TPhe paper deals with the Carathe´odory solutions of the system (1) x' (t) = f(x(t)) + g_i(x(t))u'_ i(t), where the controls u=(u_i) are vector valued functions with bounded variation. The approach is based on studying the continuity properties of the input-output map F of (1). It is shown that the restriction of F to the set of controls satisfying a uniform Lipschitz condition is continuous in the C-topology. If ui are discontinuous, but have bounded variation, a Lipschitz continuous parametrization of the graph of controls is proposed. The generalized solution thus obtained coincides almost everywhere with the limit of classical solutions of (1) corresponding to so-called mollified controls (this concept is introduced in the paper).
ON DIFFERENTIAL-SYSTEMS WITH VECTOR-VALUED IMPULSIVE CONTROLS
RAMPAZZO, FRANCO
1988
Abstract
TPhe paper deals with the Carathe´odory solutions of the system (1) x' (t) = f(x(t)) + g_i(x(t))u'_ i(t), where the controls u=(u_i) are vector valued functions with bounded variation. The approach is based on studying the continuity properties of the input-output map F of (1). It is shown that the restriction of F to the set of controls satisfying a uniform Lipschitz condition is continuous in the C-topology. If ui are discontinuous, but have bounded variation, a Lipschitz continuous parametrization of the graph of controls is proposed. The generalized solution thus obtained coincides almost everywhere with the limit of classical solutions of (1) corresponding to so-called mollified controls (this concept is introduced in the paper).
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