Frank Lempio

Difference methods for differential inclusions: a survey

The main objective of this survey is to study convergence properties of difference methods applied to differential inclusions. It presents, in a unified way, a number of results scattered in the li...

Differential stability in infinite-dimensional nonlinear programming

In this paper stability properties of the extremal value function are studied for infinite-dimensional nonlinear optimization problems with differentiable perturbations in the objective function and in the constraints. In particular, upper and lower bounds for the directional derivative of the extremal value function as well as necessary and sufficient conditions for the existence of the directional derivative are given.

Approximations of linear control problems with bang-bang solutions

We analyse the Euler discretization to a class of linear optimal control problems. First we show convergence of order h for the discrete approximation of the adjoint solution and the switching function, where h is the mesh size. Under the additional assumption that the optimal control has bang-bang structure we show that the discrete and the exact controls coincide except on a set of measure O(h). As a consequence, the discrete optimal control approximates the optimal control with order 1 w.r.t. the L 1-norm and with order 1/2 w.r.t. the L 2-norm. An essential assumption is that the slopes of the switching function at its zeros are bounded away from zero which is in fact an inverse stability condition for these zeros. We also discuss higher order approximation methods based on the approximation of the adjoint solution and the switching function. Several numerical examples underline the results.

Transformation of Quadratic Forms to Perfect Squares for Broken Extremals

In this paper we study a quadratic form which corresponds to an extremal with piecewise continuous control in variational problems. This form, compared with the classical one, has some new terms connected with the set Θ of all points of discontinuity of the control. Its positive definiteness is a sufficient optimality condition for broken extremals. We show that if there exists a solution to corresponding Riccati equation satisfying some jump condition at each point of the set Θ, then the quadratic form can be transformed to a perfect square, just as in the classical case. As a result we obtain sufficient conditions for positive definiteness of the quadratic form in terms of the Riccati equation and hence, sufficient optimality conditions for broken extremals.

Difference Methods for Differential Inclusions

First we introduce differential inclusions by means of several model problems. These model problems shall illustrate the significance of differential inclusions for a wide range of applications, e.g. dynamic systems with discontinuous state equations, nonlinear programming, and optimal control.

Stability and Convergence of Euler's Method for State-Constrained Differential Inclusions

A discrete stability theorem for set-valued Eulers method with state constraints is proved. This theorem is combined with known stability results for differential inclusions with so-called smooth state constraints. As a consequence, order of convergence equal to 1 is proved for set-valued Eulers method, applied to state-constrained differential inclusions.

Difference methods with selection strategies for differential inclusions

The objective of this paper is to investigate convergence properties of multistep methods applied to differential inclusions. These multistep methods are combined with selection strategies, especially strategies based on optimization, forcing convergence to solutions with additional differentiability properties. For selection with respect to a reference trajectory an error estimate is proved.

Lineare Optimierung in unendlichdimensionalen Vektorräumen

ZusammenfassungMit Hilfe allgemeiner Trennungssätze für konvexe Mengen werden notwendige Optimalitätskriterien und starke Dualitätssätze angegeben für lineare Optimierungsprobleme mit unendlich vielen Nebenbedingungen.SummaryBy means of general separation theorems for convex sets necessary conditions for optimality and strong duality theorems are given for linear optimization problems with infinitely many restrictions.

Bemerkungen zur Lagrangeschen Funktionaldifferentialgleichung

In [4] haben wir den Beweis einer Lagrangeschen Multiplikatorenregel skizziert fur das folgende sehr allgemeine

Set-Valued Interpolation, Differential Inclusions, and Sensitivity in Optimization

Set-valued interpolation and integration methods are introduced with special emphasis on error representations and error estimates with respect to Hausdorff distance. The connection between order of convergence results and sensitivity properties of finite-dimensional convex optimization problems is discussed. The results are applied to the numerical approximation of reachable sets of linear control problems by quadrature formulae and interpolation techniques for set-valued mappings.