Successive approximation method for quasilinear impulsive differential equations with control

We introduce a technique to define successive approximations to solutions of the control problem with implulse actions on surfaces dx/dt = A(t)x(t) + C(t)u + f(t) + mu g(t, x, u, mu), t not equal zeta(i), Delta x(zeta(i)) = B(i)x + D(i)v(i) + J(i) + mu W-i ( x, v(i), mu), i = 1,2,...,p, x(alpha) = a, x(beta) = b, where mu is a small positive parameter, zeta(i) = theta(i) + mu tau(i)(x(zeta(i)), mu), x is an element of R-n and Delta x(theta) := x(theta+) - x(theta). A sequence of piecewise continuous functions with discontinuities of the first kind that converges to a solution of the above problem is constructed. (C) 2000 Elsevier Science Ltd. All rights reserved.