By Xiaoxin Liao, Pei Yu

ISBN-10: 1402084811

ISBN-13: 9781402084812

ISBN-10: 140208482X

ISBN-13: 9781402084829

Following the hot advancements within the box of absolute balance, Prof. Xiaoxin Liao, along with Prof. Pei Yu, has created a moment variation of his seminal paintings at the topic. Liao starts with an creation to the Lurie challenge and Lurie keep an eye on method, prior to relocating directly to the easy algebraic enough stipulations for absolutely the balance of independent and non-autonomous ODE platforms, in addition to numerous precise sessions of Lurie-type structures. the focal point of the ebook then shifts towards the recent effects and learn that experience seemed within the decade because the first version used to be released. This e-book is aimed for use by way of undergraduates within the components of utilized arithmetic, nonlinear keep watch over platforms, and chaos keep an eye on and synchronisation, yet can also be important as a reference for researchers and engineers. The e-book is self-contained, although a uncomplicated wisdom of calculus, linear approach and matrix thought, and usual differential equations is a prerequisite.

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**Additional info for Absolute Stability of Nonlinear Control Systems, 2nd Edition (Mathematical Modelling: Theory and Applications)**

**Example text**

Fn j (x j ) j=1 T . t. partial variables [68]. A11 A12 is symmetric, where A11 , A12 , A21 , Let us assume that A(ai j )n×n = A21 A22 and A22 are m × m, m × p, p × m, and p × p matrices, respectively, and m + p = n. 27. t. y if there are constants εi > 0 (i = 1, 2, . . , m) such that m m xT A x ≥ ∑ εi x2i xT A x ≤ − ∑ εi x2i . 28. t. y if and only if there exists a constant ε > 0 such that positive semi-deﬁnite an m × m matrix. A11 − ε Im×m A12 is A21 A22 A11 + ε Im×m A12 is negative semi-deﬁnite , where Im×m is A21 A22 24 2 Principal Theorems on Global Stability Proof.

M) and c j ≥ 0 ( j = m + 1, . . , n) such that ⎧ n ⎪ ⎪ −c j | f j j (x j )| + ∑ ci | f j j (x j )| < 0 for x j = 0, j = 1, . . , m, ⎪ ⎨ ⎪ ⎪ ⎪ ⎩ −c j | f j j (x j )| + i=1,i= j n ∑ ci | f j j (x j )| ≤ 0, j = m+1, . . t. the partial variable y. Proof. We construct the Lyapunov function n V (x) = ∑ ci |xi |. i=1 Clearly, m V (x) ≥ ∑ ci |xi | := ϕ (y) → +∞ y → +∞, as i=1 and ϕ (y) is positive deﬁnite. 18) ≤ ≤ n ∑ − c j | f j j (x j )| + ∑ − c j | f j j (x j )| + j=1 m j=1 <0 n ∑ ci | fi j (x j )| ∑ ci | fi j (x j )| i=1,i= j n i=1,i= j if y = 0.

T. the partial variable y. Proof. 29 and is omitted. 42. 23) satisfy the following conditions: 1. fii (t, xi ) xi < 0 for xi = 0, i = 1, 2, . . , m, and fii (t, xi ) xi ≤ 0, i = m + 1, 2, . . , n; 2. There exist functions Fii (xi ) on (−∞, +∞) (i = 1, 2, . . , n), which are continuous or have only ﬁnite discontinuous points of the ﬁrst or third kind such that ⎧ Fii (xi ) xi > 0 f or xi = 0, i = 1, 2, . . , m, ⎪ ⎪ ⎪ ⎪ ⎨ Fii (xi ) xi ≥ 0, i = m + 1, m + 2, . . , n, ⎪ ⎪ ⎪ ⎪ ⎩ ±∞ 0 Fii (xi ) dxi |Fii (xi )| ≤ | fii (t, xi )|, 3.

### Absolute Stability of Nonlinear Control Systems, 2nd Edition (Mathematical Modelling: Theory and Applications) by Xiaoxin Liao, Pei Yu

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