Skip to Main content Skip to Navigation
Journal articles

Smooth regularization of bang-bang optimal control problems

Abstract : Consider the minimal time control problem for a single-input control-affine system $\dot{x}=X(x) + u_1 Y_1 (x)$ in $\R^{n}$, where the scalar control $u_1(\cdot)$ satisfies the constraint $|u_1(\cdot)| \leq 1$. When applying a shooting method for solving this kind of optimal control problem, one may encounter numerical problems due to the fact that the shooting function is not smooth whenever the control is bang-bang. In this article we propose the following smoothing procedure. For $\varepsilon > 0$ small, we consider the minimal time problem for the control system $\displaystyle \dot{x} = X(x) + u_1^{\varepsilon} Y_1(x)+ \varepsilon \sum_{i=2}^m u_i^{\varepsilon} Y_i \left(x\right)$, where the scalar controls $u_i^\varepsilon(\cdot)$, $i=1,\ldots, m$, with $m \geq 2$, satisfy the constraint $\displaystyle \sum_{i=1}^m \left(u_i^{\varepsilon}(t) \right)^2 \leq 1$. We prove, under appropriate assumptions, a strong convergence result of the solution of the regularized problem to the solution of the initial problem.
Complete list of metadatas

Cited literature [26 references]  Display  Hide  Download

https://hal.archives-ouvertes.fr/hal-00414680
Contributor : Emmanuel Trélat <>
Submitted on : Friday, February 19, 2010 - 6:17:33 PM
Last modification on : Thursday, May 3, 2018 - 3:32:06 PM
Document(s) archivé(s) le : Thursday, September 23, 2010 - 5:40:22 PM

File

reg.pdf
Files produced by the author(s)

Identifiers

  • HAL Id : hal-00414680, version 3

Collections

Citation

Cristiana J. Silva, Emmanuel Trélat. Smooth regularization of bang-bang optimal control problems. IEEE Transactions on Automatic Control, Institute of Electrical and Electronics Engineers, 2010, 55 (11), pp.2488--2499. ⟨hal-00414680v3⟩

Share

Metrics

Record views

397

Files downloads

997