Robert E. O'Malley

Singular perturbations and singular arcs. I

Singular perturbation theory is applied to obtain the asymptotic solution for the nearly singular optimal control of a constant linear system on a finite time interval. In the limit as the control cost is reduced to zero, the initial control is found to have an impulse-like behavior, while the outer solution agrees asymptotically with the familiar solution for a singular arc. The detailed structure of the impulse is provided by the asymptotic solution. (auth)

Cheap Control of the Time-Invariant Regulator

The asymptotic solution of the linear quadratic state regulator problem is obtained as the cost of the control tends to zero. Matrix Riccati gains are obtained via singular perturbations theory and are used to asymptotically calculate the optimal control and the corresponding trajectories. Several cases are distinguished and applications are discussed.

The numerical solution of boundary value problems for stiff differential equations

Abstract : The numerical solution of boundary value problems for certain stiff ordinary differential equations is studied. The methods developed use singular perturbation theory to construct approximate numerical solutions which are valid asymptotically; hence, they have the desirable feature of becoming more accurate as the equations become stiffer. Several numerical examples are presented which demonstrate the effectiveness of these methods.

Numerical Methods for Stiff Systems of Two-Point Boundary Value Problems

We develop numerical procedures for constructing asymptotic solutions of certain nonlinear singularly perturbed vector two-point boundary value problems having boundary layers at one or both endpoints. The asymptotic approximations are generated numerically and can either be used as is or to furnish a general purpose two-point boundary value code with an initial approximation and the nonuniform computational mesh needed for such problems. The procedures are applied to a model problem that has multiple solutions and to problems describing the deformation of a thin nonlinear elastic beam that is resting on an elastic foundation.

On Multiple Solutions of Singularly Perturbed Systems in the Conditionally Stable Case

Publisher Summary This chapter discusses the multiple solutions of singularly perturbed systems in the conditionally stable case. Problems concerning singularly perturbed systems can be considerably more complicated when eigenvalues of g cross or approach the imaginary axis. The strict eigenvalue stability assumptions can be weakened in boundary layer regions. The two-point problems arise naturally in optimal control theory, among many other applications. Moreover, the asymptotic behavior of solutions is extremely helpful in developing schemes for the numerical solution of stiff boundary value problems. With the assumed hyperbolic splitting, solutions to feature nonuniform convergence as e 0, that is, boundary layers near both endpoints must be expected. The limiting boundary conditions for initial values can be solved with two boundary layer correction terms, which are represented by u0 and v0. However, the solutions must lie on the stable manifolds I and T for the corresponding boundary layer systems. This will certainly be impossible when the limiting boundary conditions A(x(0),y(0),0 ) or B(x(1),y(1),0 ) are independent of y.