Necessary optimality condition in one discrete control problem from nondifferentiable control cost

In this article we consider the problem of necessary optimality condition in one discrete control problem but assuming nondifferenti-ability quality functional. We derive a necessary condition for optimality in terms of directional derivatives. Let managed object described by a system of difference equations z(t +1,x) = f (t,x,z(t,x),u(t)) , t = t₀,t₀ + 1,...,t₁ -1; x = x₀,x₀ +1,...,x₁, (1) with the initial condition z (t₀, x) = y (x), x = x₀,x₀ + 1,...,x₁, (2) where n - dimensional vector function y (x) is a solution of y( +) = g(xy()()) x = x0,x0 +1,...,x1 -Ь y(0)= y₀. Here f (t, x, z,u) (g (x,y,v)) is the given n - dimensional vector-function continuous with respect to all variables together with the partial derivatives with respect to z (y), y₀ is a given constant vector, t₀, t₁, x₀, x₁ are given numbers, the differences t₁ -1₀ and x₁ - x₀ are natural numbers, u (t) (v ( x )) is r(q)-dimensional vector of control actions with values from a specified non-empty, bounded set u(t) e U с R , t e T = {t₀,t₀ +1,...,t₁ -1}, v(x) e V сR , xeX ={x₀,x₀ +1,...,x₁ -1} . (4) Our goal is to minimize the functional .1 -1 S ()=ф1 (y (1))+ X ф2 (.)) under the constraints (1)-(4). Here ф₁ (y) (ф₂ (x, z)) is the given scalar function satisfying the Lipschitz condition with respect to y (z) and having derivatives with respect to y (z) in any direction.

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