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Stability and sliding modes for a class of nonlinear time delay systems

Vladimir B. Răsvan (2011)

Mathematica Bohemica

The following time delay system x ˙ = A x ( t ) + ∑ 1 r b q i * x ( t - τ i ) - b ϕ ( c * x ( t ) ) is considered, where ϕ : ℝ → ℝ may have discontinuities, in particular at the origin. The solution is defined using the “redefined nonlinearity” concept. For such systems sliding modes are discussed and a frequency domain inequality for global asymptotic stability is given.

Stability rates for patchy vector fields

Fabio Ancona, Alberto Bressan (2004)

ESAIM: Control, Optimisation and Calculus of Variations

This paper is concerned with the stability of the set of trajectories of a patchy vector field, in the presence of impulsive perturbations. Patchy vector fields are discontinuous, piecewise smooth vector fields that were introduced in Ancona and Bressan (1999) to study feedback stabilization problems. For patchy vector fields in the plane, with polygonal patches in generic position, we show that the distance between a perturbed trajectory and an unperturbed one is of the same order of magnitude...

Stability rates for patchy vector fields

Fabio Ancona, Alberto Bressan (2010)

ESAIM: Control, Optimisation and Calculus of Variations

This paper is concerned with the stability of the set of trajectories of a patchy vector field, in the presence of impulsive perturbations. Patchy vector fields are discontinuous, piecewise smooth vector fields that were introduced in Ancona and Bressan (1999) to study feedback stabilization problems. For patchy vector fields in the plane, with polygonal patches in generic position, we show that the distance between a perturbed trajectory and an unperturbed one is of the same order of magnitude...

The Picard-Lindelöf Theorem and continuation of solutions for measure differential equations

Gastón Beltritti, Stefania Demaria, Graciela Giubergia, Fernando Mazzone (2025)

Czechoslovak Mathematical Journal

We obtain, by means of Banach's Fixed Point Theorem, convergence for the Picard iterations associated to a general nonlinear system of measure differential equations. We study the existence of left-continuous solutions defined on maximal intervals and we establish some properties of these maximal solutions.

Uniqueness and weak stability for multi-dimensional transport equations with one-sided Lipschitz coefficient

Francois Bouchut, Francois James, Simona Mancini (2005)

Annali della Scuola Normale Superiore di Pisa - Classe di Scienze

The Cauchy problem for a multidimensional linear transport equation with discontinuous coefficient is investigated. Provided the coefficient satisfies a one-sided Lipschitz condition, existence, uniqueness and weak stability of solutions are obtained for either the conservative backward problem or the advective forward problem by duality. Specific uniqueness criteria are introduced for the backward conservation equation since weak solutions are not unique. A main point is the introduction of a generalized...

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