Stability Notions for a Class of Nonlinear Systems with Measure Controls

Aneel Tanwani 1 Bernard Brogliato 1 Christophe Prieur 2
1 BIPOP - Modelling, Simulation, Control and Optimization of Non-Smooth Dynamical Systems
Inria Grenoble - Rhône-Alpes, LJK - Laboratoire Jean Kuntzmann, INPG - Institut National Polytechnique de Grenoble
2 GIPSA-SYSCO - SYSCO
GIPSA-DA - Département Automatique
Abstract : We consider the problem of stability in a class of differential equations which are driven by a differential measure associated with inputs of locally bounded variation. After discussing some existing notions of solution for such systems, we derive Lyapunov-based conditions on the system's vector fields for asymptotic stability under a specific class of inputs. These conditions are based on the stability margin of the Lebesgue-integrable and the measure-driven components of the system. For more general inputs which do not necessarily lead to asymptotic stability, we then derive conditions such that the maximum norm of the resulting trajectory is bounded by some function of the total variation of the input, which generalizes the notion of integral input-to-state stability in measure-driven systems.
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Aneel Tanwani, Bernard Brogliato, Christophe Prieur. Stability Notions for a Class of Nonlinear Systems with Measure Controls. Mathematics of Control, Signals, and Systems, Springer Verlag, 2015, 27 (2), pp.245-275. ⟨10.1007/s00498-015-0140-7⟩. ⟨hal-01074144⟩

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