The structure of extremals of a class of second order variational problems

Abstract

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Generalized monotonicity from global minimization in fourth-order ordinary differential equations

“…The second example that we discuss is taken from the theory of thermodynamic equilibrium states of so-called 'second-order materials' [17][18][19][20][21][22][23][24][25][26]. We introduce these models by briefly dwelling on the phenomenon of necking in polymer fibres.…”

“…Other examples of information that has been derived are boundedness in C 1 (lR) (see [24] and [20, proposition 3.1]), boundedness and equidistribution of 'local energy' L 1 [24) and uniqueness of minimizers of (11) for 'generic' functions F [25].…”

Structure of solutions of nonautonomous optimal control problems in metric spaces

“…These problems arise in engineering [1,32], in models of economic growth [2,9,12,17,21,24,25,[27][28][29]31], in infinite discrete models of solid-state physics related to dislocations in one-dimensional crystals [3,26] and in the theory of thermodynamical equilibrium for materials [13,16].…”

“…The interest in these discrete-time optimal problems stems from the study of various optimization problems which can be reduced to this framework, e. g., continuoustime control systems which are represented by ordinary differential equations whose cost integrand contains a discounting factor [12], the study of the discrete FrenkelKontorova model related to dislocations in one-dimensional crystals [3,26] and the analysis of a long slender bar of a polymeric material under tension in [13,16]. Similar optimization problems are also considered in mathematical economics [9,17,24,28,29,31].…”

Structure of approximate solutions of variational problems with extended-valued convex integrands

“…(See, for example, [7,[15][16][17] and the references mentioned therein.) For these classes of problems a turnpike is not necessarily a singleton but may instead be a nonstationary trajectory (in the discrete time nonautonomous case) or an absolutely continuous function on the interval [0, ∞) (in the continuous time nonautonomous case) or a compact subset of the space R n (in the autonomous case).…”