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Bang-bang property for Bolza problems in two dimensions

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Abstract

Consider the following Bolza problem:

$$\begin{gathered} \min \int {h(x,u) dt,} \hfill \\ \dot x = F(x) + uG(x), \hfill \\ \left| u \right| \leqslant 1, x \in \Omega \subset \mathbb{R}^2 , \hfill \\ x(0) = x_0 , x(1) = x_1 . \hfill \\ \end{gathered} $$

We show that, under suitable assumptions onF, G, h, all optimal trajectories are bang-bang. The proof relies on a geometrical approach that works for every smooth two-dimensional manifold. As a corollary, we obtain existence results for nonconvex optimization problems.

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Additional information

The authors thank Professor Arrigo Cellina for suggesting the problem and useful advice.

Communicated by R. Conti

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Crasta, G., Piccoli, B. Bang-bang property for Bolza problems in two dimensions. J Optim Theory Appl 83, 155–165 (1994). https://doi.org/10.1007/BF02191766

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Key Words

  • Control theory
  • Bolza problems
  • bang-bang property
  • nonconvex optimization problems