We consider the evolution of open planar curves by the steepest descent flow of a geometric functional, under different boundary conditions. We prove that, if any set of stationary solutions with fixed energy is finite, then a solution of the flow converges to a stationary solution as time goes to infinity. We also present a few applications of this result.

A class of neural networks that solve linear programming problems is analyzed. The neural networks considered are modeled by dynamic gradient systems that are constructed using a parametric family of exact (nondifferentiable) penalty functions. It is proved that for a given linear programming problem and sufficiently large penalty parameters, any trajectory of the neural network converges in fi...

Consider the mutually catalytic branching process with finite branching rate γ. We show that as γ → ∞, this process converges in finite-dimensional distributions (in time) to a certain discontinuous process. We give descriptions of this process in terms of its semigroup in terms of the infinitesimal generator and as the solution of a martingale problem. We also give a strong construction in ter...

We investigate the dynamics of an oscillator subject to dry friction via the following differential inclusion (S) ẍ(t) + ∂Φ(ẋ(t)) + ∇f(x(t)) 3 0, t ≥ 0, where f : R → R is a smooth potential and Φ : R → R is a convex function. The friction is modelized by the subdifferential term −∂Φ(ẋ). When 0 ∈ int(∂Φ(0)) (dry friction condition), it is shown in [1] that the unique solution of (S) converges i...

We consider the Gt/Mt/st + GIt many-server fluid queue with time-varying parameters, intended as an approximation for the corresponding stochastic queueing model, having general arrival process (the Gt), Markovian service (the Mt) and customer abandonment with mutually independent times to abandon with a general distribution (the +GIt) when there are many servers (the st) and the system experie...

We previously introduced and analyzed the Gt/Mt/st + GIt many-server fluid queue with time-varying parameters, intended as an approximation for the corresponding stochastic queueing model when there are many servers and the system experiences periods of overload. In this paper, we establish an asymptotic loss of memory (ALOM) property for that fluid model, i.e., we show that there is asymptotic...

In this paper, we propose a new fixed-lag smoother that estimates the fixed-delayed state for a continuous-time stochastic system. The estimation error variance of the proposed smoother is minimized under the constraint that the estimated state converges to the real state exactly in finite time after noises or uncertainties disappear. For numerical computing, the proposed smoother is represente...

We consider the large time behavior of three-dimensional Navier-Stokes flow around a rotating rigid body. Assume that angular velocity body gradually increases until it reaches small terminal one at certain finite and is fixed afterwards. then show fluid motion converges to steady solution as t → ∞.