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Zbl 0636.73047
Huang, Falun
On the holomorphic property of the semigroup associated with linear elastic systems with structural damping. (English)
[J] Acta Math. Sci. 5, 271-277 (1985). ISSN 0252-9602

{\it G. Chen} and {\it D. L. Russell} [(*) A mathematical model for linear elastic systems with structural damping, MRC Technical summary Report 2089, Math. Res. Center. Univ. Wisconsin-Madison (1980)] studied the following linear elastic systems with structural damping $$ \cases \ddot y+B\dot y+Ay=0, \\ y(0)=y\sb 0, \dot y(0) = y\sb 1, \endcases $$ where $\cdot$ means $\frac{d}{dt}$, $y,y\sb 0,y\sb 1\in H$, a Hilbert space with inner product ($\cdot,\cdot)$ and associated with $\Vert \cdot \Vert$, A and B are positive definite self-adjoint unbounded linear operators on H and B is $A\sp{1/2}$-bounded. Letting $x\sb 1=A\sp{1/2}y$, $x\sb 2=\dot y$, we get the equivalent first order linear systems $$ \cases \frac{d}{dt}\pmatrix x\sb 1 \\ x\sb 2 \endpmatrix = \pmatrix 0 & A\sp{1/2} \\ -A\sp{1/2} & -B\endpmatrix \pmatrix x\sb 1 \\ x\sb 2\endpmatrix = {\cal A}\pmatrix x\sb 1 \\ x\sb 2\endpmatrix \\ x\sb 1(0)=A\sp{1/2}y\sb 0,\qquad x\sb 2(0)=y\sb 1. \endcases $$ G. Chen and D. L. Russel (*) have proved that ${\cal A} = \pmatrix 0 & A\sp{1/2} \\ - A\sp{1/2} & -B \endpmatrix$ generates a holomorphic semigroup, if some addition conditions are satisfied. Moreover, they still conjectured that if A and B are positive definite self-adjoint unbounded linear operators with the domain ${\cal D}(B)\supset {\cal D}(A\sp{1/2})$, such that $$ (I)\quad \beta\sb 1(A\sp{1/2}x,x)\le (Bx,x)\le \beta\sb 2(A\sp{1/2}x,x),\quad x\in {\cal D}(A\sp{1/2}), $$ or $$ (II)\quad \beta\sb 1(Ax,x)\le (B\quad 2x,x)\le \beta\sb 2(Ax,x),\quad x\in {\cal D}(A), $$ then ${\cal A} = \pmatrix 0 & A\sp{1/2} \\ -A\sp{1/2} & -B \endpmatrix$ should generate a holomorphic semi-group, where $\beta\sb 1$ and $\beta\sb 2$ are positive constant numbers with $\beta\sb 1\le \beta\sb 2$. Also some partial results for conjecture (I) and (II) are shown in (*). Recently, the author gave an answer to the conjecture (I) affirmatively [A problem for linear elastic systems with structural damping (to appear)]. In this paper we will show that conjecture (II) is also true.

MSC 2000:: *74H45 Vibrations
47D03 (Semi)groups of linear operators
47B25 Symmetric and selfadjoint operators (unbounded)
20M05 Free semigroups

Keywords: linear elastic systems; structural damping; Hilbert space; positive definite self-adjoint unbounded linear operators; first order linear systems; holomorphic semigroup

Citations: Zbl 0515.73033

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