Optimal Control of Longitudinal Vibrations of Composite Rods with the Same Wave Propagation Time in Each Partстатья
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Дата последнего поиска статьи во внешних источниках: 25 сентября 2013 г.
Аннотация:Abstract — We consider longitudinal elastic vibrations of a composite rod and find closed form 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.
DOI: 10.1134/S0012266113050091
The boundary control problem for elastic vibrations considered in the present paper was studied in a series of papers by V.A. Il’in and E. I. Moiseev, where optimal boundary controls for a homogeneous rod were obtained with the use of boundary controls of the first and second kind for certain values of the control time T. The obtained results can be found in [1]. Time-optimal controls were obtained in [2] for a rod consisting of two parts with the same signal propagation time, and a similar result was derived in [3] without the constraint on the signal propagation time. The control problem was also solved for arbitrary sufficiently large time intervals for homogeneous rods (e.g., see [4, 5]).
Consider a rod consisting of n parts. We number the parts successively from 1 to n and denote their endpoints by xi (i = 0, . . . , n) so that the ith part lies on the interval [xi−1, xi]. The set of points xi consists of the boundary points x0 and xn and the junction points x1, x2, . . . , xn−1.
We assume that the linear density of the rod on the ith part is i, the Young modulus is ki, and the signal propagation velocity is ai = ki/i.
We subject the rod to the restrictive requirement that signal passage time is the same for each part. The time of passage through one part is denoted by s,
s = (xi − xi−1)/ai, i= 1, . . . , n.