Geodesic
Hamilton--Jacobi inequalities in control problems for impulsive dynamical systems
Informatics and Automation, Differential equations and topology. II, Tome 271 (2010), pp. 93-110

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We propose definitions of strong and weak monotonicity of Lyapunov-type functions for nonlinear impulsive dynamical systems that admit vector measures as controls and have trajectories of bounded variation. We formulate infinitesimal conditions for the strong and weak monotonicity in the form of systems of proximal Hamilton–Jacobi inequalities. As an application of strongly and weakly monotone Lyapunov-type functions, we consider estimates for integral funnels of impulsive systems as well as necessary and sufficient conditions of global optimality corresponding to the approach of the canonical Hamilton–Jacobi theory. 
@article{TRSPY_2010_271_a7,
     author = {V. A. Dykhta and O. N. Samsonyuk},
     title = {Hamilton--Jacobi inequalities in control problems for impulsive dynamical systems},
     journal = {Informatics and Automation},
     pages = {93--110},
     publisher = {mathdoc},
     volume = {271},
     year = {2010},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/TRSPY_2010_271_a7/}
}
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V. A. Dykhta; O. N. Samsonyuk. Hamilton--Jacobi inequalities in control problems for impulsive dynamical systems. Informatics and Automation, Differential equations and topology. II, Tome 271 (2010), pp. 93-110. http://geodesic.mathdoc.fr/item/TRSPY_2010_271_a7/