Geodesic
Approximation Theorem for a Nonlinear Control System with Sliding Modes
Informatics and Automation, Dynamical systems and optimization, Tome 256 (2007), pp. 102-114

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We consider the question of validity of the extension of a nonlinear control system by introducing the so-called sliding modes (i.e., by convexifying the set of admissible velocities) in the presence of constraints imposed on the endpoints of trajectories. We prove that a trajectory of the extended system can be approximated by trajectories of the original system if the equality constraints of the extended system are nondegenerate in the first order. The proof is based on a nonlocal estimate for the distance to the zero set of the nonlinear operator corresponding to the extended system, and involves a specific iteration process of corrections. 
@article{TRSPY_2007_256_a5,
     author = {A. V. Dmitruk},
     title = {Approximation {Theorem} for a {Nonlinear} {Control} {System} with {Sliding} {Modes}},
     journal = {Informatics and Automation},
     pages = {102--114},
     publisher = {mathdoc},
     volume = {256},
     year = {2007},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2007_256_a5/}
}
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A. V. Dmitruk. Approximation Theorem for a Nonlinear Control System with Sliding Modes. Informatics and Automation, Dynamical systems and optimization, Tome 256 (2007), pp. 102-114. http://geodesic.mathdoc.fr/item/TRSPY_2007_256_a5/