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One-dimensional infinite-horizon variational problems arising in continuum mechanics. (English) Zbl 0672.73010

The authors study a variational problem for real valued functions defined on an infinite semiaxis of the line. They seek a “minimal solution” to the following problem: minimize the functional

(1)I(w(·))= 0 f(w(s),w ˙(s),w ¨(s))ds,
(2)wA x ={vW loc 2,1 (0,):(v(0),v ˙(0))=x ̲}·

Here W loc 2,1 C 1 denotes the Sobolev space of functions possessing a locally integrable second derivative, and f=f(w,p,r) is a smooth function satisfying f rr 0, f(w,p,r)a|w| α -b|p| β +c|r| γ -d; a,b,c,d>0; α,γ(1,), β[1,), α>β, γ>β and x ̲ is a given value. The organization of the paper is as follows: In Section 1 the connection of the variational problem (1),(2) with the equilibrium problem of a long slender bar of polymetric material under tension is discussed. In Section 2 a fixed endpoint variational problem is analyzed. In Section 3 a criterion is given for the solution of (1),(2) to be minimal. Section 4 gives the proof the existence of a minimal energy solution. The Section 5 is containing the main result of the paper: there always exists a periodic minimal solution of (1),(2). In Section 7 an interesting analytic result is established.

Reviewer: I.Ecsedi

MSC:
74S30Other numerical methods in solid mechanics
74A99Generalities, axiomatics, foundations of continuum mechanics of solids
49J27Optimal control problems in abstract spaces (existence)
49K27Optimal control problems in abstract spaces (optimality conditions)