We consider an optimal control problem for a system of the form = f(x,u), with a running cost L. We prove an interior sphere property for the level sets of the corresponding value function V. From such a property we obtain a semiconcavity result for V, as well as perimeter estimates for the attainable sets of a symmetric control system.
We find conditions, in multi-objective convex programming with nonsmooth functions, when the sets of efficient (Pareto) and properly efficient solutions coincide. This occurs, in particular, when all functions have locally flat surfaces (LFS). In the absence of the LFS property the two sets are generally different and the characterizations of efficient solutions assume an asymptotic form for problems with three or more variables. The results are applied to a problem in highway construction, where...
We establish a local Lipschitz regularity result for local minimizers of asymptotically convex variational integrals.
We establish a local Lipschitz regularity result for local minimizers of asymptotically convex variational integrals.
The object of this paper is to prove existence and regularity results for non-linear elliptic differential-functional equations of the form over the functions that assume given boundary values ϕ on ∂Ω. The vector field satisfies an ellipticity condition and for a fixed x, F[u](x) denotes a non-linear functional of u. In considering the same problem, Hartman and Stampacchia [Acta Math.115 (1966) 271–310] have obtained existence results in the space of uniformly Lipschitz continuous functions...
We prove a lower semicontinuity result for variational integrals associated with a given first order elliptic complex, extending, in this general setting, a well known result in the case .
We consider some metric regularity properties of order q for set-valued mappings and we establish several characterizations of these concepts in terms of Hölder-like properties of the inverses of the mappings considered. In addition, we show that even if these properties are weaker than the classical notions of regularity for set-valued maps, they allow us to solve variational inclusions under mild assumptions.
In a recent paper [Forum Math., 2008] the authors established some global, up to the boundary of a domain Ω ⊂ ℝⁿ, continuity and Morrey regularity results for almost minimizers of functionals of the form . The main assumptions for these results are that g is asymptotically convex and that it satisfies some growth conditions. In this article, we present a specialized but significant version of this general result. The primary purpose of this paper is provide several applications of this simplified...
The Young measures, used widely for relaxation of various optimization problems, can be naturally understood as certain functionals on suitable space of integrands, which allows readily various generalizations. The paper is focused on such functionals which can be attained by sequences whose “energy” (=th power) does not concentrate in the sense that it is relatively weakly compact in . Straightforward applications to coercive optimization problems are briefly outlined.
We discuss variational problems on two-dimensional domains with energy densities of linear growth and with radially symmetric data. The smoothness of generalized minimizers is established under rather weak ellipticity assumptions. Further results concern the radial symmetry of solutions as well as a precise description of their behavior near the boundary.
The research on a class of asymptotic exit-time problems with a vanishing Lagrangian, begun in [M. Motta and C. Sartori, Nonlinear Differ. Equ. Appl. Springer (2014).] for the compact control case, is extended here to the case of unbounded controls and data, including both coercive and non-coercive problems. We give sufficient conditions to have a well-posed notion of generalized control problem and obtain regularity, characterization and approximation results for the value function of the problem....