Structure of extremals for one-dimensional variational problems arising in continuum mechanics. (English) Zbl 0881.49001

In this paper, the structure of optimal solutions of the variational problem \[ \int^T_0 f(w(t),w'(t),w''(t))dt\to \min,\quad w\in A^T_{x,y}, \] is investigated, where \(T>0\), \((x,y)\in\mathbb{R}^2\), \[ A^T_{x,y}= \{v\in W^{2,1}([0,T]): x= (v(0),v'(0)), y=(v(T),v'(T))\}, \] \(W^{2,1}[(0,T)]\subset C^1\) is the Sobolev space of functions having second derivative and \(f\) belongs to a space of certain properties of functions. The main results of the paper are the proof of the existence of good configurations and the turnpike property of the integrand \(f\).


49J05 Existence theories for free problems in one independent variable
74S30 Other numerical methods in solid mechanics (MSC2010)
74P10 Optimization of other properties in solid mechanics
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