# Control Theory and Optimization I by M.I. Zelikin, S.A. Vakhrameev

By M.I. Zelikin, S.A. Vakhrameev

The one monograph at the subject, this booklet issues geometric equipment within the conception of differential equations with quadratic right-hand aspects, heavily with regards to the calculus of diversifications and optimum keep an eye on conception. in line with the author’s lectures, the publication is addressed to undergraduate and graduate scholars, and medical researchers.

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L) the following inequalities hold: ⎧ ˆ ˆ u, ˆ t) ≤ L x − xˆ + ω¯ d(u, u) ⎨ ϕ(x, u, t) − ϕ(x, ∀t ∈ [0, T ], x, xˆ ∈ Rn , u, uˆ ∈ U, ⎩ ϕ(0, u, t) ≤ L ∀u, t ∈ U × [0, T ]. 7) (A3) The maps f, h, h0 and gl (l = 1, . . , L) are of type C 1 in x and there exists a continuity modulus ω¯ : [0, ∞) → [0, ∞) such that for ϕ = f (x, u, t), h(x, u, t), h0 (x, u, t), gl (x) (l = 1, . . 1 Optimal Control Problem 11 the following inequalities hold: ∂ ∂ ϕ(x, u, t) − ϕ(x, ˆ u, ˆ t) ≤ ω¯ x − xˆ + d(u, u) ˆ ∂x ∂x ∀t ∈ [0, T ], x, xˆ ∈ Rn , u, uˆ ∈ U.

Let us take the point y˜ ∈ y 0 , y˜ such that x 0 , y˜ ⊥ y 0 , y˜ ˜ and y˜ and show that y˜ is not a point of K¯ 2 close to x 0 . Indeed, the points y 0 , y, belong to the same line and y˜ ∈ int H . 3 Appendix 39 (the shortest distance is one smaller than any other one). At the same time, y˜ ∈ ˜ so (y 0 , y), x 0 − y˜ < x 0 − y 0 . Also we have 0 ∈ H since, if this were not so, the line [0, ∞), crossing y 0 and belonging to K¯ 2 , should necessarily have points in common with int H . Proof of the Lagrange Principle Now we are ready to give the proof of the main claim.

34) is not valid, that is, L νl = 0, ψ(T ) + μ + l=1 which implies ψ(T ) = 0, μ = νl = 0 (l = 1, . . 29) and Gronwall’s Lemma it follows that ψ(t) = 0 for all t ∈ [0, T ]. So, H ψ(t), x(t), u(t), t = 0 u∗ (t)). for any u(t) (not only for This means that the application of any admissible control keeps the cost function unchanged and this corresponds to the trivial situation of an “uncontrollable” system. So the nontriviality condition is proven as well. 3 The Regular Case In the so-called regular case, when μ > 0 (this means that the nontriviality condition holds automatically), the variable ψ(t) and constants νl may be normalized and ˜ changed to ψ(t) := ψ(t)/μ and ν˜ l := νl /μ.