Abstract

A computational method for the approximation of reachable sets for non-linear dynamic systems is suggested. The method is based on a discretization of the interesting region and a projection onto grid points. The projections require to solve optimal control problems which are solved by a direct discretization approach. These optimal control problems allow a flexible formulation and it is possible to add non-linear state and/or control constraints and boundary conditions to the dynamic system. Numerical results for non-convex reachable sets are presented. Possible applications include robust optimal control problems.

Highlights

  • The subject of this paper is the description of an algorithm for the approximation of reachable sets of non-linear control problems

  • The occurring optimal control problems are not solved theoretically by use of the Pontryagin’s maximum principle as in [23] but numerically by suitable discretization methods. This approach turns out to be powerful in practice and allows to include control and/or state constraints and even boundary conditions

  • We suggest a computational method which allows to approximate the reachable set of a nonlinear dynamic system at a fixed time point T by using optimal control techniques

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Summary

INTRODUCTION

The subject of this paper is the description of an algorithm for the approximation of reachable sets of non-linear control problems. The occurring optimal control problems are not solved theoretically by use of the Pontryagin’s maximum principle as in [23] but numerically by suitable discretization methods. This approach turns out to be powerful in practice and allows to include control and/or state constraints and even boundary conditions. Consider a suitable one-step discretization scheme, e.g. an explicit Runge-Kutta method, with increment function Φ on an equidistant time grid with time points ti = t0 + ih, i = 0, 1, .

THE ALGORITHM
NUMERICAL EXAMPLES
Example 1
Example 3
CONCLUSIONS AND FUTURE WORKS
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