different planes. We are interested in conditions ensuring their simultaneous observability and in estimating the sufficient observability time. There have been many results during the last twenty years on the simultaneous observability and controllability of systems of strings and beams, see e.g., [1]-[6], [11]-[12], [20], [25]. In all earlier papers the vibrations were assumed to take place in a common vertical plane. Here, we still assume that each string or beam is vibrating in some plane, but these planes may differ from one another. This leads to important new difficulties, requiring vectorial generalizations of clasical Ingham type theorems. Our approach also allows us to consider infinite systems of strings or beams, which requires a deeper study of the overall density of the union of all corresponding eigenfrequencies. 2020 Mathematics Subject Classification. Primary: 93B07; Secondary: 35L05, 74K10, 42A99. Key words and phrases. Fourier series, non-harmonic analysis, wave equation, strings, beams, observability. The first author was supported by the Visiting Professor Programme, Sapienza Universit di Roma. ∗ Corresponding author: Paola Loreti. 633 634 VILMOS KOMORNIK, ANNA CHIARA LAI AND PAOLA LORETI For a general introduction to the controllability of PDE's we refer to [22, 23] or [18]. The approach of the present paper is based on some classical results of Ingham [14], Beurling [9] and Kahane [17] on nonharmonic analysis. Some of the first applications to control thery were given in the papers of Ball and Slemrod [7] and Haraux [13]. We refer to [19] for a general introduction. The paper is organized as follows. Section2 is devoted to the statement of our main results. In Section3 we briefly recall out harmonic analysis tools on which the proofs of our main theorems are based. The remaining part of the paper is devoted to the proofs of the results. In particular, the theorems concerning the observability of string systems (Theorems 2.1, 2.2 and 2.3) are proved in Sections 4 and5. In Section6 we prove Theorem 2.4 on the observability of infinite beam systems under some algebraic conditions on the lengths of the beams. Finally in Section7 we prove Proposition1 providing many examples where the hypotheses of Theorem 2.4 are satisfied. 2. Main results. In what follows we state our main results on the simultaneous observability of string and beam systems. 2.1. Simultaneous observability of string systems. We consider a system of J < 1 vibrating strings of length `j, 1 ≤ j ≤ J. The problem is set in the spherical π π coordinate system (r; '; θ) 2 [0; 1) × R × (− 2 ; 2 ]. The direction of the j-th string 3 is dj := (`j;'j; θj): this, together with an unit vector vj in R orthogonal to dj, is fixed by initial data. The transversal displacement of the j-th string at time t and at the point (r; 'j; θj) for r 2 [0; `j] is denoted by uj(t; r; 'j; θj)vj. We consider the following uncoupled system: 8 >uj;tt − uj;rr = 0 in R × (0; `j); > <uj(t; 0;'j; θj) = uj(t; `j;'j; θj) = 0 for t 2 R; >uj(0; r; 'j; θj) = uj0(r) and uj;t(0; r; 'j; θj) = uj1(r) for r 2 (0; `j); > :j = 1; : : : ; J: (1) (As usual, the subscripts t and r denote differentiations with respect to these vari- ables.) It is well-known that the system is well posed for 1 2 uj0 2 H0 (0; `j) and uj1 2 L (0; `j); j = 1; : : : ; J; and the corresponding functions t 7! uj;r(t; 0;'j; θj) are locally square integrable (\hidden regularity"). We seek conditions ensuring that the linear map J X (u10; u11; : : : ; uJ0; uJ1) 7! uj;r(·; 0;'j; θj)vj (2) j=1 QJ 1 2 2 3 of the Hilbert space H := j=1 H0 (0; `j) × L (0; `j) into Lloc(R; R ) is one-to-one. If this is so, then we would also like to obtain more precise, quantitative norm estimates. Setting kπ !j;k := `j SIMULTANEOUS OBSERVABILITY OF INFINITELY MANY STRINGS AND BEAMS 635 for brevity, the solutions of (1) are given by the formulas 1 X i!j;kt −i!j;kt uj(t; r; 'j; θj) = bj;ke + bj;−ke sin(!j;kr); j = 1;:::;J k=1 with suitable complex coefficients bj;±k, and J J 1 ! X X X i!j;kt −i!j;kt uj;r(t; 0;'j; θj)vj = !j;k bj;ke + bj;−ke vj: j=1 j=1 k=1 The linear map (2) is not always one-to-one. Indeed, if there exists a real number ! such that the set of vectors vj : there exists a kj satisfying !j;kj = ! is linearly dependent, then denoting by J 0 the set of the corresponding indices j and choosing a nontrivial (finite) linear combination X αjvj = 0; j2J 0 the functions ( i!t 0 αje sin(!r) if j 2 J , uj(t; r; 'j; θj) := 0 if j 2 f1;:::;Jg n J 0 define a non-trivial solution of (1) satisfying J X uj;r(t; 0;'j; θj)vj = 0 for all t 2 R; j=1 so that the linear map (2) on H is not one-to-one. A positive observability result is the following: Theorem 2.1. Assume that `j=`m is irrational for all j 6= m: (3) Then there exists a number T0 2 [2 max f`1; : : : ; `J g ; 2(`1 + ··· + `J )] such that the restricted linear map J X (u10; u11; : : : ; uJ0; uJ1) 7! uj;r(·; 0;'j; θj)vjjI j=1 where u1; : : : ; uJ are solutions of (1), is one-to-one for every interval I of length > T0, and for no interval I of length < T0. Remark 1. The proof of Theorem 2.1 yields a more precise estimation of T0 in some special cases (see Remark7 below). We give three examples. (i) If J = 3 and v1; v2; v3 are mutually orthogonal, then T0 = 2 max f`1; `2; `3g, see Figure1-(i). (ii) If J = 3 and v1 ? v2 = v3, then T0 = 2 max f`1 + `2; `3g , see Figure1-(ii). 636 VILMOS KOMORNIK, ANNA CHIARA LAI AND PAOLA LORETI d3 p3 p1 = p3 d3 p3 p1 p1 p2 d3 d2 d2 d1 d1 d1 d2 p2 p2 (i) (ii) (iii) Figure 1. A system of three strings with vibration planes pj spanned by dj := (`j; φj; θj) and vj ? dj, j = 1; 2; 3. In (i) `1 = `2 = `3 = 1, the vj's are pairwise orthogonal, and v1 = d3, v2 = d1, v3 = d2. We have T0 = 2 maxp f`1; `2; `3g = 2. In (ii), we have `1 = `3 = 1, `2 = 2=(2 + 2) and v1 = d3 ? d1 = v2 = v3. Then T0 = 2 max f`1 + `2; `3gp ≈ 3:1715. In the planarp case (iii) we have `1 = 1, `2 = 2=(2 + 2) and `3 = 2=(4 + 2), so that T0 = 2 max f`1 + `2 + `3g ≈ 3:9103. (iii) If all vectors vj are equal (we call this the planar case), then T0 = 2(`1 + ··· + `J ){ see Figure1-(iii). Thus we recover an earlier theorem in [5,6] by using Fourier series. It was also proved in [11] by a method based on d'Alembert's formula. The method of Fourier series enables us to consider more general equations of the type uj;tt −uj;rr +ajuj = 0 in (1) with arbitrary nonnegative constants (d'Alembert's formulas cannot be generalized to the case where aj 6= 0). Moreover, we may even consider infinitely many strings. Incidentally note that if the vectors vj are linearly independent then we do not need the assumption (3). But this may only happen if we have at most three vectors, while the interest of the present paper is to have many, maybe even infinitely many strings or beams, and thus many vectors vj. Under some further assumptions on the lengths of the strings we may also get explicit norm estimates. We adopt the following notations. For each fixed j ≥ 1, let q kπx ej;k(x) := 2=`j sin ; k = 1; 2;::: `j 2 s be the usual orthonormal basis of L (0; `j). We denote by D (0; `j) the Hilbert spaces obtained by completion of the linear hull of (ej;k) with respect to the Eu- clidean norm 1 1 2s !1=2 X X kπ 2 ckej;k := jckj : `j s k=1 D (0;`j ) k=1 SIMULTANEOUS OBSERVABILITY OF INFINITELY MANY STRINGS AND BEAMS 637 2 Note that, identifying L (0; `j) with its dual, we have 0 2 1 1 −1 −1 D (0; `j) = L (0; `j);D (0; `j) = H0 (0; `j) and D (0; `j) = H (0; `j) with equivalent norms.
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