This paper presents an alternative approach for obtaining a converse Lyapunov theorem for discrete-time systems. The proposed approach is constructive, as it provides an explicit Lyapunov function. The developed converse theorem establishes existence of global Lyapunov functions for globally exponentially stable (GES) systems and semi-global practical Lyapunov functions for globally asymptotically stable systems. Furthermore, for specific classes of systems, the developed converse theorem can be used to establish non-conservatism of a particular type of Lyapunov functions. Most notably, a proof that conewise linear Lyapunov functions are non-conservative for GES conewise linear systems is given and, as a by-product, tractable construction of polyhedral Lyapunov functions for linear systems is attained.

In this paper ISS small-gain theorems for discrete-time systems are stated, which do not require input-to-state stability (ISS) of each subsystem. This approach weakens conservatism in ISS small-gain theory, and for the class of exponentially ISS systems we are able to prove that the proposed relaxed small-gain theorems are non-conservative in a sense to be made precise. The proofs of the small-gain theorems rely on the construction of a dissipative finite-step ISS Lyapunov function which is introduced in this work. Furthermore, dissipative finite-step ISS Lyapunov functions, as relaxations of ISS Lyapunov functions, are shown to be sufficient and necessary to conclude ISS of the overall system.

No abstract

This paper addresses characterizations of integral input-to-state stability (iISS) for hybrid systems. In particular, we give a Lyapunov characterization of iISS unifying and generalizing the existing theory for pure continuous-time and pure discrete-time systems. Moreover, iISS is related to dissipativity and detectability notions. Robustness of iISS to sufficiently small perturbations is also investigated. As an application of our results, we provide a maximum allowable sampling period guaranteeing iISS for sampled-data control systems with an emulated controller. This work was mostly done while N. Noroozi was at Shiraz University. for all (t k , k) ∈ dom w with k ∈ {1, . . . , J 1 }. Let w(t J 1 , J 1 ) = 0, otherwise according to (86) and (87), w(t, k) ≡ 0 for all (t, j) ∈ domw such that (t, j) ≥ (t J 1 , J 1 ); so the proof is complete. Assume that t J 1 = t J 1 +1 , so w(t, J 1 ) flows for all t in the second flow interval of domw. Then using Lemma 31, we getIt follows from (93) and the fact thatβIt follows with the fact that ρ 2 (w(t J 1 , J 1 )) ≥ ρ 2 (w 0 ) thatBy application of (85), we get w(t, J 1 ) ≤ β(w 0 , ρ 2 (w 0 )(t + J 1 )).By reapplication of (85), we haveNow J 2 jumps happen in a row. Given that Lemma 32 for any (t k , k) ∈ domw with k ∈ {J 1 + 1, . . . , J 2 + J 1 } gives w(t k , k) ≤ β(w(t J 1 +1 , J 1 ), ρ 2 (w(t J 1 +1 , J 1 ))(k − J 1 )).It follows from (94) and the fact thatβ(w 0 , t J 1 +1 , J 1 ) = β(w 0 , ρ 2 (w 0 )(t J 1 +1 + J 1 )) that w(t k , k) ≤ β(β(w 0 , ρ 2 (w 0 )(t J 1 +1 + J 1 )), ρ 2 (w(t J 1 +1 , J 1 ))(k − J 1 )).From ρ 2 (w(t J 1 +1 , J 1 )) ≥ ρ 2 (w 0 ), we have w(t k , k) ≤ β(β(w 0 , ρ 2 (w 0 )(t J 1 +1 + J 1 )), ρ 2 (w 0 )(k − J 1 )).By application of (85), we haveBy reapplication of (85) and the fact that t J 1 +1 = · · · = t J 1 +J 2 , we getfor all (t k , k) ∈ dom w with k ∈ {J 1 + 1, . . . , J 2 + J 1 }.By repeated application of the above arguments (i.e. concatenating flows and jumps, the fact that ρ 2 (w(t, j)) ≥ ρ 2 (w 0 ) and exploiting (85)) yield w(t, j) ≤β(w 0 , t, j) ∀(t, j) ∈ domw.It is easy to see that when w starts with jumps the above arguments essentially hold. Eventually, the properties (89)-(91) immediately follow from the very definition ofβ and exploiting (85). This completes the proof.

International audienceWe provide a homotopy algorithm that computes a decay point of a monotone operator, i.e., a point whose image under the monotone operator is strictly smaller than the preimage. For this purpose we use a fixed point algorithm and provide a function whose fixed points correspond to decay points of the monotone operator. This decay point plays a crucial role in checking, in a semi-global fashion, the local input-to-state stability of an interconnected system numerically and in the numerical construction of local input-to-state stability (LISS) Lyapunov functions. We give some improvements of this algorithm and show the advantage to an earlier approach based on the algorithm of Eaves