Nonselfadjoint operators generated by the equation of a nonhomogeneous damped string

Abstract

Abstract. We consider a one-dimensional wave equation, which governs the vibrations of a damped string with spatially nonhomogeneous density and damping coefficients. We introduce a family of boundary conditions depending on a complex parameter h. Corresponding to different values of h, the problem describes either vibrations of a finite string or propagation of elastic waves on an infinite string. Our main object of interest is the family of nonselfadjoint operators A h in the energy space of two-component initial data. These operators are the generators of the dynamical semigroups corresponding to the above boundary-value problems. We show that the operators A h are dissipative, simple, maximal operators, which differ from each other by rank-one perturbations. We also prove that the operator A 1 (h = 1) coincides with the generator of the Lax-Phillips semigroup, which plays an important role in the aforementioned scattering problem. The results of this work are applied in our two forthcoming papers both to the proof of the Riesz basis property of the eigenvectors and associated vectors of the operators A h and to establishing the exact and approximate controllability of the system governed by the damped wave equation.

Criterion of Bari basis property for $2 \times 2$ Dirac-type operators with strictly regular boundary conditions

“…Combining Proposition 7.1 with our previous results for 2 × 2 Dirac type operators we obtain the Riesz basis property and analogous of Bari basis property for the dynamic generator L of the non-canonical initial-boundary value problem (7.1)-( 7.3) for a damped string equation. The part (i) of the following result improves known results in the literature on the Riesz basis property for the operator L in the case −β 1 = β 2 , a 1 ≡ 0, h 0 = 0 (see [9,10,44,45,6,14,18,38] and references therein). The part (ii) shows the application of one of our main results Theorem 1.3.…”

“…For example, Riesz basis property of the root vectors system of L guarantees the exponential stability of the corresponding C 0 -semigroup. The Riesz basis property and behavior of the spectrum of the operator L have been studied in numerous papers (see [9,10,44,45,6,14,18,38] and references therein).…”

Criterion of Bari basis property for 2 × 2 Dirac‐type operators with strictly regular boundary conditions

“…□ Combining Proposition 7.1 with our previous results for 2 × 2 Dirac-type operators, we obtain the Riesz basis property and analogous of Bari basis property for the dynamic generator  of the noncanonical initial-BVP (7.1)-( 7.3) for a damped string equation. Part (i) of the following result improves known results in the literature on the Riesz basis property for the operator  in the case −𝛽 1 = 𝛽 2 , 𝑎 1 ≡ 0, ℎ 0 = 0 (see [6,9,10,14,18,40,47,48] and references therein). Part (ii) shows the application of one of our main results, Theorem 1.3.…”

The damped string problem revisited

“…Under additional smoothness assumptions on α and ρ, and typically for separated boundary conditions in T * T , T T * , and/or in the case of damping at the end points x = 0, 1, the Riesz basis property (without parentheses) for G T,α/ρ 2 has been established in [23], [24], [47], [86], [87], [88], [90], [91], [92]. In this context we recall that the existence of a Riesz basis (without parentheses) for D + B, or equivalently, G T,α/ρ 2 , is equivalent to both operators being unbounded spectral operators in the sense of Dunford (cf.…”

“…We note that the area of damped wave equations remains incredibly active up to this day. Since we cannot possibly describe the recent developments in detail in this paper, we refer, for instance, to [6], [7], [8], [9], [11], [13], [15], [16], [17], [19], [20], [21], [22], [23], [24], [31], [32], [34], [36], [37], [38], [39], [46], [53], [54], [55], [56], [57], [58], [59], [60], [66], [74], [75], [76], [77], [78], [79], [86], [87], [88], [90], [91], [92], [94], [101], [103], and the references therein, which lead the interested reader into a variety of directions.…”