Optimality conditions for multidimensional control problems with polyconvex gradient restrictions
نویسنده
چکیده
The motivation for a closer investigation of Dieudonné-Rashevsky type problems (P) is two-fold. First, due to its close affinity to the basic problem of multidimensional calculus of variations, the problem (1.3) − (1.5) is well-suited as a model problem in order to ascertain how the proof of optimality conditions is influenced through the weakening of the convexity properties of the data. Since the classical proof of Pontryagin’s principle is based on an implicit convexification of the integrand as well as of the set of feasible controls, 01) an answer to this question is of conceptual interest. On the other hand, Dieudonné-Rashevsky type problems find applications in such different areas as convex geometry, 02) material sciences, 03) population dynamics 04) and mathematical image processing, 05) thus proving considerable practical importance.
منابع مشابه
Pontryagin’s principle for Dieudonné-Rashevsky type problems with polyconvex data. Revised version
with n, m > 2, Ω ⊂ R, m < p <∞ and a compact set A ⊂ R with nonempty interior. In the case of a convex integrand f(s, ξ, · ) and a convex restriction set A = K, the global minimizers of (1.1) − (1.3) satisfy optimality conditions in the form of Pontryagin’s principle 01) even though the usual regularity condition for the equality operator (1.2) fails. 02) The question arises whether the Pontrya...
متن کاملPontryagin’s principle for Dieudonné-Rashevsky type problems with polyconvex integrands
with n, m > 2, Ω ⊂ R, m < p < ∞ and a compact set K ⊂ R with nonempty interior. In the case of a convex integrand f(s, ξ, · ) and a convex restriction set K, the global minimizers of (1.1) − (1.3) satisfy optimality conditions in the form of Pontryagin’s principle 01) even though the usual regularity condition for the equality operator (1.2) fails. 02) The question arises whether the Pontryagin...
متن کاملOn Sequential Optimality Conditions without Constraint Qualifications for Nonlinear Programming with Nonsmooth Convex Objective Functions
Sequential optimality conditions provide adequate theoretical tools to justify stopping criteria for nonlinear programming solvers. Here, nonsmooth approximate gradient projection and complementary approximate Karush-Kuhn-Tucker conditions are presented. These sequential optimality conditions are satisfied by local minimizers of optimization problems independently of the fulfillment of constrai...
متن کاملMathematik in den Naturwissenschaften Leipzig Quasiconvex relaxation of multidimensional control problems with integrands f ( t
We prove a general relaxation theorem for multidimensional control problems of Dieudonné-Rashevsky type with nonconvex integrands f(t, ξ, v) in presence of a convex control restriction. The relaxed problem, wherein the integrand f has been replaced by its lower semicontinuous quasiconvex envelope with respect to the gradient variable, possesses the same finite minimal value as the original prob...
متن کاملGeometrical properties of polyconvex polytopes
Motivated by existence and semicontinuity problems in multidimensional calculus of variations, a number of generalized convexity notions for functions has been introduced since the 50s of the last century. In its hierarchical order, the most important of these notions are polyconvexity, quasiconvexity and rank-one convexity. 01) However, the necessity to employ these semiconvexity notions as we...
متن کاملذخیره در منابع من
با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید
عنوان ژورنال:
دوره شماره
صفحات -
تاریخ انتشار 2013