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Necessary conditions of second order optimality for systems with three-point boundary conditions
(Maltepe Üniversitesi, 2009) Sharifov, Yagub A.; Djabrailov, S. I.
In this report the object of investigation is an optimal control problem in systems of nonlinear first order ordinary differential equations with three-point boundary conditions: x? = f(x, u, t), x(t) ? R n , t ? T = [t0, t1], (1) Ax(t0) + Bx(t1) + Dx(t2) = c, t1 ? (t0, t2), (2) Here f ? R n is continuous by collection of variables together with its partial derivatives with respect to x and u up to the second order inclusive, A, B, D ? R n×n, c ? R n×1 are constant matrices. It is supposed that control action satisfy restriction u(t) ? V, t ? T , where V is a convex compact set from R r . The goal of optimal control problem is optimization of the functional: J(u) = ?(x(t0), x(t2)) (3) defined on the solutions of boundary problem (1)-(2) at admissible controls where it is supposed that function ?(x, y) is continuous by x and y up to the second order inclusive. The formula of the second order increment of functional (3) is calculated. On the basis of control variations there are obtained new necessary conditions of optimality for quasi-singular controls for systems which are described by a set of differential equations with three-point boundary conditions.