By Xiaoxin Liao, Pei Yu
Following the hot advancements within the box of absolute balance, Prof. Xiaoxin Liao, along side Prof. Pei Yu, has created a moment version of his seminal paintings at the topic. Liao starts off with an advent to the Lurie challenge and Lurie regulate process, prior to relocating directly to the straightforward algebraic adequate stipulations for absolutely the balance of independent and non-autonomous ODE platforms, in addition to numerous exact periods of Lurie-type structures. the point of interest of the ebook then shifts towards the hot effects and examine that experience seemed within the decade because the first version was once released. This ebook is aimed for use by way of undergraduates within the parts of utilized arithmetic, nonlinear keep watch over structures, and chaos regulate and synchronisation, yet can also be helpful as a reference for researchers and engineers. The ebook is self-contained, even though a uncomplicated wisdom of calculus, linear procedure and matrix idea, and traditional differential equations is a prerequisite.
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Additional resources for Absolute Stability of Nonlinear Control Systems, 2nd Edition (Mathematical Modelling: Theory and Applications)
18) satisﬁes 1. fii (xi ) xi < 0 for xi = 0, i = 1, 2, . . , m; fii (xi ) xi ≤ 0, i = m + 1, . . , n; 2. ±∞ fii (xi ) dxi 0 = −∞, i = 1, 2, . . , m; 3. There are constants ci > 0 (i = 1, 2, . . , m), c j ≥ 0 ( j = m + 1, . . , n), ε > 0 such that ε Im×m 0 A(ai j )n×n + is negative semi − deﬁnite, 0 0 n×n where ⎧ ⎨ 1 ci fi j (x j ) c j f ji (xi ) + , xi x j = 0, ai j (x) = i, j = 1, 2, . . t. the partial variable y. 26 2 Principal Theorems on Global Stability Proof. We construct the Lyapunov function n V (x) = − ∑ xi i=1 0 ci fii (xi ) dxi .
The set M if for any ε > 0, there exists δ (ε ) such that d(x0 , M) < δ (ε ) implies d(x(t,t0 ; x0 ), M) < ε for all t ≥ t0 , and for any x0 ∈ Rn , lim d(x(t,t0 ; x0 ), M) = 0. 24. Suppose that V (x) ∈ C[Rn , R1 ] and that V (x) satisﬁes ϕ1 (d(x, M)) ≤ V (x) ≤ ϕ2 (d(x, M)), dV dt ≤ −ψ (d(x, M)), ϕ1 , ϕ2 ∈ KR, ψ ∈ K. t. the set M. Proof. For any ε > 0, choosing δ (ε ) := ϕ2−1 (ϕ1 (ε )), we write ϕ1 (d(x(t,t0 ; x0 ), M)) ≤ V (x(t,t0 ; x0 )) ≤ V (x0 ) ≤ ϕ2 (d(x0 , M)) < ϕ2 (δ (ε )) if d(x0 , M) < δ (ε ).
The partial variable y. Proof. t. the partial variable y. 33. 37. 18) satisﬁes 1. fi (xi ) xi > 0 for xi = 0 aii < 0, i = m + 1, . . , n, and fi (xi )xi ≥ 0, aii ≤ 0, i = 1, 2, . . , n; 2. There exist constants ci > 0 (i = 1, 2, . . , m), c j ≥ 0 ( j = m + 1, . . , n) such that ⎧ n ⎪ ⎪ −c |a | + ci |ai j | < 0, j = 1, . . , m, ⎪ j j j ∑ ⎨ ⎪ ⎪ ⎪ ⎩ −c j |a j j | + i=1,i= j n ∑ ci |ai j | ≤ 0, j = m + 1, . . t. the partial variable y. Proof. We construct the Lyapunov function n V (x) = ∑ ci |xi |.
Absolute Stability of Nonlinear Control Systems, 2nd Edition (Mathematical Modelling: Theory and Applications) by Xiaoxin Liao, Pei Yu