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 $$f$$.

MSC:

 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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