On the vibrations described by the telegraph equation in the case of a system consisting of several parts of different densities and elasticities

We consider mixed initial-boundary value problems for longitudinal vibrations described by the telegraph equation in the case of a system consisting of several parts with different densities and elasticities but with equal impedances. We consider the cases of control by displacements at both endpoints of the rod, by elastic forces at both endpoints, and by an elastic force at one endpoint and a displacement at the other endpoint. We find closed-form expressions for the solutions of these mixed problems.     

Mixed problems for the equation of longitudinal vibrations of a heterogeneous rod and for the equation of transverse vibrations of a heterogeneous string consisting of two segments with different densities and elasticities

Doklady Mathematics -     

Mixed problems for the equation of longitudinal vibrations of a heterogeneous rod with a free or fixed right end consisting of two segments with different densities and elasticities

Doklady Mathematics -     

Optimization of the boundary displacement control of vibrations of a rod consisting of two dissimilar parts

We optimize the boundary displacement control that is applied at one end of a rod consisting of two dissimilar parts and brings the rod vibrations from a given initial state to a given terminal state for the case in which the other end of the rod is fixed.     

Optimization of elasticity-induced boundary control for the vibrations of a rod consisting of two different segments

Doklady Mathematics -     

Discrete transparent boundary conditions for the equation of rod transverse vibrations

Local perturbations of an infinitely long rod travel to infinity. On the contrary, in the case of a finite length of the rod, the perturbations reach its boundary and are reflected. The boundary conditions constructed here for the implicit difference scheme imitate the Cauchy problem and provide almost no reflection. These boundary conditions are non-local with respect to time, and their practical implementation requires additional calculations at every time step. To minimise them, a special rational approximation, similar to the Hermite - Padé approximation is used. Numerical experiments confirm the high “transparency” of these boundary conditions and determine the conditional stability regions for finite-difference scheme.     

A terminal-boundary value problem that describes the process of damping the vibrations of a rod consisting of two segments with different densities and elasticity coefficients but with identical wave travel times

An explicit analytic expression is obtained for optimal boundary controls exercised on one end of a string by a displacement or by an elastic force under a model nonlocal boundary condition of one of four types.     

Numerical solution of the optimal boundary control of transverse vibrations of a beam

Boundary control is an effective means for suppressing excessive structural vibrations. By introducing a quadratic index of performance in terms of displacement and velocity, as well as the control force, and an adjoint problem, it is possible to determine the optimal control. This optimal control is expressed in terms of the adjoint variable by utilizing a maximum principle. With the optimal control applied, the determination of the corresponding displacement and velocity is reduced to solving a set of partial differential equations involving the state variable, as well as the adjoint variable, subject to boundary, initial, and terminal conditions. The set of equations may not be separable and analytical solutions may only be found in special cases. Furthermore, the computational effort to determine an analytic solution may also be excessive. Herein a numerical algorithm is presented, which easily solves the optimal boundary control problem in the space‐time domain. An example of a continuous system is analyzed. This is the case of the vibrating cantilever beam. Using a finite element recurrence scheme, numerical solutions are obtained, which compare the behavior of the controlled and uncontrolled systems. Also, the analytic solution to the problem is compared with the results obtained using the numerical scheme presented. © 1999 John Wiley & Sons, Inc. Numer Methods Partial Differential Eq 15: 558–568, 1999     

On the optimal boundary control of vibrations of a spherical layer

We seek the optimal boundary control of vibrations of a spherical layer in the spherically symmetric case. This paper continues the series of papers by V.A. Il’in and his students and, unlike the previous papers, uses a more general control optimality criterion. We obtain closed formulas for the controls; these formulas are consistent with Il’in’s earlier results.