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Infeasible and Critically Feasible Optimal Control

Regina S. Burachik, C. Yalçın Kaya, Walaa M. Moursi

Abstract

We consider optimal control problems involving two constraint sets: one comprised of linear ordinary differential equations with the initial and terminal states specified and the other defined by the control variables constrained by simple bounds. When the intersection of these two sets is empty, typically because the bounds on the control variables are too tight, the problem becomes infeasible. In this paper, we prove that, under a controllability assumption, the ``best approximation'' optimal control minimizing the distance (and thus finding the ``gap'') between the two sets is of bang--bang type, with the ``gap function'' playing the role of a switching function. The critically feasible control solution (the case when one has the smallest control bound for which the problem is feasible) is also shown to be of bang--bang type. We present the full analytical solution for the critically feasible problem involving the (simple but rich enough) double integrator. We illustrate the overall results numerically on various challenging example problems.

Infeasible and Critically Feasible Optimal Control

Abstract

We consider optimal control problems involving two constraint sets: one comprised of linear ordinary differential equations with the initial and terminal states specified and the other defined by the control variables constrained by simple bounds. When the intersection of these two sets is empty, typically because the bounds on the control variables are too tight, the problem becomes infeasible. In this paper, we prove that, under a controllability assumption, the ``best approximation'' optimal control minimizing the distance (and thus finding the ``gap'') between the two sets is of bang--bang type, with the ``gap function'' playing the role of a switching function. The critically feasible control solution (the case when one has the smallest control bound for which the problem is feasible) is also shown to be of bang--bang type. We present the full analytical solution for the critically feasible problem involving the (simple but rich enough) double integrator. We illustrate the overall results numerically on various challenging example problems.
Paper Structure (16 sections, 9 theorems, 78 equations, 3 figures)

This paper contains 16 sections, 9 theorems, 78 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Best Approximation Solution to the Infeasible Problem
  4. Properties of Problem (Pf)
  5. Maximum Principle for Problem (Pf)
  6. Controllability
  7. Best approximation solution
  8. Critical Feasibility
  9. Maximum Principle for Problem (Pcf2)
  10. Solution to the critically feasible problem
  11. A double integrator problem
  12. Numerical Experiments
  13. Double integrator
  14. Damped oscillator
  15. Machine tool manipulator
  16. ...and 1 more sections

Key Result

Lemma 1

The constraint set of Problem (Pf) is convex and (strongly and weakly) closed (i.e., closed w.r.t. the norm topology and w.r.t. the weak topology in $L^2$).

Figures (3)

  • Figure 1: Double Integrator: (a) Critically feasible solution and (b)--(d) Best approximation solutions (infeasible case).
  • Figure 2: Damped oscillator: (a) Critically feasible solution and (b)--(d) Best approximation solutions (infeasible case).
  • Figure 3: Machine tool manipulator: (a) Critically feasible solution and (b)--(d) Best approximation solutions (infeasible case).

Theorems & Definitions (14)

  • Lemma 1
  • Lemma 2
  • Theorem 1: Controllability via a Gramian Test Matrix Rugh1996
  • Theorem 2: Controllability via a More Easily Computable Test Matrix
  • Theorem 3: Gap Vector and the Best Approximation Control in ${\cal B}$
  • Remark 1: The Best Approximation Control in ${\cal A}$
  • Remark 2: Time-invariant Systems
  • Remark 3
  • Theorem 4: Critically Feasible Control
  • Remark 4
  • ...and 4 more