Accurate Approximating Solution of the Differential Inclusion Based on the Ordinary Differential Equation

Numerous problems in applied mathematics can be transformed and described by the differential inclusion ẋ 2 f(t, x) − NQx involving NQx, which is a normal cone for a closed convex set Q 2 ℝn at x 2 Q. We study the Cauchy problem for this inclusion. Since the variations of x lead to changing NQx, the solution of the analyzed inclusion becomes extremely complicated. We consider an ordinary differential equation containing a control parameter K. If K is sufficiently large, then the indicated equation gives a solution approximating the solution of the original inclusion. We also prove the theorem on approximation of these solutions with arbitrarily small errors (the errors can be controlled by increasing the parameter K).