By Michael Malisoff, Frédéric Mazenc

The development of strict Lyapunov capabilities is a demanding challenge that's of important ongoing learn curiosity. even if communicate Lyapunov functionality idea promises the lifestyles of strict Lyapunov services in lots of occasions, the Lyapunov services that communicate conception offers are usually summary and nonexplicit, and hence would possibly not lend themselves to engineering purposes. frequently, even if a procedure is understood to be reliable, one nonetheless wishes specific Lyapunov capabilities; even though, as soon as a suitable strict Lyapunov functionality has been built, many robustness and stabilization difficulties should be solved shortly via common suggestions designs or robustness arguments. against this, non-strict Lyapunov features are frequently with no trouble built, e.g., from passivity, backstepping, or forwarding (especially within the time various context), or through the use of the Hamiltonian in Euler–Lagrange systems.

*Constructions of Strict Lyapunov Functions* includes a extensive repertoire of Lyapunov structures for nonlinear structures, targeting tools for remodeling non-strict Lyapunov services into strict ones. Many vital sessions of dynamics are coated: Jurdjevic–Quinn structures; time-varying platforms pleasurable LaSalle or Matrosov stipulations; slowly and speedily time-varying structures; adaptively managed dynamics; and hybrid platforms. The explicitness and ease of the structures lead them to appropriate for suggestions layout, and for quantifying the consequences of uncertainty. Readers will enjoy the authors’ mathematical rigor and unifying, design-oriented strategy, in addition to the varied labored examples, overlaying a number of functions which are of compelling curiosity together with the adaptive keep watch over of chemostats and the stabilization of underactuated ships.

Researchers from applied-mathematical and engineering backgrounds operating in nonlinear and dynamical structures will locate this monograph to be most respected and for graduate scholars of keep watch over idea it's going to even be an authoritative resource of data on a crucial subject.

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**Extra resources for Constructions of Strict Lyapunov Functions**

**Sample text**

The constants depend on the function V selected. 7 Lyapunov Functions with Bounded Gradients 47 Lyapunov function V does not allow us to determine an approximate upper bound for this gain. We illustrate this using the simple one-dimensional nonlinear system x˙ = −x − x3 + u y=x. 64) holds for all x. 64) holds for all x. 66) which cannot be satisﬁed for all values of x. 55). This provides yet another reason for wanting explicit Lyapunov functions. 55) can be deduced from the mere existence of a strict Lyapunov function, as provided by the converse Lyapunov theorem.

30), which will lead to explicit functions β and γ in the ISS estimate. 26) satisﬁes max i=1,2 ∂Vs (t, x) ≤ 17|x|1 ∀x = (x1 x2 ) ∈ R2 and t ≥ 0. 32) 1 ≤ − 12 |x|2 + 3 × 1022 δ12 (t), by the triangle inequality pq ≤ 1 2 p 2ε + 2ε q 2 applied to the terms in braces with p = |x|, q = 102|δ1 (t)|, and ε = 6. 30). As we saw in Sect. 3, explicit ISS Lyapunov functions lead to explicit expressions for the functions β ∈ KL and γ ∈ K∞ in the ISS estimate. 30). 27) gives 1 V˙ s (t, x) ≤ − 204 Vs (t, x) + 3 × 1022δ12 (t).

42) satisﬁes the asymptotic gain property. 42) satisﬁes lim sup |x(t, x0 , u)| ≤ γ(|u|∞ ) t→+∞ for all x0 ∈ Rn and u ∈ M(Rm ). 42) under disturbances that converge to 0. 42) is a linear time-invariant system x˙ = Ax + Bu. In fact, 0-GAS linear time-invariant systems satisfy the converging-input converging state (CICS) property which says that trajectories converge to zero when the inputs do [165]. 6 Comments 21 noted in [165], the system x˙ = −x + (x2 + 1)u has divergent solutions when u(t) = (2t + 2)−1/2 .