On Stability of Measure Driven Differential Equations

Aneel Tanwani (1) , Bernard Brogliato (1) , Christophe Prieur (1, 2)

1 BIPOP - Modelling, Simulation, Control and Optimization of Non-Smooth Dynamical Systems
2 GIPSA-SYSCO - GIPSA - Systèmes non linéaires et complexité

We consider the problem of stability in a class of differential equations which are driven by a differential measure associated with the inputs of locally bounded variation. After discussing some existing notions of solution for such systems, we derive conditions on the system's vector fields for asymptotic stability under a specific class of inputs. These conditions present a trade-off between the Lebesgue-integrable and the measure-driven components of the system. In case the system is not asymptotically stable, we derive weaker conditions such that the 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.

Keywords: Stability; Input-To-State Stability; Lyapunov Methods

Domains: Automatic

Aneel Tanwani, Bernard Brogliato, Christophe Prieur. On Stability of Measure Driven Differential Equations. NOLCOS 2013 - 9th IFAC Symposium on Nonlinear Control Systems, Sep 2013, Toulouse, France. pp.241-246, ⟨10.3182/20130904-3-FR-2041.00135⟩. ⟨hal-00926112⟩