Stabilization by unbounded‐variation noises

Abstract

In this paper, we claim the availability of deterministic noises for stabilization of the origins of dynamical systems, provided that the noises have unbounded variations. To achieve the result, we first consider the system representations of rough systems based on rough path analysis; then, we provide the notion of asymptotic stability for rough systems to analyze the stability for the systems. In the procedure, we also confirm that the system representations include stochastic differential equations; we also found that asymptotic stability for rough systems is the same property as uniform almost sure asymptotic stability provided by Bardi and Cesaroni. After the discussion, we confirm that there is a case that deterministic noises are capable of making the origin become asymptotically stable for rough systems while stochastic noises do not achieve the same stabilization results. ‡ In this paper, the term 'noises' is used when deterministic noises and stochastic noises are stated in a bundle. The deterministic noises are non-probabilistic signals having high-frequency oscillations, and the stochastic noises are probabilistic vibration signals such as Wiener processes.STABILIZATION BY UNBOUNDED-VARIATION NOISES 4127 subsets of Euclidean spaces. The stochastic stability property 'almost the same' as AS is uniform almost sure asymptotic stability (UASAS) [13,14]; all the sublevel sets of the related Lyapunov functions are invariant sets with probability one. However, achieving the property is generally difficult because the necessary and sufficient conditions are restrictive. Furthermore, there is a negative result that the non-AS origin never becomes the UASAS origin by the addition of any diffusion term [15]; that is, as long as employing the stabilization by noise with the use of Wiener processes, we should allow the possibility of all sample paths traveling to points very far away from the origin. Nevertheless, this paper claims that the stabilization by noise has still possibility to provide the non-AS origins with AS, provided that the resulting systems are neither ordinary nor stochastic differential equations. The assertion will be confirmed by the addition of deterministic noises having unbounded variations; this plan needs to employ a system representation by using rough path analysis [16][17][18][19] because the theory enables considering the dynamics with unbounded-variation functions such as Wiener processes and a particular kind of deterministic processes. The key point of the analysis is to classify external inputs by calculating the orders of the variations that is finite. Based on the information of the orders, rough paths and their dynamics-generally said to be rough differential equations, and rough systems in this paper-are obtained.Recently in [20], the stability notions for rough systems are defined with some important discussions; however, the claims have not been proven completely. Therefore, in this paper, we provide the proof of a concrete system representation of rough systems, show that ...

Simplification of control design for driftless nonholonomic systems based on rough path anlaysis

“…In the conference paper version [16], there is a mistake on the reference of (25). It is referred as in [11] despite the condition of p ∈ [1,3); in contrast, this paper considers p ∈ [1,4). Furthermore, the representation (23) provides more strict formulation of rough differential equations than [11] because it enables clarifying the relationship between rough integrals over dU 1 and their elements du 1 , du 2 and du 3 .…”

“…The proof is started from considering Theorem 4 in [11]; that is, we obtain the second-order geometric rough path U as…”

“…Recently, the investigation of dynamical systems driven by periodic signals with infinite frequencies is developed [11,12]. The works are based on rough path analysis, which deal with dynamical systems including thumping vibrations such as white noises [13,14].…”

“…The analysis considers the basis of the vibrations as unbounded-variation functions such as Wiener and Poisson processes, and then clarifies the seamless connection between ordinary and stochastic differential equations despite the unboundedness of the processes. The first authors' previous works [11,12] focuse attention on applying the rough path analysis into non-probabilistic systems driven by highly-oscillatory signals (rough signals) so that the approximate models concerned with Lie bracket motions become the true dynamics. As a result, the models are identified as strict representations of rough differential equations, which are extended notions of ordinary differential equations allowing the existence of unbounded-variation functions.…”

“…In rough path analysis, the notion of the order is provided for rough signals. The previous works [11,12] deal with just the second-order rough signals, which generate the simplest Lie bracket motions. Therefore, to generalize control analysis for nonholonomic systems via rough path analysis, we should derive concrete representations for rough differential equations including higher-order rough signals.…”