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.

We present a constructive proof of the fact that the set of algebraic Pfaff equations without algebraic solutions over the complex projective plane is dense in the set of all algebraic Pfaff equations of a given degree.

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...

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...

We give an estimate for the distance between a given approximate solution for a Lipschitz differential inclusion and a true solution, both depending continuously on initial data.

The purpose of this article is to find the 7th order formulas with rational parameters. The formulas are of the 11th stage. If we compare the coefficients of the development ${\sum }_{i=1}^{\infty }\frac{h^i}{i!}\frac{d^{i-1}}{d{x}^{i-1}}𝐟\left[x,𝐲\left(x\right)\right]$ up to $h^7$ with the development given by successive insertion into the formula $h.f_i(k_0,k_1,...,k_{i-1})$ for $i=1,2,...,10$ and $k={\sum }_{i=0}^{10}{p}_i,k_i$ we obtain a system of 59 condition equations with 65 unknowns (except, the 1st one, all equations are nonlinear). As the solution of this system we get the parameters of the 7th order Runge-Kutta formulas as rational numbers.

The linear homogeneous differential equation with variable delays $ẏ\left(t\right)={\sum }_{j=1}^n{\alpha }_j\left(t\right)[y\left(t\right)-y(t-{\tau }_j\left(t\right))]$ is considered, where ${\alpha }_j\in C(I,ℝ͞͞⁺)$, I = [t₀,∞), ℝ⁺ = (0,∞), ${\sum }_{j=1}^n{\alpha }_j\left(t\right)>0$ on I, ${\tau }_j\in C(I,ℝ⁺),$ the functions $t-{\tau }_j\left(t\right)$, j=1,...,n, are increasing and the delays ${\tau }_j$ are bounded. A criterion and some sufficient conditions for convergence of all solutions of this equation are proved. The related problem of nonconvergence is also discussed. Some comparisons to known results are given.