Singular quadratic variational problems

The purpose of this paper is to show that the general theory of quadratic forms developed earlier by the author is applicable to singular variational problems as well as to nonsingular ones. In particular, this general theory is applicable to the singular variational problems associated with Legendre polynomials, associated Legendre polynomials, Jacobi polynomials, and Tchebysheff polynomials.     

Singular perturbations of elliptic problems

Summary The paper treates applications of singular perturbations of variational inequalities, to differential problems. Some informations on the boundary layer phenomenon are obtained. Entrata in Redazione il 20 ottobre 1971.     

Singular perturbations for variational inequalities with time-dependent constraints

We study an abstract hyperbolic variational inequality with a (small) parameter and with time-dependent subdifferentials in a real Hilbert space. We prove that its solution converges to a solution of a parabolic variational inequality as the parameter tends to zero. We also explain how to apply the abstract theory to concrete unilateral problems.     

Singular perturbations of some nonlinear problems

We deal with singular perturbations of nonlinear problems depending on a small parameter ε > 0. First we consider the abstract theory of singular perturbations of variational inequalities involving some nonlinear operators, defined in Banach spaces, and describe the asymptotic behavior of these solutions as ε → 0. Then these abstract results are applied to some boundary value problems. Bibliography: 15 titles.     

Singular perturbations for a class of nonlinear hyperbolic-hyperbolic problems

We show how, for nonlinear equations satisfying certain symmetry conditions, the determining equations reduce for solutions taken in some invariant subspace.     

Singular perturbations in a class of problems of optimal control with integral convex criterion

The problem of optimal control is investigated with a linear law of motion and convex quality criterion. A small positive parameter appears in front of the derivatives of some of the unknowns in the law of motion. The behaviour of the optimal solution is studied when the small parameter approaches zero with some assumptions that are different from thos encountered in the literature.     

On the phase transition temperature in a variational problem of elasticity theory for two-phase media

We obtain sufficient conditions for the noncoincidence of the phase transition temperatures and illustrate this result by examples of problems in two-phase elastic media. We also indicate some cases where equilibrium states exist or not depending on the values of the temperature lying between the lower and upper phase transition temperatures. Bibliography: 10 titles. Dedicated to Nina Nikolaevna Uraltseva Translated from Problemy Matematicheskogo Analiza, 41, May 2009, pp. 37–48.     

On variational perturbations of control problems: Minimum-time problem and minimum-effort problem

In order to obtain numerical solutions for an abstract optimal control problem, one approximates the abstract operations in a computationally feasible manner. After having found an approximate optimal solution, the question is whether a sequence of these approximate optimal solutions converges to an optimal solution of the original problem. In this work, we are concerned with this type of convergence on the time-optimal control problem for a class of linear systems with distributed parameters and on the minimum-effort problem.     

Finite rate of propagation of perturbations in multidimensional problems of two-phase filtration

One proves that the solutions of the system of equations of the two-phase filtration in the multidimensional case describe a process which propagates with a finite velocity. This statement is proved also for a certain class of degenerate second-order parabolic equations.     

A new class of variational problems arising in the modeling of elastic multi-structures

Summary It is shown howelastic multi-structures that comprise substructures of possibly different dimensions (three-dimensional structures, plates, rods) are modeled bycoupled, pluri-dimensional, variational problems of a new type. Following recent work by the author, H. LeDret, and R. Nzengwa, we describe here in detail one such problem, which is simultaneously posed over a threedimensional open set with a slit and a two-dimensional open set. The numerical analysis of such problems is also discussed and finally, some numerical results are presented.Dedicated to R. S. Varga on the occasion of his sixtieth birthdayInvited lecture,Conference on Approximation Theory and Numerical Linear Algebra, in honor of Richard S. Varga on the occasion of his 60th birthday, March 30–April 1, 1989, Kent State University, Kent, USALaboratoire du Centre National de la Recherche Scientifique associé à l'Université Pierre et Marie Curie     

The limiting behavior of the value-function for variational problems arising in continuum mechanics

In this paper we study the limiting behavior of the value-function for one-dimensional second order variational problems arising in continuum mechanics. The study of this behavior is based on the relation between variational problems on bounded large intervals and a limiting problem on [0,∞)$[0,\infty )$.     

The Levitin-Polyak well-posedness by perturbations for systems of general variational inclusion and disclusion problems

In this paper, the notions of the Levitin-Polyak well-posedness by perturbations for system of general variational inclusion and disclusion problems (shortly, (SGVI) and (SGVDI)) are introduced in Hausdorff topological vector spaces. Some sufficient and necessary conditions of the Levitin-Polyak well-posedness by perturbations for (SGVI) (resp., (SGVDI)) are derived under some suitable conditions. We also explore some relations among the Levitin-Polyak well-posedness by perturbations, the existence and uniqueness of solution of (SGVI) and (SGVDI), respectively. Finally, the lower (upper) semicontinuity of the approximate solution mappings of (SGVI) and (SGVDI) are established via the Levitin-Polyak well-posedness by perturbations.     

Singular perturbations of abstract wave equations

Given, on the Hilbert space H₀, the self-adjoint operator B and the skew-adjoint operators C₁ and C₂, we consider, on the Hilbert space H?D(B)⊕H₀, the skew-adjoint operator