The second order necessary optimality conditions in a control problem with variable structure
The paper deals with the problem of optimal control by minimizing functional of a terminal type I (u, v ) - Ф1 (x ('i )) + ф2 (y ('2 )) (<sup>1</sup>) with constraints u (')eU с R<sup>r</sup>, I eli =['0, | ], v(')eV сR<sup>q</sup>, ' eT<sub>2</sub> =[',h], t x(|) = 1 f (',s,x(s),u(s))ds, ' e T, (3) y = g(t,y,v),teT<sub>2</sub> (4) y (I,) = G(x(',)). (5) Here f (t,s,x,u), (g(t,y,v)) are give n (ra)-dimensional vector-functions, respectively, continuous with respect to all the variables together with partial derivatives with respect to (x,u) ((y,v)) up to the second order inclusive, G(x) is га-dimensional twice continuously differentiable vector-function, t<sub>0</sub>, tj, t<sub>2</sub> are given and t<sub>0</sub> < tj < t<sub>2</sub>, и(t) (v(t)) are r (g)-dimensional piecewise-continu-ous (with a finite number of points of discontinuity of the first kind) vector-functions of control actions, values from a given nonempty, bounded, and open sets U (V), ф (x), ф (y) are given twice continuously differentiable in R<sup>n</sup> (R<sup>m</sup>) scalar functions. Here ф (x) and ф (y) are given m-dimensional vector functions, respectively, continuous with respect to all the variables together with partial derivatives up to the second order inclusive.