G. G. Magaril-Il’yaev

How best to recover a function from its inaccurately given spectrum

Consider the problem of optimal recovery of a function and its derivatives on the line from the Fourier transform of the function known approximately on a set of finite measure. We find an optimal recovery method and an optimal set on which we must measure the Fourier transform with given error.

On the Reconstruction of Convolution-Type Operators from Inaccurate Information

We address the problem of optimal reconstruction of the values of a linear operator on ℝd or ℤd from approximate values of other operators. Each operator acts as the multiplication of the Fourier transform by a certain function. As an application, we present explicit expressions for optimal methods of reconstructing the solution of the heat equation (for continuous and difference models) at a given instant of time from inaccurate measurements of this solution at other time instants.

Newton’s method, differential equations, and the lagrangian principle for necessary extremum conditions

We show how one can use a modified Newton’s method to prove existence and uniqueness theorems for solutions of differential equations and theorems on the continuous and differentiable dependence of these solutions on the initial data and parameters and to derive necessary conditions for an extremum in various extremum problems (from the origins to our days).

An implicit-function theorem for inclusions

We consider the question of the solvability of an inclusion F(x, σ) ∈ A, i.e., of determining a mapping (implicit function) σ ↦ x(σ) defined on a set such that F(x(σ), σ) ∈ A for any σ from this set. Results of this kind play a key role in the different branches of analysis and, especially, in the theory of extremal problems, where they are the main tool for deriving conditions for an extremum.

An Implicit Function Theorem for Inclusions Defined by Close Mappings

The paper deals with the question of the solvability of inclusions F(x, σ) ∈ Q for mappings F close, in some metrics, to a given mapping ̂F.

Exactness and optimality of methods for recovering functions from their spectrum

Optimal methods are constructed for recovering functions and their derivatives in a Sobolev class of functions on the line from exactly or approximately defined Fourier transforms of these functions on an arbitrary measurable set. The methods are exact on certain subspaces of entire functions. Optimal recovery methods are also constructed for wider function classes obtained as the sum of the original Sobolev class and a subspace of entire functions.

Pontryagin maximum principle, relaxation, and controllability

The relations between the necessary minimum conditions in an optimal control problem (Pontryagin maximum principle), the minimum conditions in the corresponding relaxation (weakened) problem, and sufficient conditions for the local controllability of the controlled system specifying the constraints in the original formulation are studied. An abstract optimization problem that models the basic properties of the optimal control problem is considered.

The Pontryagin maximum principle. Ab ovo usque ad mala

A proof of the Pontryagin maximum principle for a sufficiently general optimal control problem is presented; the proof is based on the implicit function theorem and the theorem on the solvability of a finite-dimensional system of nonlinear equations. The exposition is self-contained: all necessary preliminary facts are proved. These facts are mainly related to the properties of solutions to differential equations with discontinuous right-hand side and are derived as corollaries to the implicit function theorem, which, in turn, is a direct consequence of Newton’s method for solving nonlinear equations.