The Euler and Weierstrass conditions for nonsmooth variational problems
The central problem in this paper is to find an absolutely continuous function \(x: [0,1 ]\to \mathbb{R}^n\) that minimizes \[ J[x]= l(x(0), x(1))+ \int^1_0 L(t, x(t), \dot x(t))dt. \] Under hypotheses of smoothness on \(l\) and \(L\) much weaker than those of the classical case, the authors show that a weak local minimizer (i.e., local in the \(W^{1,1}\) norm) must satisfy appropriate versions of the Euler-Lagrange equation and the natural boundary conditions. For strong local minimizers (i.e., local in the uniform norm), the Weierstrass condition is incorporated as well. The central issues are (i) the correct extension of the Euler-Lagrange equation, (ii) the complete absence of the convexity hypothesis \[ \text{the function } v\mapsto L(t, x, v) \text{ is convex for each } t,x, \tag \(*\) \] and (iii) the successful treatment of weak local minimizers. The sharpened Euler-Lagrange inclusion is the one introduced in \textit{P. D. Loewen} and \textit{R. T. Rockafellar} [SIAM J. Control Optimization 32, No. 2, 442-470 (1994; Zbl 0823.49016)], where only strong local minimizers were studied and condition \((*)\) was central. See also \textit{B. S. Mordukhovich} [SIAM J. Control Optimization 33, No. 3, 882-915 (1995)], where the same form of Euler's equation is established without \((*)\), but for local minimizers of a different sort. The proofs rely on new techniques in nonsmooth analysis, which are of independent interest: fuzzy calculus of proximal subgradients and a close analysis of subgradients under convexification. The advances reported here are in some respects complementary to those cited above. Although the treatment of weak local minimizers without convexity conditions is new, the results at hand do not constitute a strict generalization of the work mentioned earlier. The hypotheses on the integrand \(L\), while mild in comparison with their classical antecedents, include a local finiteness condition that eliminates from consideration any problem having only abnormal extremals. Thus, problems where differential constraints are enforced through infinite penalties are beyond the reach of the main theorem.
scientific article; zbMATH DE number 826771
scientific article