The fractional variational calculus is a recent fifield, where classical problems are considered, but in the presence of derivatives. Since there several defifinitions derivatives, it logical to think different types optimality conditions. For this reason, order solve problems, two theorems necessary conditions well known: an Euler-Lagrange equation which involves Caputo and Riemann-Liouville o...

The paper treats functions which are defined on closed subsets of [0, 1] and which are k times Peano differentiable. A necessary and sufficient condition is given for the existence of a k times Peano differentiable extension of such a function to [0, 1]. Several applications of the result are presented. In particular, functions defined on symmetric perfect sets are studied.

In this paper, we establish a nessary condition for nonlipschitz constrained optimization problems in terms of Fréchet subdifferentials and normal cones in Asplund spaces.

Key scientific and application questions concern the relationships between individual-level activities and their effects on broader human phenomena, such as transportation systems and cities. Continuing advances in geographic information science, location-aware technologies, and geosimulation methods offer great potential for observational and simulation studies of human activities at high leve...

In this paper we employ the topological concepts of index of an equilibrium point of a vector eld and of the Conley index of an isolated invariant set of a vector eld with the purpose of deriving necessary conditions for the design of global control dynamics. Our results are a generalization of those of Brockett, Coron and others, which were derived for the special case of stabilization.

In this paper we consider parameterized Lyapunov inequalities arising in stability analysis of discrete-discrete and continuous-discrete 2D systems. A necessary condition for their feasibility is presented. This condition is based on an optimization method published earlier by the author. Its properties include being reasonably easy to check, and its coverage being adjustable via the algorithm’...

In this paper we present a method for determining some variations of a singular trajectory of an affine control system. These variations provide necessary optimality conditions which may distinguish between maximizing and minimizing problems. The generalized Legendre-Clebsch conditions are an example of these type of conditions.

Ω [F (∇v (x)) + g (x, v (x))] dx with F a convex function defined on RN , it has been conjectured in [2] that the suitable form of the Euler-Lagrange equations satisfied by a solution u should be ∃p(·) ∈ L(Ω), a selection from ∂F (∇u(·)), such that div p(·) = gv(·, u(·)) in the sense of distributions. Equivalently, the condition can be expressed as ∃p(·) ∈ L(Ω) : div p(·) = gv(·, u(·)) and, for...