5,644 research outputs found
Distributed Lagrangian Methods for Network Resource Allocation
Motivated by a variety of applications in control engineering and information
sciences, we study network resource allocation problems where the goal is to
optimally allocate a fixed amount of resource over a network of nodes. In these
problems, due to the large scale of the network and complicated
inter-connections between nodes, any solution must be implemented in parallel
and based only on local data resulting in a need for distributed algorithms. In
this paper, we propose a novel distributed Lagrangian method, which requires
only local computation and communication. Our focus is to understand the
performance of this algorithm on the underlying network topology. Specifically,
we obtain an upper bound on the rate of convergence of the algorithm as a
function of the size and the topology of the underlying network. The
effectiveness and applicability of the proposed method is demonstrated by its
use in solving the important economic dispatch problem in power systems,
specifically on the benchmark IEEE-14 and IEEE-118 bus systems
Linearized Asymptotic Stability for Fractional Differential Equations
We prove the theorem of linearized asymptotic stability for fractional
differential equations. More precisely, we show that an equilibrium of a
nonlinear Caputo fractional differential equation is asymptotically stable if
its linearization at the equilibrium is asymptotically stable. As a consequence
we extend Lyapunov's first method to fractional differential equations by
proving that if the spectrum of the linearization is contained in the sector
\{\lambda \in \C : |\arg \lambda| > \frac{\alpha \pi}{2}\} where
denotes the order of the fractional differential equation, then the equilibrium
of the nonlinear fractional differential equation is asymptotically stable
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