### A barrier method for quasilinear ordinary differential equations of the curvature type

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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...

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 $$\left|x\left(t\right)-y\left(t\right)\right|\u2a7dr\left(t\right)=\left|{x}_{0}-y\left({t}_{0}\right)\right|{e}^{{\int}_{{t}_{0}}^{t}l\left(s\right)ds}+{\int}_{{t}_{0}}^{t}\gamma \left(s\right){e}^{{\int}_{s}^{t}l\left(\tau \right)d\tau}ds,$$ 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...

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.