Franco Blanchini

Giuseppe Belgioioso, Filippo Fabiani, Franco Blanchini et al.

We adopt an operator-theoretic perspective to study convergence of linear fixed-point iterations and discrete- time linear systems. We mainly focus on the so-called Krasnoselskij-Mann iteration x(k+1) = ( 1 - α(k) ) x(k) + α(k) A x(k), which is relevant for distributed computation in optimization and game theory, when A is not available in a centralized way. We show that convergence to a vector in the kernel of (I-A) is equivalent to strict pseudocontractiveness of the linear operator x -> Ax. We also characterize some relevant operator-theoretic properties of linear operators via eigenvalue location and linear matrix inequalities. We apply the convergence conditions to multi-agent linear systems with vanishing step sizes, in particular, to linear consensus dynamics and equilibrium seeking in monotone linear-quadratic games.

Filippo Fabiani, Giuseppe Belgioioso, Franco Blanchini et al.

State convergence is essential in several scientific areas, e.g. multi-agent consensus/disagreement, distributed optimization, monotone game theory, multi-agent learning over time-varying networks. This paper is the first on state convergence in both continuous- and discrete-time linear systems affected by polytopic uncertainty. First, we characterize state convergence in linear time invariant systems via equivalent necessary and sufficient conditions. In the presence of uncertainty, we complement the canonical definition of (weak) convergence with a stronger notion of convergence, which requires the existence of a common kernel among the generator matrices of the difference/differential inclusion (strong convergence). We investigate under which conditions the two definitions are equivalent. Then, we characterize weak and strong convergence by means of Lyapunov and LaSalle arguments, (linear) matrix inequalities and separability of the eigenvalues of the generator matrices. We also show that, unlike asymptotic stability, state convergence lacks of duality.

Franco Blanchini, Filippo Fabiani, Sergio Grammatico

Merging two Control Lyapunov Functions (CLFs) means creating a single "new-born" CLF by starting from two parents functions. Specifically, given a "father" function, shaped by the state constraints, and a "mother" function, designed with some optimality criterion, the merging CLF should be similar to the father close to the constraints and similar to the mother close to the origin. To successfully merge two CLFs, the control-sharing condition is crucial: the two functions must have a common control law that makes both Lyapunov derivatives simultaneously negative. Unfortunately, it is difficult to guarantee this property a-priori, i.e., while computing the two parents functions. In this paper, we propose a technique to create a constraint-shaped "father" function that has the control-sharing property with the "mother" function. To this end, we introduce a partial control-sharing, namely, the control-sharing only in the regions where the constraints are active. We show that imposing partial control-sharing is a convex optimization problem. Finally, we show how to apply the partial control-sharing for merging constraint-shaped functions and the Riccati-optimal functions, thus generating a CLF with bounded complexity that solves the constrained linear-quadratic stabilization problem with local optimality.