By Xiao-Xin Liao

Following the new advancements within the box of absolute balance, Professor Xiaoxin Liao, at the side of Professor Pei Yu, has created a moment variation of his seminal paintings at the topic. Liao starts with an creation to the Lurie challenge and the Lurie keep watch over method, sooner than relocating directly to the straightforward algebraic enough 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 publication then shifts towards the recent effects and examine that experience seemed within the decade because the first variation was once released. This contains nonlinear keep an eye on structures with a number of controls, period keep watch over platforms, time-delay and impartial Lurie keep an eye on structures, structures defined by way of practical differential equations, absolutely the balance for neural networks, in addition to purposes to chaos keep an eye on and chaos synchronization.

This booklet is aimed toward undergraduates and teachers within the parts of utilized arithmetic, nonlinear keep an eye on structures and chaos regulate and synchronisation, yet can also be valuable as a reference paintings for researchers and engineers. The booklet is self-contained, although a uncomplicated wisdom of calculus, linear approach and matrix concept, and traditional differential equations is needed to achieve an entire knowing of the workings and methodologies mentioned inside.

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

**Sample text**

The research on Lurie’s problem has so far resulted in a number of monographs and several hundred of scientiﬁc publications. In fact, the pose of the Lurie problem actually initiated the research on the robust control and robust stability for nondeterministic systems or multivalued differential equations. It promoted the application and development of stability theory. In recent years, the developments in chaos control and chaos synchronization, neural networks, electrical systems have revealed that these research areas are closely related to Lurie control systems.

X, f (σ )), which is, however, impossible as shown above. Therefore, cT b + d TB−1 d ≥ 0. 11 following the order (1) =⇒ (2) =⇒ (3) =⇒ (1). Proof. 11 holds. 15) that ∀x0 = 0, U(x0 ) > 0 when cT x0 > 0. 4 Several Equivalent SANC for Negative Deﬁnite Derivatives 51 where m is a constant. Thus, cT x¯ = cT B−1 d − cT B−1 d − m T −1 c B c = m. cT B−1 c So for all m ∈ (0, +∞), U(x) ¯ > 0. However, on the other hand, U(x) ¯ = x¯T B x¯ − 2 d T x¯ − cT b, cT B−1 d − m T −1 T c (B ) cT B−1 c cT B−1 d − m −1 B c × B B−1 d − cT B−1 c cT B−1 d − m −1 B c − cTb.

T. the partial variable y. 22. 7 Systems with Separable Variables It will be shown later that by topological transformations a number of automatic control systems of different forms can be reduced to systems with separable variables or to systems with generalized separable variables. In this section, we discuss the systems with separable variables in detail. Consider a nonlinear system with separable variables [68]: n n j=1 j=1 ∑ f1 j (x j ), . . 17) where the dot denotes differentiation with respect to time t, fi j (x j ) ∈ C[R1 , R1 ], fi j (0) = 0, i, j = 1, 2, .