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 )) (¹) with constraints u (')eU с R^r, I eli =['0, | ], v(')eV сR^q, ' eT₂ =[',h], t x(|) = 1 f (',s,x(s),u(s))ds, ' e T, (3) y = g(t,y,v),teT₂ (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, t₀, tj, t₂ are given and t₀ < tj < t₂, 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 Rⁿ (R^m) 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.

Keywords

Authors

References