Necessary optimality conditions in optimal control problems by discrete-continuous systems

C0nsider a minimizati0n pr0blem 0f the functi0nal S (u, v ) = ф( x (ti )) + ф( у (T )), t x(t +1) = 2 f (t,x,x(x),u (x)) , t e T ₁ = {t ₀, t ₀ +1, t ₀ + 2,...,t ₁ -1}, x = tr X 0 0 t у (t) = J g (t, x, у (x), v (x)) dx, t e T ₂ = [ T], ti у (ti ) = G (x (ti)). Here t ₀, t ₁, t ₂, x ₀ are the given values, the difference t ₁ -1 ₀ is a natural number, ф(x), ф(у) are the given c0ntinu0usly-differentiable scalar functi0ns, f (t, x, x, u) , (g (t, x, y, v)) are the given n(m)-dimensi0nal vector- functi0ns c0ntinu0us in the aggregate 0f variables together with partial derivatives with respect (x, u) ((y, v)) , G(x) is the given c0ntinu0usly-differentiable m-dimensi0nal vector-functi0n, u (t) (v (t)) are r(q) - dimenstonal vectors 0f c0ntrol acti0ns with the values from the given n0n-empty, b0unded, and 0pen set U(V), i.e. u (t) e U с R , t e Ti, v (t)eV с R , t e T ₂. We call the pair (u (t), v (t)) with the ab0ve menttoned properties an admissible c0ntrol, the c0rresp0nding pr0cess (u (t), v (t), x (t), у (t)) - an admissible process. Our g0al is t0 derive a necessary 0ptimality c0nditi0n in the problem under ab0ve c0nsiderati0ns.

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