By Xiao-Xin Liao

Following the new advancements within the box of absolute balance, Professor Xiaoxin Liao, along with Professor Pei Yu, has created a moment version of his seminal paintings at the topic. Liao starts with an creation to the Lurie challenge and the Lurie regulate method, sooner than relocating directly to the straightforward algebraic adequate stipulations for absolutely the balance of self sufficient and non-autonomous ODE structures, in addition to a number of certain sessions of Lurie-type structures. the focal point of the booklet then shifts towards the recent effects and examine that experience seemed within the decade because the first variation used to be released. This comprises nonlinear keep watch over platforms with a number of controls, period regulate platforms, time-delay and impartial Lurie regulate platforms, structures defined by means of sensible differential equations, absolutely the balance for neural networks, in addition to functions to chaos keep an eye on and chaos synchronization.

This publication is geared toward undergraduates and teachers within the parts of utilized arithmetic, nonlinear keep watch over structures and chaos keep an eye on and synchronisation, yet can also be important as a reference paintings for researchers and engineers. The publication is self-contained, although a uncomplicated wisdom of calculus, linear procedure and matrix idea, and usual differential equations is needed to achieve an entire knowing of the workings and methodologies mentioned inside of.

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**Additional info for Absolute stability of nonlinear control systems**

**Example text**

Xm )T , z = (xm+1 , xm+2 , . . , xn )T , fi j (t, x j ) ∈ C[I × R, R], fi j (0) = 0, i, j = 1, 2, . . , n. 16) is unique. 41. If there exist functions ϕi (xi ) on (−∞, +∞) (i = 1, 2, . . , at the discontinuous points, the left- and right-hand limits of ϕ (xi ) exist) such that 1. ϕi (xi ) xi > 0 for xi = 0, i = 1, 2, . . , m; ϕ (xi ) xi ≥ 0, i = m + 1, . . , n; 2. 0±∞ ϕi (xi ) dxi = +∞, i = 1, 2, . . , m; 3. t. the partial variable y. Proof. 29 and is omitted. 42. 23) satisfy the following conditions: 1.

T. the partial variable y. Proof. 26 to complete the proof. 33. 18) satisﬁes 1. 31; 2. There exist n functions ci > 0 (i = 1, 2, . . , 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.

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. 34. 18) satisﬁes the following conditions: 1. 31; f (x ) 2. , i = j, i, j = 1, . . , n; 3. ⎤ ⎡ 1 −b21 · · · −bn1 ⎢ −b21 1 · · · −bn2 ⎥ A˜ 11 A˜ 12 ⎥ ⎢ , A˜ := ⎢ . ..