By Michael Malisoff, Frédéric Mazenc
The development of strict Lyapunov capabilities is a hard challenge that's of important ongoing examine curiosity. even though communicate Lyapunov functionality thought promises the lifestyles of strict Lyapunov features in lots of occasions, the Lyapunov capabilities that communicate conception offers are usually summary and nonexplicit, and accordingly won't lend themselves to engineering functions. frequently, even if a method is understood to be solid, one nonetheless wishes specific Lyapunov features; although, as soon as a suitable strict Lyapunov functionality has been built, many robustness and stabilization difficulties may be solved presently via commonplace suggestions designs or robustness arguments. against this, non-strict Lyapunov services are usually with ease 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 features a extensive repertoire of Lyapunov structures for nonlinear platforms, concentrating on tools for remodeling non-strict Lyapunov capabilities into strict ones. Many very important sessions of dynamics are lined: Jurdjevic–Quinn structures; time-varying structures pleasurable LaSalle or Matrosov stipulations; slowly and quickly time-varying platforms; adaptively managed dynamics; and hybrid structures. The explicitness and ease of the buildings cause them to compatible for suggestions layout, and for quantifying the results of uncertainty. Readers will enjoy the authors’ mathematical rigor and unifying, design-oriented process, in addition to the varied labored examples, protecting a number of functions which are of compelling curiosity together with the adaptive keep an eye on of chemostats and the stabilization of underactuated ships.
Researchers from applied-mathematical and engineering backgrounds operating in nonlinear and dynamical platforms will locate this monograph to be most beneficial and for graduate scholars of keep watch over concept it is going to even be an authoritative resource of knowledge on a crucial subject.
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Additional resources for Constructions of Strict Lyapunov Functions
This proves the proposition. 3. 82) ∂x for all x ∈ R2 . Proof. Suppose the contrary. 67) satisﬁes V˙ ≤ C|d(t)| . 77) give V˙ ≤ C|x2 (0)e−t | . 85) for all t ≥ 0. 2, x1 (t) = et x1 (0) for all t ≥ 0, so lim V (x(t)) = +∞. 85) yield a contradiction. 2 Dependence on Coordinates We next show that the property of having no strict Lyapunov function with a bounded gradient is coordinate dependent. 4. 69). 67) with d ≡ 0 into Z˙ 1 = −Z1 Z˙ 2 = −2Z2 . 67) is globally linearizable. Proof. Routine calculations yield ∂Z1 ∂x1 (x) = eP (x1 x2 ) [1 + x1 x2 P (x1 x2 )] = eP (x1 x2 ) 1 + 1 + g(x1 x2 ) 2 − g(x1 x2 ) = eP (x1 x2 ) 3 >0.
Also, when X = Rn , we use the term radially bounded to mean properness, which in this case gives the condition that lim inf V (t, x) = +∞. 2) provided the following hold: 1. V is radially unbounded; and 2. Lϕ1 V (x) ≥ 0 ⇒ [x = 0 or Lϕ2 V (x) = 0]. 2) has the small control property provided: For each ε > 0, there is a δ > 0 such that if 0 = |x| < δ, then there is a u ∈ U such that |u| < ε and ∇V (x)ϕ1 (x) + ∇V (x)ϕ2 (x)u < 0. A special case of Artstein’s Theorem  says the following: Let V (x) be a positive deﬁnite radially unbounded function.
Analogous arguments can be carried out for the iISS case; see . This motivates our search for explicit iISS Lyapunov functions as well. 2 Non-strict Lyapunov Function Our main building blocks for strict Lyapunov functions will be non-strict Lyapunov functions (which are also called weak Lyapunov functions). Nonstrict Lyapunov functions V are deﬁned in exactly the same way as strict Lyapunov functions except instead of the decay condition V˙ < 0 outside the equilibrium state, we have V˙ ≤ 0. A positive deﬁnite function V that satisﬁes all requirements for being a (non-)strict Lyapunov function except for properness is called a (non-)strict Lyapunov-like function.
Constructions of Strict Lyapunov Functions by Michael Malisoff, Frédéric Mazenc