# A Course in Combinatorial Optimization by Schrijver A. By Schrijver A.

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78) Φ(t0 , t0 ) = I and verifies the group property Φ(t, t0 ) = Φ(t, s)Φ(s, t0 ) ∀s ∈ (t0 , t). 77) implies x(t) ˙ = d Φ(t, t0 )x0 = A(t)Φ(t, t0 )x0 = A(t)x(t). 76). 79) follows from the fact that x(t) = Φ(t, s)xs = Φ(t, s)Φ(s, t0 )xt0 = Φ(t, t0 )xt0 . 78), then t det Φ(t, t0 ) = exp tr A(s) ds . , Φ˙ j,n (t, t0 ). 80). 81) s=t0 then for any t ∈ [t0 , T ] det Φ(t, t0 ) > 0. 80). 84) s=t0 where Φ −1 (t, t0 ) exists for all t ∈ [t0 , T ] and satisfies d −1 Φ (t, t0 ) = −Φ −1 (t, t0 )A(t), dt Φ −1 (t0 , t0 ) = I.

96) for 0 < m < 1 and v(t) ≤ c 1 − (1 − m)cm−1 1 − m−1 t τ =t0 ξ(τ ) dτ t for m > 1 and τ =t0 ξ(τ ) dτ < 1 . 97) then for any t ∈ [t0 , ∞) the following inequality holds: t v(t) ≤ c exp ξ(s) ds . 98) s=t0 This result remains true if c = 0. 98). 98) on applying c → 0. 4 The Lagrange Principle in Finite-Dimensional Spaces Let us recall several simple and commonly used definitions. 7 A set C lying within a linear set X is called convex if, together with any two points x, y ∈ C, it also contains the closed interval [x, y] := z : z = αx + (1 − α)y, α ∈ [0, 1] .

31) for the Bolza Problem has the form H (ψ, x, u, t) := ψ T f (x, u, t) − μh x(t), u(t), t , t, x, u, ψ ∈ [0, T ] × Rn × Rr × Rn . 18). Indeed, the representation in the Mayer form, x˙n+1 (t) = h x(t), u(t), t , 20 2 The Maximum Principle implies ψ˙ n+1 (t) = 0 and, hence, ψn+1 (T ) = −μ. 42) ∂x ⎪ ⎪ ⎪ L ⎪ ∂ ∂ ⎪ ⎪ ⎪ ψ(T ) = −μ h0 x ∗ (T ) − νl gl x ∗ (T ) . 29). 2), where f = f x(t), u(t) , h = h x(t), u(t) . 43) It follows that for all t ∈ [t0 , T ] H ψ(t), x ∗ (t), u∗ ψ(t), x ∗ (t) = const. Proof One can see that in this case H = H ψ(t), x(t), u(t) , that is, ∂ H = 0.