Weaker Notion Of Stability Research Articles

Weak-subdifferentials for vector functions and applications to multiobjective semi-infinite optimization problems

ABSTRACTA notion of generalized weak-subdifferentials for vector functions on a linear space is proposed. Properties of the weak-subdifferentials are investigated. A notion of weak-stability for a nonconvex optimization problems is introduced. Finally, relations between weak-stability and strong duality of the problem are established.

On the notion of weak stability and related issues of hybrid diffusion systems

This work is concerned with the weak stability of hybrid diffusion systems, which consist of a number of diffusions modulated by a jump process. First, we show that when the jump component is nearly completely decomposable into a number of ergodic jump processes, the overall system is still weakly stable (or positive recurrent). Next we show that even if the process contains transient states, the positive recurrence can still be preserved. Then, we examine the asymptotic distribution when only one ergodic group of states is involved in the jump process and the state space of the continuous state belongs to a compact set. Our attention is devoted to asymptotic distribution in this case. The distribution is obtained by utilizing a spectrum gap property of the underlying process.

Authors' reply [to comments on "control over noisy channels" by S. Tatikonda and S. Mitter

The authors reply to the comment by A.S. Matveev and A.V. Savkin (see ibid., p.2105-10, 2005). We acknowledge the error pointed out in the comments, but wish to state that this was an oversight also missed by the referee of the paper. That one cannot observe an unstable process over a noisy channel in this strong sense has been known to us since at least 2000. Sahai's Ph.D. dissertation (A. Sahai, 2001), written under the direction of one of the authors, investigates the information theory needed for unstable processes. There, he discusses the need for a weaker notion of stability (an appropriate moment stability). This led to the development of his anytime capacity framework. An explicit counterexample is given by A. Sahai (2001), showing that the result in question cannot be true. These ideas were further discussed by A. Sahai (2000; 2001) and Sahai and Mitter (2000). For plants without disturbances, S. Tatikonda and S. Mitter (2004) shows that the traditional Shannon capacity rate condition is sufficient to ensure stability over an erasure channel. The addition of plant disturbances requires a weaker notion of stability.

A dual to Lyapunov's stability theorem

Lyapunov's second theorem is a standard tool for stability analysis of ordinary differential equations. Here we introduce a theorem which can be viewed as a dual to Lyapunov's result. From existence of a scalar function satisfying certain inequalities it follows that “almost all trajectories” of the system tend to zero. The scalar function has a physical interpretation as the stationary density of a substance that is generated in all points of the state space and flows along the system trajectories. If the stationary density is bounded everywhere except at a singularity in the origin, then almost all trajectories tend towards the origin. The weaker notion of stability allows for applications also in situations where Lyapunov's theorem cannot be used. Moreover, the new criterion has a striking convexity property related to control synthesis.