On spectrum of a general Petrovsky type equation and Riesz basis of \(N\)-connected beams with linear feedback at joints
In this very important and useful paper the general Petrovsky-type system in a space variable in normal form \[ \begin{cases} \frac{\partial}{\partial t}\left[\begin{matrix} {\mathbf u}(x,t)\\{\mathbf v}(x,t)\end{matrix}\right]=K\,\frac{\partial^2}{\partial x^2}\left[\begin{matrix} {\mathbf u}(x,t)\\{\mathbf v}(x,t)\end{matrix} \right], \quad x\in(0,1),\;t>0,\\ [A,B][{\mathbf u}_x(0,t),{\mathbf v}_x(0,t),{\mathbf u}(0,t),{\mathbf v}(0,t)]^T=0,\\ [E,F][{\mathbf u}_x(1,t),{\mathbf v}_x(1,t),{\mathbf u}(1,t),{\mathbf v}(1,t)]^T=0,\end{cases}\tag{1} \] is studied. Here \({\mathbf u}(x,t)\), \({\mathbf v}(x,t)\) are column vectors in \(\mathbb{R}^n\), \(A\), \(B\), \(E\), and \(F\) are real, constant \((2n\times 2n)\)-matrices; \(K\) is a constant \((2n\times 2n)\)-matrix of the form: \[ K=\left[\begin{matrix} 0 & \Upsilon \\-\Upsilon & 0\end{matrix}\right],\quad \Upsilon =\text{diag}[\ell^{-2}_1,\ell^{-2}_2,\dots(-1)^{n+1}\ell^{-2}_n], \quad\ell_j>0, \quad j=1,2,\dots,n. \] The characteristic equation of system (1) is derived and the eigenfunctions are found. The general treatment of the asymptotic expansions of the eigenvalues is presented. The authors consider also a system of \(N\)-connected beams with the governing equation \[ \begin{cases} y_{tt}(x,t)+y_{xxxx}(x,t)=0,\quad L_{j-1}<x<L_j, & j=2,\dots,n \\ y(0)=y_{xx}(0)=0,\;y_x(L_n)=y_{xxx}(L_n)=0.\end{cases}\tag{2} \] Main result: A general approach is presented for the analysis of the distribution of eigenvalues of system (1). The asymptotics of eigenpairs of an \(N\)-connected beam equation with linear joints are proposed. The Riesz basis for the special system (2) is obtained. All results are presented in a systematic manner, precise proofs of the theorems being sufficiently detailed. Finally, some remarks on some unsolved problems are given.
scientific article; zbMATH DE number 2093310