Peter Kloeden

Eventual stability in general control systems

The concept of eventual stability of a point was introduced by La Salle and Rath [3] in 1963 for ordinary differential equations to overcome the fact that for many perturbation and adaptive control problems the point in question may not be an equilibrium (invariant) point. There was at first some controversy as to whether or not eventual stabilities were a generalization of Lyapunov stabilities, as the latter imply that the set or point is invariant. This was settled by Lakshmikantham and Leela [2] with the generalisation of invariance to asymptotic invariance; eventually stable sets are asymptotically invariant, which enables us to regard Lyapunov stability as a special case of eventual stabilities. Axiomatically defined general control systems [l, 4, 5, 6, 71 provide an

Optimization Under Uncertainty

Classical optimization: Select a solution s∗ from a set of possible solutions S (e.g., st-paths, matchings, ...) that minimizes an objective function f : S → Q+0 Situation here: Problem generator PG generates instances I : S → Q+0 that differ due to noise; no knowledge about PG or the type of the noise Goal: Given I1, I2 generated by PG, find a solution that is likely to be good for other instances from PG Approach: • For each instance I, consider its ρ-approximation set Aρ(I) = {s ∈ S | I(s) ≤ ρ ·min s∈S I(s)} • For a well-tuned parameter ρ∗, select a solution from Aρ∗(I1) ∩ Aρ∗(I2) uniformly at random • Finding the “right” ρ∗ is nontrivial ; Use ρ∗ that maximizes the Similarity of I1 and I2, |Aρ∗(I1) ∩ Aρ∗(I2)| |Aρ∗(I1)||Aρ∗(I2)|