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This research is devoted to the asymptotic and spectral analysis of a coupled Euler–Bernoulli and Timoshenko beam model. The model is governed by a system of two coupled hyperbolic partial differential equations and a four-parameter family of the boundary conditions modeling the action of self-straining actuators. The aforementioned system of equations of motion together with the boundary conditions is equivalent to a single operator evolution equation in the energy space. That equation defines a semigroup of bounded operators. The dynamics generator of the semigroup is our main object of interest. For each set of boundary parameters, the dynamics generator has a compact inverse. If all four boundary parameters are not purely imaginary numbers, then the dynamics generator is a nonself-adjoint operator in the energy space. We present two main results in this research. As the first one, we calculate the spectral asymptotics of the dynamics generator. We find that the complex spectrum lies in a strip parallel to the real axis, and is asymptotically close to the axis. The latter fact means that the system is stable, but is not uniformly stable. As the second main result, we prove that the set of the root vectors of the dynamics generator forms a Riesz basis in the energy space. The results obtained in the present paper allow us to conclude that the dynamics generator is a Riesz spectral operator in the sense of N. Dunford. [ABSTRACT FROM AUTHOR]

We extend the classical concept of transformation operators to the one‐dimensional wave equation with spatially nonhomogeneous coefficients containing the first order damping term. The equation governs the vibrations of a damped string. Our results hold in the cases of an infinite, semi‐infinite or a finite string. Transformation operators were introduced in the fifties by I.M. Gelfand, B.M. Levitan and V.A. Marchenko in connection with the inverse scattering problem for the one‐dimensional Schrödinger equation. In the classical case, the transformation operator maps the exponential function (stationary wave function of a free particle) into the so‐called Jost solution of the perturbed Schrödinger equation. In our case, it is natural to introduce two transformation operators, which we call outgoing and incoming transformation operators respectively. (The terminology is motivated by an analog with the Lax–Phillips scattering theory.) The first of them is related to the nonselfadjoint quadratic operator pencil generated by the original problem, and the second one is related to the adjoint pencil. We introduce a pair of asymptotically exponential solutions for each of the pencils and show that our transformation operators map certain exponential type functions to these solutions. Our main results are the proof of the existence of transformation operators (which have the forms of the identity operator plus certain Volterra integral operators) and estimates for their kernels. To obtain these results, we derive a pair of integral equations for the kernels of the transformation operators. These equations are the generalizations of the corresponding classical equation which is valid in the case of a wave equation without damping term. One of possible applications of the method developed in this paper is given in our forthcoming work. In that work, we use the transformation operators to prove the fact that the dynamics generator of a finite string with damping both in the equation and in the boundary conditions is a Riesz spectral operator. The latter result provides a class of nontrivial examples of nonselfadjoint operators which admit an analog of the spectral decomposition. The result also has significant applications in the control theory of distributed parameter systems. [ABSTRACT FROM AUTHOR]

We consider an infinite sequence of radial wave equations obtained by the separation of variables in the spherical coordiantes from the 3‐dimensional damped wave equation with spacially nonhomogeneous spherically symmetric coefficients. The nonconservative boundary conditions are given on the sphere |x|=a. Our main objects of interest are the nonselfadjoint operators in the energy space of 2‐component initial data, which are the dynamics generators for the systems governed by the aforementioned equations and boundary conditions. Our main results are precise asymptotic formulas for the complex eigenvalues and eigenfunctions of these operators and the corresponding nonselfadjoint quadratic operator pencils. Based on the asymptotic results of the present work, we will show in a forthcoming paper that the sets of root vectors of the above operators and of the dynamics generator corresponding to the full 3‐dimensional damped wave equation form Riesz bases in the appropriate energy spaces. Therefore, we will show that all these operators are spectral in the sense of Dunford. [ABSTRACT FROM AUTHOR]