Download Constructions of Strict Lyapunov Functions by Michael Malisoff, Frédéric Mazenc PDF

By Michael Malisoff, Frédéric Mazenc

The development of strict Lyapunov services is a tough challenge that's of important ongoing study curiosity. even though speak Lyapunov functionality concept promises the life of strict Lyapunov services in lots of events, the Lyapunov capabilities that speak idea offers are frequently summary and nonexplicit, and as a result would possibly not lend themselves to engineering purposes. usually, even if a approach is understood to be good, one nonetheless wishes specific Lyapunov services; besides the fact that, as soon as a suitable strict Lyapunov functionality has been developed, many robustness and stabilization difficulties should be solved shortly via regular suggestions designs or robustness arguments. against this, non-strict Lyapunov features are usually comfortably 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 wide repertoire of Lyapunov structures for nonlinear platforms, targeting equipment for reworking non-strict Lyapunov capabilities into strict ones. Many vital sessions of dynamics are lined: Jurdjevic–Quinn structures; time-varying platforms pleasant LaSalle or Matrosov stipulations; slowly and swiftly time-varying platforms; adaptively managed dynamics; and hybrid platforms. The explicitness and straightforwardness of the structures cause them to compatible for suggestions layout, and for quantifying the consequences of uncertainty. Readers will enjoy the authors’ mathematical rigor and unifying, design-oriented procedure, in addition to the various labored examples, masking a number of purposes which are of compelling curiosity together with the adaptive regulate 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 dear and for graduate scholars of keep watch over idea it's going to even be an authoritative resource of knowledge on an important subject.

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V˙ 2 (t, x) is non-zero definite on {x ∈ RBn : W (x) = 0}. 4) that remains in RBn for all t ≥ t0 . For the proof, see [145]. 2 Non-strict Lyapunov Function 33 . where |x|M = inf{|x − q| : q ∈ M } is the distance of x from M . When G is independent of t, non-zero definiteness simply says that G is bounded away from zero on the part of any annulus around 0 that is close enough to M . 19) evolving on an open set X provided for each x0 ∈ M , the corresponding solution t → x(t, x0 ) remains in M for all times t ≥ 0.

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. 87). 2. 87) that has a bounded gradient. 3 Strictification It is tempting to surmise that non-strict Lyapunov functions with globally bounded gradients can be transformed into strict Lyapunov functions with globally bounded gradients. 5. 90) for all t ∈ R and x ∈ Rn . Assume that ∂f ∂x is bounded. 89) admits a strict Lyapunov function U such that ∂U (t, x) ≤ 1 and ∂t for all t ∈ R and x ∈ Rn .

The proof of this result also follows from the LaSalle Invariance Principle. However, it is far from clear how to construct CLFs for systems satisfying the Weak Jurdjevic Quinn Conditions. We address this CLF construction problem in Chap. 4. 21) with equilibrium state 0 by replacing the continuous time t ≥ 0 with the discrete time k ∈ {0, 1, 2, . 22) in the conditions defining Lyapunov functions. 22). 21), the sequence V (k, xk ) is decreasing in k, as long as xk = 0. If instead V is only a non-strict Lyapunov function, then V (k, xk ) is nonincreasing.

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