QUASI-SINGULAR CONTROL IN DISCRETE SYSTEMS CONTROL PROBLEM WITH NONLOCAL BOUNDARY CONDITIONS

Consider a discrete control system x(t + 1)=f(t,x(t),u(t)),t is an element of T, (1) with boundary conditions Phi(x (t(0)), x(t(1))) = l . (2) Here T = {t(0),t(0) +1,...,t(1)-1} is a finite set of consecutive natural numbers, at that t(0) and t(1) is given, Phi (x(0), x(1)) is the given twice continuously differentiable with respect to the set of variables n-dimensional vector-valued function, l is the given constant vector, x(t) is a state vector, u(t) is a control actions vector, f (t, x,u) is the given n-dimensional vector-valued function continuous with respect to the set of variables together with the partial derivatives with respect to (x,u) up to the second order inclusive. Let U be the given non-empty, bounded, and convex set in R-r. Each control function u (t) satisfying the condition u(t) is an element of U subset of R-r, t is an element of T (3) will be called admissible control. We consider the problem of the minimum of the functional S(u)=phi(x(t(0)),x(t(1))) under constraints (1)-(3). Here phi (x(0) ,x(1)) is the twice continuously differentiable scalar function with respect to the set of variables. A necessary condition for the optimality of quasi-singular controls is established.