A. B. Vasil’eva

Singularly perturbed problems with boundary and internal layers

This paper is an expanded version of the talk given by the authors at the International Conference “Differential Equations and Topology” dedicated to the centenary of the birth of L.S. Pontryagin. We present a brief survey and describe new ideas and methods of analysis in the asymptotic theory of solutions with internal layers, which is one of the topical fields of singular perturbation theory.

Singular Perturbations and Some Optimal Control Problems

Abstract Some results on singular perturbations in optimal control problems of Soviet authors are surveyed. Asymptotic solution of the two-point singular perturbed boundary value problems, appearing in the optimal control theory with an unbounded control, is constructed by the method of boundary layer functions. The state feedback matrix of a linear system with respect to a quadratic performance index is the solution of the matrix Riccati differential equation with singular initial values. The asymtotic solution of this initial value problem has certain peculiarities. Some statements regarding a strong (weak) convergence of solutions of perturbed problems to corresponding solutions of reduced-order problems are established for certain linear and nonlineer optimal control problems with control constraints. Analogous results take place for optimal control problems with differential equations with a small time delay and for discrete optimal control problems with a small step.

Singularly perturbed boundary value problems with a power-law boundary layer

We consider boundary value problems for second-order singularly perturbed equations whose solution has a power-law boundary layer that occurs because the degenerate equation has multiple roots.

Boundary layers in the solution of singularly perturbed boundary value problem with a degenerate equation having roots of multiplicity two

The occurrence of exponential boundary layers in a second-order ordinary differential equation due to the fact that the degenerate equation has a root of multiplicity two is briefly reviewed.

Alternating boundary layer type solutions of some singularly perturbed periodic parabolic equations with Dirichlet and Robin boundary conditions

In this paper, we continue the analysis of alternating boundary layer type solutions to certain singularly perturbed parabolic equations for which the degenerate equations (obtained by setting small parameter multiplying derivatives equal to zero) are algebraic equations that have three roots. Here, we consider spatially one-dimensional equations. We address special cases where the following are true: (a) boundary conditions are of the Dirichlet type with different values of unknown functions specified at different endpoints of the interval of interest; (b) boundary conditions are of the Robin type with an appropriate power of a small parameter multiplying the derivative in the conditions. We emphasize a number of new features of alternating boundary layer type solutions that appear in these cases. One of the important applications of such equations is related to modeling certain types of bioswitches. Special choices of Dirichlet and Robin type boundary conditions can be used to tune up such bioswitches.

Two-point boundary value problem for a singularly perturbed equation with a reduced equation having multiple roots

Contrast steplike structures for a singularly perturbed equation are considered in the case when the corresponding reduced equation has multiple roots.

On parabolic equations with a small parameter

A singularly perturbed parabolic equation with a nonlinear right-hand side of a special form is examined. A numerical analytical study of such equations is performed.

On certain alternating boundary-layer solutions to singularly perturbed parabolic equations

The solution to a special singularly perturbed parabolic problem with a right-hand independent of the spatial variable is studied numerically and analytically.

Solutions to a singularly perturbed parabolic equation with internal and boundary layers depending on stretched variables of different orders

The solution to a singularly perturbed parabolic equation with internal and boundary layers whose stretched variables may depend on different powers of the perturbation parameter is considered. An asymptotic representation of the solution is constructed and substantiated, and its stability is proved.

On a system of singularly perturbed second-order quasilinear ordinary differential equations in the critical cases

A singularly perturbed system of second-order quasilinear ordinary differential equations with a small parameter multiplying the second derivatives is examined in the case where the coefficient matrix of the first derivatives is singular and does not depend on the unknown functions.