Geodesic
On the penalty function method in the problem of constructing reachable sets for control systems with state constraints
Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 19 (2013) no. 1, pp. 81-86

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The paper is devoted to the problem of constructing reachable sets for a nonlinear control system with state constraints given in the form of inequalities. An analog of the penalty function method is proposed; it consists in replacing the original system with state constraints by an auxiliary system without state constraints. The right-hand side of the auxiliary system is a modification of the right-hand side of the original system and depends on a scalar parameter (a penalty coefficient). Under certain conditions, we show that a reachable set of the original system can be approximated from within in the Hausdorff metric by reachable sets of auxiliary systems as the penalty coefficient tends to infinity.
Keywords: control system, reachable set, state constraints, penalty function method. 
M. I. Gusev. On the penalty function method in the problem of constructing reachable sets for control systems with state constraints. Trudy Instituta matematiki i mehaniki, Trudy Instituta Matematiki i Mekhaniki UrO RAN, Tome 19 (2013) no. 1, pp. 81-86. http://geodesic.mathdoc.fr/item/TIMM_2013_19_1_a7/
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