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A continuation method for motion-planning problems

Yacine Chitour (2006)

ESAIM: Control, Optimisation and Calculus of Variations

We apply the well-known homotopy continuation method to address the motion planning problem (MPP) for smooth driftless control-affine systems. The homotopy continuation method is a Newton-type procedure to effectively determine functions only defined implicitly. That approach requires first to characterize the singularities of a surjective map and next to prove global existence for the solution of an ordinary differential equation, the Wazewski equation. In the context of the MPP, the aforementioned...

A continuation method for motion-planning problems

Yacine Chitour (2005)

ESAIM: Control, Optimisation and Calculus of Variations

We apply the well-known homotopy continuation method to address the motion planning problem (MPP) for smooth driftless control-affine systems. The homotopy continuation method is a Newton-type procedure to effectively determine functions only defined implicitly. That approach requires first to characterize the singularities of a surjective map and next to prove global existence for the solution of an ordinary differential equation, the Wazewski equation. In the context of the MPP, the aforementioned...

A model for gene activation.

Oppenheimer, Seth F., Fan, Ruping, Pruett, Stephan (2009)

Electronic Journal of Differential Equations (EJDE) [electronic only]

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 note on existence theorem of Peano

Oleg Zubelevich (2011)

Archivum Mathematicum

An ODE with non-Lipschitz right hand side has been considered. A family of solutions with L p -dependence of the initial data has been obtained. A special set of initial data has been constructed. In this set the family is continuous. The measure of this set has been estimated.

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