By Hidenori Kimura

1 Introduction.- 2 components of Linear structures Theory.- three Norms and Factorizations.- four Chain-Scattering Representations of the Plant.- five J-Lossless Conjugation and Interpolation.- 6 J-Lossless Factorizations.- 7 H-infinity keep an eye on through (J, J')-Lossless Factorization.- eight State-Space recommendations to H-infinity keep an eye on Problems.- nine constitution of H-infinity regulate

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**Sample text**

3). (a) For every x ∈ U , Φ(0, x) = x. (b) For every s, t ∈ R and x ∈ U , Φ(s + t, x) = Φ(s, Φ(t, x)). (c) Φ is a continuous function. Proof. Statement (a) follows from the deﬁnition of Φ. 4(b). 3. In the classical point of view, the objective of the theory of diﬀerential equations is to ﬁnd explicit expressions for the ﬂow Φ(s, x). , the behaviour of the orbits when s tends to ±∞. 3). 4. 3) and take p ∈ U . By the continuous dependence of the solutions on the initial conditions and parameters, the function Φp : R → U given by Φp (s) := Φ(s, p) is continuously ˙ p (s) = f (Φp (s)), if there exists s0 such that diﬀerentiable.

Basic elements of the qualitative theory of ODEs A be the matrix representation of T . In the sequel we will identify the linear map T with its matricial representation A, and write A ∈ L(Rn ). , det(A) = 0, we will write A ∈ GL(Rn ). If A ∈ L(Rn ) we denote by t or trace(A) the trace of A, and by d or det(A) the determinant of A. This explains our use of the variable s, instead of the more usual one t, to denote the time in the diﬀerential equation. Let A ∈ L(Rn ). Then for every s ∈ R we deﬁne the exponential matrix of the matrix sA as the formal power series ∞ k k s A , esA := k!

X, y¯) = Consider the diﬀerentiable function hx : R2 → R2 deﬁned by hx (¯ (¯ x, x ¯ y¯). Using the Jacobian matrix of hx and the vector ﬁeld f we can deﬁne a vector ﬁeld fx on R2 satisfying the equality x, y¯)) = f (hx (¯ x, y¯)) = f (¯ x, x¯ y¯). Dhx (fx (¯ From here, one obtains the following expression for fx when x¯ = 0 x, y¯) = fx (¯ P (¯ x, x ¯ y¯), Q(¯ x, x ¯y¯) − y¯P (¯ x, x ¯y¯) x ¯ . 9) can be extended to x ¯ = 0 to yield an analytic vector ﬁeld on R2 . Such a vector ﬁeld is called a blow-up in the x-direction.