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

Doklady Mathematics -     

Optimization scheme for boundary control by displacements at two ends of oscillations of a rod consisting of two dissimilar segments

Doklady Mathematics -     

Analytic form of optimal boundary controls of transverse vibrations of a rod consisting of several parts with different densities and elasticities but with the same impedances

We consider the problem of optimal boundary displacement control (control minimizing the boundary energy functional) of an elastic rod consisting of several parts with different densities and elasticities but with the same impedances. We obtain a closed-form expression for the optimal control bringing the rod from an arbitrarily given initial state to an arbitrarily given terminal state in a given time T.     

Critical-time boundary displacement control at two ends of an inhomogeneous rod in the case of coincidence of the wave propagation times through each of two homogeneous parts

We study a boundary displacement control at two ends of an inhomogeneous rod that has two parts of distinct densities and elasticities in the case of coinciding wave propagation times over these parts. The control acts on a time interval of critical length. We obtain a closed analytical form of the boundary displacement control bringing the rod in critical time from the initial state of rest into a given terminal state specified by given terminal displacement and terminal velocity.     

Optimal boundary displacement control of string vibrations with a nonlocal evenness condition of the second kind

We study the behavior of a string with the nonlocal boundary condition u _x (l, t) = u _x ($ x^\circ $ x^\circ , t). A displacement control u(0, t) = μ(t) bringing the string from an arbitrarily given initial state to an arbitrarily given terminal state is applied at the left endpoint of the string. For the initial and terminal functions, we find necessary and sufficient conditions for the controllability of the string. Under these conditions, we carry out optimization; i.e., of all admissible controls, we choose a control minimizing the boundary energy integral.     

Optimal boundary displacement control of string vibrations with a nonlocal oddness condition of the first kind

We consider the problem of optimal boundary control by the displacement at left endpoint of a string in the case of a nonlocal oddness boundary condition of the first kind. We obtain a necessary and sufficient condition for the problem controllability under arbitrary initial and terminal conditions and construct a closed analytical form of the control itself under these conditions. In addition, we consider the problem of optimal boundary control by the displacement at one endpoint of the string for a given displacement mode at the other endpoint.     

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.     

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 -     

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.     

Flexural vibrations of a moving rod

The problem of the transverse natural vibrations of part of a rod between two coaxially fixed guides, moving with an arbitrary constant velocity, is investigated. The conditions of rigid clamping are taken as the boundary conditions. Additional shear stresses, due to longitudinal tension or compression are taken into account. Relations defining the natural frequencies and forms are constructed in an exact formulation by Fourier method. The dependence of the natural frequencies and forms of the lowest vibration modes on the rate of displacement, unknown in the literature, are constructed, and their features are established. A modelling and animation of unusual wave motions of the rod are presented. The main characteristics for the higher vibration modes are constructed.     

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 -     

Optimal boundary control of forced vibrations by the displacement at one end of the string with the other end fixed

For a string vibration process described by an inhomogeneous wave equation, we consider the problem of boundary control at one end of the string with the other end being fixed. For any time interval T > 2l, where l is the string length, we find a function u(0, t) = µ(t) bringing the vibration system from a given initial state into a given terminal state and minimizing the boundary energy integral.     

Optimization of the boundary control by a displacement or by an elastic force on one end of a string under a model nonlocal boundary condition

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.     

Exact control of vibrations of a rectangular plate by boundary forces

The problem of control of vibrations of a right-angled plate is considered with restrictions posed on the absolute value of the control forces. The purpose of control is to bring the plate into the rest state in finite time.