Quasi-singular control in discrete systems control problem with nonlocal boundary conditions

Consider a discrete control system with boundary conditions x {t +1) = f [t, x {t),M (t)), t e T , (1) ф(^x(t,)^,x(ti)) = I . ⁽²⁾ Here T = {t),^ + 1,- -,A “1} is a finite set of consecutive natural numbers, at that t₀ and t₁ is given, Ф(Х),X) is the given twice continuously differentiable with respect to the set of variables и-dimensional vector-valued function, l is the given constant vector, x(t) is a state vector, m (t) is a control actions vector, f (t, x, m) is the given и-dimensional vector-valued function continuous with respect to the set of variables together with the partial derivatives with respect to (x, и) up to the second order inclusive. Let U be the given non-empty, bounded, and convex set in R' . Each control function и (t) satisfying the condition и(t)e U c R', t e T (3) will be called admissible control. We consider the problem of the minimum of the functional S (^M ) = ф(^x (^t„)^{, x} (^ti)) under constraints (1)-(3). Here 9(x, x) 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.

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