相关论文: On sufficient conditions in the classical problem …
In the paper, we offer a method for studying an extremal in the classical calculus of variation in the presence of various degenerations. This method is based on introduction of Weierstrass type variations characterized by a numerical…
This article explores minimum of an extremal in the variational problem with delay under the degeneracy of the Weierstrass condition. Here for study the minimality of extremal, variations of the Weierstrass type are used in two forms: in…
Some classic second-order sufficient optimality conditions in the calculus of variations are shown to be equivalent, while also introducing a new equivalent second-order condition which is extremely easy to apply: simply integrate a linear…
We introduce the notion of strong local minimizer for the problems of the calculus of variations on time scales. Simple examples show that on a time scale a weak minimum is not necessarily a strong minimum. A time scale form of the…
This paper addresses both necessary and relevant sufficient extremum conditions for a variational problem defined by a smooth Lagrangian, involving higher derivatives of several variable vector valued functions. A general formulation of…
We introduce a discrete-time fractional calculus of variations. First and second order necessary optimality conditions are established. Examples illustrating the use of the new Euler-Lagrange and Legendre type conditions are given. They…
In this paper, we study problems of minimization of a functional depending on the fractional Caputo derivative of order $0<\alpha \leq 1$ and the fractional Riemann- Liouville integral of order $\beta > 0$ at fixed endpoints. A fractional…
We give a proper fractional extension of the classical calculus of variations. Necessary optimality conditions of Euler-Lagrange type for variational problems containing both classical and fractional derivatives are proved. The fundamental…
This paper develops a novel approach to necessary optimality conditions for constrained variational problems defined in generally incomplete subspaces of absolutely continuous functions. Our approach involves reducing a variational problem…
We give a necessary and sufficient condition for the existence of a local solution of the inverse problem of calculus of variations in terms of the identical vanishing of the variation of a functional on an extended space (with the number…
We study more general variational problems on time scales. Previous results are generalized by proving necessary optimality conditions for (i) variational problems involving delta derivatives of more than the first order, and (ii) problems…
In this paper the necessary conditions of optimality in the form of maximum principle are derived for a very general class of variational problems. This class includes problems with any optimization criteria and constraints that can be…
For an optimal control problem, the concept of a strong local infimum is introduce, for which necessary conditions consisting of some family of "maximum principles" are formulated. If a function delivers a strong local minimum in this…
Summary]{In this paper, we study problems of minimization of a functional depending on the fractional Caputo derivative of order $0<\alpha \leq 1$ and the fractional Riemann- Liouville integral of order $\beta > 0$ at fixed endpoints. A…
In this paper we discuss the existence of solutions to vectorial differential inclusions. We investigate sufficient conditions for existence, more flexible than those available in the literature, so that important applications can be fitted…
This paper provides necessary and sufficient conditions of optimality for variational problems that deal with a fractional derivative with respect to another function. Fractional Euler--Lagrange equations are established for the fundamental…
Invariant conditions for conformable fractional problems of the calculus of variations under the presence of external forces in the dynamics are studied. Depending on the type of transformations considered, different necessary conditions of…
Fractional operators play an important role in modeling nonlocal phenomena and problems involving coarse-grained and fractal spaces. The fractional calculus of variations with functionals depending on derivatives and/or integrals of…
The present paper is a continuation of the author's previous works, in which necessary and sufficient local extrema at a stationary point of a polynomial or a power series (and thus of an analytic function) are given. It is known that for…
We systematically introduce an approach to the analysis and (numerical) solution of a broad class of nonlinear unconstrained optimal control problems, involving ordinary and distributed systems. Our approach relies on exact representations…