Boundary control in critical time by elastic forces applied at two ends of an inhomogeneous rod for the case in which the wave propagation time over each of the inhomogeneity parts is the same

We study boundary control in critical time by elastic forces at two ends of an inhomogeneous rod consisting of two parts of distinct densities and elasticities for the case in which the wave propagation time over each of these parts is the same. We present a closed-form expression for the boundary control by elastic forces bringing the originally quiescent stick into a given terminal state specified by a given terminal displacement and a given terminal velocity in a given critical time.     

Boundary control by displacements at one endpoint of an inhomogeneous rod for fixed other endpoint in the case of coinciding times of passage of the wave through either of the inhomogeneity parts

In the present paper, we consider a boundary control by displacements at one endpoint of an inhomogeneous rod that has two parts of different densities and elasticities with fixed other endpoint for the case in which the times of passage of the wave through each of these inhomogeneity parts coincide. We find a closed analytic form of the boundary control by displacements bringing the stick from the original rest state into a given terminal state specified by given terminal displacement and velocity.     

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.     

Boundary control by an elastic force at one end of an inhomogeneous rod with the other end free for the case in which the wave travels either of the homogeneous parts in the same time

We study the boundary control by an elastic force at one end of an inhomogeneous rod that has two parts of different densities and elasticities and whose other end is free. The case in which the wave travels either of the homogeneous parts in the same time is considered. We present a closed-form analytical expression for the boundary control by an elastic force that brings the rod from the initial quiescent state to a given terminal state specified by given terminal displacement and terminal velocity.     

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

Doklady Mathematics -     

On the complete damping of oscillations of an inhomogeneous rod by means of a boundary control at one end

The process of damping oscillations of a rod consisting of two dissimilar segments and fixed at one end is considered. The damping is carried out by means of a boundary control at the other end of the rod.     

Optimization of the control by elastic boundary forces at two ends of a string in an arbitrarily large time interval

We further develop the method, devised earlier by the authors, which permits finding closed-form expressions for the optimal controls by elastic boundary forces applied at two ends, x = 0 and x = l, of a string. In a sufficiently large time T, the controls should take the string vibration process, described by a generalized solution u(x, t) of the wave equation

$$u_{tt} (x,t) - u_{tt} (x,t) = 0,$$

from an arbitrary initial state

$$\{ u(x,0) = \varphi (x), u_t (x,0) = \psi (x)$$

to an arbitrary terminal state

$$\{ u(x,T) = \hat \varphi (x), u_t (x,T) = \hat \psi (x).$$

     

Singularities of wave fronts at the boundary between two media

Moscow Aviation Institute. Translated from Funktsional'nyi Analiz i Ego Prilozheniya, Vol. 28, No. 2, pp. 63–66, April–June, 1994.     

Optimal control of longitudinal vibrations of composite rods with the same wave propagation time in each part

We consider longitudinal elastic vibrations of a composite rod and find closedform expressions that describe optimal boundary controls bringing the rod from the quiescent state into a state with given displacement function φ(t) and velocity function ψ(t) in time T. We assume that the wave propagation time through each part of the rod is the same and T is a multiple of that time.     

Methods of boundary control theory in the nonstationary inverse problem for an inhomogeneous string

An approach to inverse problems based on the boundary control theory is developed. The dynamic problem to recover a density of an inhomogeneous string via its free endopoint oscillations generated by an instantaneous force source is proposed. The problem is to determine the coefficient ρ(x)>0 in the equation ρ(x)u_tt(x, t)−u_xx(x, t)=0(x, t>0) with the conditions u|_<0=0, u_x(0, t)=δ(t) by using a known function (response) u(0, t)=r(t) (t>0). The authors propose an algorithm based upon the approach and demonstrate its numerical efficiency in the test problems including those for nonmonotone ρ(x)'s. Bibliography: 12 titles. Translated fromZapiski Nauchnykh Seminarov POMI, Vol. 186, pp. 37–49, 1990. Translated by T. N. Surkova.     

The methods of boundary control theory in the inverse spectral problem for an inhomogeneous string

The inverse spectral problem for an inhomogeneous string is solved by means of the methods of boundary control theory. A numerical algorithm is described and computations of the density of the string are presented for a number of model examples.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova, Vol. 179, pp. 14–22, 1989.     

Observation of two soliton propagation in an erbium doped inhomogeneous lossy fiber with phase modulation

We consider optical pulse propagation in an Erbium doped inhomogeneous lossy optical fiber with time dependent phase modulation, which is governed by a system of Generalized Inhomogeneous Nonlinear Schrödinger Maxwell–Bloch (GINLS–MB) equation. Multi-soliton propagation is studied analytically by means of deriving associated Lax pair and the soliton solutions are obtained using Darboux transformation. By suitably adjusting the group velocity dispersion and nonlinearity parameter, we discuss various soliton dynamics such as periodic distributed amplification, pulse compression etc. In each case, we demonstrate the influence of inhomogeneous parameter. Finally we investigate the pulse compression through nonlinear tunneling.     

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

Doklady Mathematics -     

Optimal control of a reflection boundary coefficient in an acoustic wave equation

We seek to optimally control a reflection boundary coefficient for an acoustic wave equation. The goal-quantified by an objective functional- is to drive the solution close to a target by adjusting this coefficient, which acts as a control. The problem is solved by finding the optimal control, which minimizes the objective functional. Then the optimal control is used as a an approximation for an inverse “ identification” problem.