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A constructive method for solving stabilization problems

Vadim Azhmyakov (2000)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

The problem of asymptotic stabilization for a class of differential inclusions is considered. The problem of choosing the Lyapunov functions from the parametric class of polynomials for differential inclusions is reduced to that of searching saddle points of a suitable function. A numerical algorithm is used for this purpose. All the results thus obtained can be extended to cover the discrete systems described by difference inclusions.

A differential inclusion : the case of an isotropic set

Gisella Croce (2005)

ESAIM: Control, Optimisation and Calculus of Variations

In this article we are interested in the following problem: to find a map u : Ω 2 that satisfies D u E a.e. in Ω u ( x ) = ϕ ( x ) x Ω where Ω is an open set of 2 and E is a compact isotropic set of 2 × 2 . We will show an existence theorem under suitable hypotheses on ϕ .

A differential inclusion: the case of an isotropic set

Gisella Croce (2010)

ESAIM: Control, Optimisation and Calculus of Variations

In this article we are interested in the following problem: to find a map u : Ω 2 that satisfies D u E a.e. in Ω u ( x ) = ϕ ( x ) x Ω where Ω is an open set of 2 and E is a compact isotropic set of 2 × 2 . We will show an existence theorem under suitable hypotheses on φ.

A discrete contact model for crowd motion

Bertrand Maury, Juliette Venel (2011)

ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique

The aim of this paper is to develop a crowd motion model designed to handle highly packed situations. The model we propose rests on two principles: we first define a spontaneous velocity which corresponds to the velocity each individual would like to have in the absence of other people. The actual velocity is then computed as the projection of the spontaneous velocity onto the set of admissible velocities (i.e. velocities which do not violate the non-overlapping constraint). We describe here the...

A discrete contact model for crowd motion

Bertrand Maury, Juliette Venel (2011)

ESAIM: Mathematical Modelling and Numerical Analysis

The aim of this paper is to develop a crowd motion model designed to handle highly packed situations. The model we propose rests on two principles: we first define a spontaneous velocity which corresponds to the velocity each individual would like to have in the absence of other people. The actual velocity is then computed as the projection of the spontaneous velocity onto the set of admissible velocities (i.e. velocities which do not violate the non-overlapping constraint). We describe here...

A new method of proof of Filippov’s theorem based on the viability theorem

Sławomir Plaskacz, Magdalena Wiśniewska (2012)

Open Mathematics

Filippov’s theorem implies that, given an absolutely continuous function y: [t 0; T] → ℝd and a set-valued map F(t, x) measurable in t and l(t)-Lipschitz in x, for any initial condition x 0, there exists a solution x(·) to the differential inclusion x′(t) ∈ F(t, x(t)) starting from x 0 at the time t 0 and satisfying the estimation x ( t ) - y ( t ) r ( t ) = x 0 - y ( t 0 ) e t 0 t l ( s ) d s + t 0 t γ ( s ) e s t l ( τ ) d τ d s , where the function γ(·) is the estimation of dist(y′(t), F(t, y(t))) ≤ γ(t). Setting P(t) = x ∈ ℝn: |x −y(t)| ≤ r(t), we may formulate the conclusion in Filippov’s theorem...

A smooth Lyapunov function from a class- 𝒦ℒ estimate involving two positive semidefinite functions

Andrew R. Teel, Laurent Praly (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We consider differential inclusions where a positive semidefinite function of the solutions satisfies a class- 𝒦ℒ estimate in terms of time and a second positive semidefinite function of the initial condition. We show that a smooth converse Lyapunov function, i.e., one whose derivative along solutions can be used to establish the class- 𝒦ℒ estimate, exists if and only if the class- 𝒦ℒ estimate is robust, i.e., it holds for a larger, perturbed differential inclusion. It remains an open question whether...

A strong relaxation theorem for maximal monotone differential inclusions with memory

Nikolaos S. Papageorgiou (1994)

Archivum Mathematicum

We consider maximal monotone differential inclusions with memory. We establish the existence of extremal strong and then we show that they are dense in the solution set of the original equation. As an application, we derive a “bang-bang” principle for nonlinear control systems monitored by maximal monotone differential equations.

A study of second order differential inclusions with four-point integral boundary conditions

Bashir Ahmad, Sotiris K. Ntouyas (2011)

Discussiones Mathematicae, Differential Inclusions, Control and Optimization

In this paper, we discuss the existence of solutions for a four-point integral boundary value problem of second order differential inclusions involving convex and non-convex multivalued maps. The existence results are obtained by applying the nonlinear alternative of Leray Schauder type and some suitable theorems of fixed point theory.

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