On the computation of singular controls

Abstract

We consider singular optimal control problems consisting of a state equation <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\dot{x}=Ax+bu</tex> for vectors <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">x</tex> and scalars <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">u</tex> and a cost functional <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">J = \frac{1}{2} \int\min{0}\max{T}(x'Qx+\epsilon^{2}u^{2})dt</tex> to be minimized for <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">|u|\leq m</tex> and <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\epsilon=0</tex> . By considering the problem as <tex xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">\epsilon \rightarrow 0</tex> , singular perturbation concepts can be used to compute solutions consisting of bang-bang controls followed by singular arcs. The procedure further develops a numerical technique proposed by Jacobson, Gershwin, and Lele [18], as well as additional analytic methods developed by other authors.