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This paper deals with the numerical solution of optimal control problems for ODEs. The methods considered here rely on some standard optimization code to solve a discretized version of the control problem under consideration. We aim to make available to the optimization software not only the discrete objective functional, but also its gradient. The objective gradient can be computed either from forward (sensitivity) information or backward (adjoint) information. The purpose of this paper is to discuss various ways of adjoint computation. It will be shown both theoretically and numerically that methods based on the continuous adjoint equation require a careful choice of both the integrator and gradient assembly formulas in order to obtain a gradient consistent with the discretized control problem. Particular attention is given to automatic differentiation techniques which generate automatically a suitable integrator. [ABSTRACT FROM AUTHOR]

For the problem of minimizing the sum of Euclidean norms (MSN), most existing quadratically convergent algorithms require a strict complementarity assumption. However, this assumption is not satisfied for a number of MSN problems. In this paper, we present a globally and quadratically convergent algorithm for the MSN problem. In particular, the quadratic convergence result is obtained without assuming strict complementarity. Examples without strictly complementary solutions are given to show that our algorithm can indeed achieve quadratic convergence. Preliminary numerical results are reported. [ABSTRACT FROM AUTHOR]

The main purpose of this paper is to introduce and study generalized hyperconnected spaces. Various characterizations of generalized hyperconnected spaces and preservation theorems are discussed. [ABSTRACT FROM AUTHOR]

Numerical representations of choice functions allow the expression of a problem of choice as a problem of finding maxima of real-valued functions, which requires that less information be defined and which is easier to work with. In this paper, the existence of numerical representations of choice functions by imposing assumptions directly on the choice function is obtained. In particular, it is proved that (IIA) is a necessary and sufficient condition for a choice function to be representable, if the set of alternatives is countable, while the conjunction of (IIA) and a ‘continuity condition’ is sufficient to ensure it in separable topological spaces. [ABSTRACT FROM AUTHOR]

We establish necessary and sufficient conditions for the Fredholm property of a planar problem with shift and conjugation for a pair of functions and obtain a formula for the determination of its index. [ABSTRACT FROM AUTHOR]

We solve the Landau-Kolmogorov problem for a class of functions absolutely monotone on a finite interval. For this class of functions, new exact additive inequalities of the Kolmogorov type are obtained. [ABSTRACT FROM AUTHOR]

The article presents a discussion that focuses on the Kantorovich and the Durrmeyer-type modifications of the generalized favard operators. The discussion is also said to be aimed at proving an inverse approximation theorem for functions f. An analogous inequality for the rth weighted modulus of smoothness of the function f is also presented. The theorems, equations, and calculations associated with such mathematical concept are also included in the discussion. Mathematical proofs linked to such equations are also provided.

A new upper and lower solution theory is presented for the second order problem (G'(y))' + f(t, y) = 0 on finite and infinite intervals. The theory on finite intervals is based on a Leray-Schauder alternative, whereas the theory on infinite intervals is based on results on the finite interval and a diagonalization process. [ABSTRACT FROM AUTHOR]

We obtain a uniform and sufficient condition for the convergence of the Lagrange interpolation process with Jacobi nodes on a closed interval [ a, b] ⊂ (−1, 1). The condition is stated in terms of the second differences of the interpolated function and uses its values only at the interpolation nodes. Some well-known criteria for uniform convergence are obtained as a consequence of our result. [ABSTRACT FROM AUTHOR]

We consider a continuous function that changes its sign on an interval finitely many times and pose the problem of the approximation of this function by a polynomial that inherits its sign. For this approximation, we obtain (in the case where this is possible) Jackson-type estimates containing modified weighted moduli of smoothness of the Ditzian-Totik type. In some cases, constants in these estimates depend substantially on the location of points where the function changes its sign. We give examples of functions for which these constants are unimprovable. We also prove theorems that are analogous, in a certain sense, to inverse theorems of approximation without restrictions. [ABSTRACT FROM AUTHOR]

The global convergence of locally convergent and stable (quasi-stable) implicit iterations xn = T(xn, xn-1) in connected (pathwise connected) metric spaces are established. A sufficient condition for the local convergence of implicit iteration processes is also given. [ABSTRACT FROM AUTHOR]

Sufficient conditions are derived for the controllability of nonlinear neutral evolution integrodifferential systems with infinite delay in a Banach space. The results are obtained by using the resolvent operators and the Schaefer fixed-point theorem. An example is given to illustrate the results. [ABSTRACT FROM AUTHOR]

For $N=\; 2^s$, in the space of complex-valued $N$-periodic functions of an integer argument, we construct $s$ orthogonal bases such that the elements of the $r$th basis, where $r\backslash in\backslash \; 1,\backslash dots,s\backslash$, take $2^r$ values of uniformly distributed on the unit circle in the complex plane. [ABSTRACT FROM AUTHOR]