Stochastic control stabilizing unstable or chaotic maps

TL;DR

This paper introduces a stochastic control method that stabilizes unstable and chaotic maps by noise, relaxing previous assumptions and broadening the range of applicable maps and parameters, with simulations demonstrating effectiveness.

Contribution

It presents a new stochastic control approach that stabilizes chaotic maps with fewer restrictions and wider parameter ranges than previous methods.

Findings

Noise can stabilize unstable equilibria.

Stabilization effectiveness depends on map instability and initial conditions.

Bounded noise achieves stabilization without initial value limitations.

Abstract

The paper considers a stabilizing stochastic control which can be applied to a variety of unstable and even chaotic maps. Compared to previous methods introducing control by noise, we relax assumptions on the class of maps, as well as consider a wider range of parameters for the same maps. This approach allows to stabilize unstable and chaotic maps by noise. The interplay between the map properties and the allowed neighbourhood where a solution can start to be stabilized is explored: as instability of the original map increases, the interval of allowed initial conditions narrows. A directed stochastic control aiming at getting to the target neighbourhood almost sure is combined with a controlling noise. Simulations illustrate that for a variety of problems, an appropriate bounded noise can stabilize an unstable positive equilibrium, without a limitation on the initial value.

Equations

$$z_{n+1}=\mathrm{f}(z_n),n\in \mathbb{N},z_0>0,$$

$$\mathrm{f}(z)=K-(1+q)(z-K)+\varphi (z-K),|\varphi (z-K)|\le C|z-K{|}^{1+\kappa }.$$

$$z_{n+1}=\mathrm{f}(z_n)-{\alpha }_j(1+{\ell }_c{\chi }_{n+1})\frac{\mathrm{f}(z_n)-K}{|\mathrm{f}(z_n)-K|}+\ell {\zeta }_{n+1},z_n\in J^{(j)},j=0,\dots ,k-1,$$

$$z_{n+1}=\mathrm{f}(z_n)+\sigma {\xi }_{n+1}(z_n-K),n\ge \tau ,z_{\tau }\in (K-\delta ,K+\delta ).$$

$$-\lambda :=\mathbb{E}\ln |1+q-\sigma {\xi }_{n+1}|<0.$$

$$I_{\delta }:=(-\delta ,\delta ),I_u:=(-u,u),I_{u,K}:=(K-u,K+u),I_{\delta ,K}:=(K-\delta ,K+\delta ).$$

$$\sum _{i=1}^{\infty }\frac{{\theta }_i^2}{b_i^2}<\infty .$$

$$\begin{gathered} \mathbb{P}[\text{There exists }\mathcal{N}=\mathcal{N}(\bar{J})<\infty \\ \text{such that }{\xi }_{\mathcal{N}}\in (a,b],{\xi }_{\mathcal{N}+1}\in [a,b),\dots ,{\xi }_{\mathcal{N}+\bar{J}}\in (a,b]]=1. \end{gathered}$$

$$z_{n+1}=\mathrm{f}(z_n),n\in \mathbb{N},z_0>0,$$

$$\mathrm{f}(z)=K-(1+q)(z-K)+\varphi (z-K),$$

$$|\varphi (x)|\le C|x{|}^{1+\kappa },\forall x\in I_u.$$

$$x:=z-K,f(x)=\mathrm{f}(z-K)-K,\text{so }x\in I_u\text{when}z\in I_{u,K}.$$

$$\begin{array}{rl}& f(x)<0,x\in (0,u),f(x)>0,x\in (-u,0),\\ & f(x)=-(1+q)x+\varphi (x),x\in I_u,\end{array}$$

$$x_{n+1}=f(x_n)$$

$$x_{n+1}=f(x_n)+\sigma {\xi }_{n+1}{x}_n.$$

$${\Theta }_n:=1+q-\sigma {\xi }_n$$

$$f(x_n)+\sigma {\xi }_{n+1}{x}_n=-{\Theta }_{n+1}{x}_n+\varphi (x_n),$$

$$x_{n+1}=-(1+q-\sigma {\xi }_{n+1})x_n+\varphi (x_n)=-{\Theta }_{n+1}{x}_n+\varphi (x_n).$$

$$v_n:=\ln |{\Theta }_n|,$$

$$\begin{array}{rl}& 1+q-\sigma \le {\Theta }_n\le 1+q+\sigma ,\ln |1+q-\sigma |\le v_n\le \ln (1+q+\sigma ),\\ & |{\Theta }_n|\le 1+q+\sigma =:\bar{\Theta },|v_n|\le \max \left\{\left|\ln |1+q-\sigma |\right|,\ln (1+q+\sigma )\right\}=:\bar{v}.\end{array}$$

$$\mathbf{E}{v}_n=\mathbf{E}\ln |{\Theta }_n|=:-\lambda <0.$$

$$\frac{1}{n}\sum _{i=1}^n{v}_i\to -\lambda ,\text{as}n\to \infty .$$

$$-(\lambda +\epsilon )n<\sum _{i=1}^n{v}_i\le -(\lambda -\epsilon )n.$$

$$z_{n+1}=\mathrm{f}(z_n)+\sigma {\xi }_{n+1}(z_n-K),z_{\tau }\in I_{\delta ,K},n\ge \tau .$$

$$|z_m-K|\le \eta e^{-\varsigma (m-n_0)},\forall m\ge n_0.$$

$$\tau =\inf \{n:z_n\in I_{\delta ,K}\}.$$

$$C{u}^{\kappa }<\frac{1}{2}.$$

$$q<\sigma <2+q;$$

$$\mathbb{P}\{\xi \in (1-\iota ,1]\}=p_{\iota }>0,\text{mbox}foreach\iota \in \left(0,1-\frac{q}{\sigma }\right).$$

$${\underline{\Theta }}_{\iota }:=1+q-(1-\iota )\sigma ,\bar{\Theta }:=1+q+\sigma .$$

Full text

{classcode}

39A50; 37H10; 93D15; 39A30

1 Introduction

Significant interest to discrete models is stimulated by complicated types of behavior exhibited even by simple maps.

The idea of stabilizing an unstable equilibrium of differential equations by noise originates from the work of R. Khasminskii on stochastic stability [22], see also the most recent edition of the monograph [25]. In 1983, the possibility to stabilize a linear system by noise was demonstrated in the paper of L. Arnold et al [5]. This approach was expanded and developed in later works: for stochastic differential and functional differential equations see, e.g. [2, 4, 16, 26], and for stochastic difference equations in [1, 3].

Kolmogorov’s Law of Large Numbers was applied in the proof of stability of the zero equilibrium for linear and nonlinear stochastic non-homogeneous equations in [9, 10], and for systems with square nonlinearities in [23]. This approach was originated by H. Kesten (see e.g. [21] for a linear model, and [24] for convergence in probability).

Introduction of the stochasticity into the population dynamics description, as well as into controls, is quite a natural part of a model design due to many reasons. This includes an extrinsic noise, which may be described by a state-independent, or additive, stochastic perturbation. The implementation of control cannot be done precisely, leading to a state-dependent, or multiplicative, stochastic perturbation. The influence of stochasticity on population survival, chaos control and eventual cyclic behavior was investigated in [12, 13], different types of control which include stochastic perturbations were considered for the scalar case in [11, 15], see the recent papers [7, 8, 18] and references therein for systems.

In [14] it was shown that in certain cases, an additive non-decaying stochastic perturbation could completely diminish the Alee effect. We were able to prove this result, using the fact that for a given sequence $(\xi_{n})_{n\in\mathbb{N}}$ of independent identically distributed random variables, for a subinterval of their support and any number $\bar{J}\in\mathbb{N}$, there exists a random number $\mathcal{N}$, starting from which $\bar{J}$ random variables, in a row, take values in this interval with probability 1. The influence of stochastic perturbations on the Allee effect in discrete systems was also recently studied in [6, 27].

Concerning the noise component of controls in [11, 15], it slightly reduced the range of control parameters where stabilization is guaranteed. Unlike [11, 15], in the present paper noise is an important stabilizing factor: an otherwise unstable positive equilibrium becomes stable after introducing an unstructured noise, once the initial value is close enough to this equilibrium. The idea is inspired by both physical and biological models. As examples, we consider population dynamics models: logistic and Ricker. The fact that stochastic perturbations can stabilize an unstable equilibrium was discovered for physical models in the 1950s, following the well-known example of the pendulum of Kapica [20]. Here we generalize it to a wide range of population dynamics models, in a discrete setting.

We consider a deterministic difference equation

[TABLE]

where ${\rm f}:[0,\infty)\to:[0,\infty)$ is continuous and has a unique positive fixed point $K>0$. We suppose that the equilibrium $K$ is unstable, and moreover, in some interval $(K-u,K+u)$, the function $\rm f$ can be represented for some $u,q,C,\kappa>0$ as

[TABLE]

Our aim is to construct a stochastic control which stabilizes the equilibrium $K$ for initial values $x_{0}\in(K-u,K+u)$ with a given probability. Since equation (1.1) is applied to population dynamics models, we consider only bounded noises, which are supposed to be mutually independent. Compared to previous methods introducing control by noise, we relax assumptions on the class of maps and consider a wider range of parameters for the function ${\rm f}$. In particular, the original map can be chaotic. However, the stronger the map instability is, the smaller is the allowed neighbourhood where a solution can start. With chaotic maps and values coming occasionally as close to the equilibrium as required, with a probability close to one the solution eventually at least once enters the target neighbourhood. However, the situation changes if there is an attractive cycle or other orbit separated from the equilibrium point. To attain the target domain, we apply a directed stochastic control which is stopped once a solution is in the required neighbourhood.

The main type of control includes noise only, it can be considered as cost-free, if such a noise is natural. However, first an additional control should push a solution into a smaller target interval $(K-\delta,K+\delta)$. Note that to reach this interval, various control methods can be applied, for example, Prediction Based Control [11] or non-stochastic Target Oriented Control [17, 19]. As stochasticity is an intrinsic part of a control, we incorporate noise at this preliminary control stage as well. Prediction Based and Proportional Feedback controls [11, 15] included either multiplicative or additive noise. We introduce a Directed Walk Control (DWC)

[TABLE]

acting on a finite number of nested intervals $J^{(j)}\subset J^{(j-1)}$, starting from some interval $J^{(0)}=(K-u,K+u)$, until the solution $z$, after a finite number of steps which can be explicitly computed, reaches the target interval $J^{(k)}=(K-\delta,K+\delta)$. Here $(\chi_{n})_{n\in\mathbb{N}}$ and $(\zeta_{n})_{n\in\mathbb{N}}$ are sequences of independent bounded noises. In addition to its ability to reach a target interval, which any of the mentioned methods can, DWC includes, unlike [11, 15], both a multiplicative and an additive noise, which is more natural in the setting of the model. In addition, the total number of steps to reach the target interval is estimated (see the appendix), the step where the solution enters it is random, however, the maximal cost can be evaluated analytically. This is, generally, not an easy task, see, for example, [17].

As soon as, at a random moment $\tau=\inf\{n\in\mathbb{N}:z_{n}\in(K-\delta,K+\delta)\}$, the solution $z$ reaches the target interval $(K-\delta,K+\delta)$, the second type of control is launched, which eventually brings $z$ to the equilibrium. This is the main control, which is described in Section 7 and is based on the application of the Kolmogorov’s Law of Large Numbers. In this paper we call this method the Multiplicative Noise Control (MNC). This control is genuinely stochastic, and application of such a control can be viewed as an example of stabilization by noise. After applying MNC, the equation takes the form

[TABLE]

For $\rm f$ satisfying (1.2), suggested in (1.4) control stabilizes the equilibrium $K$ if

[TABLE]

For quite a variety of independent and identically distributed random variables $\xi_{n}$ inequality (1.5) is valid when $q$ and $\sigma$ are small enough, and $\sigma^{2}>2q,$ see e.g. [23, 24]. In Section 4 we present calculations of $\lambda$ in the two cases: $\xi_{n}$ are continuous uniformly distributed on $[-1,1]$ random variables and $\xi_{n}$ are Bernoulli distributed random variables taking the values of 1 and -1 with equal probabilities $p=1-p=0.5$. It appears that $q$ and $\sigma$ do not need to be too small to ensure fulfillment of condition (1.5). In particular, for Bernoulli distributed $\xi_{n}$, condition (1.5) holds for any $q\geq 0$, when $\sigma$ is chosen appropriately (see (4.3) in Section 4). The case of $q=1$, $\sigma=2.1$ is illustrated by simulation, see Fig. 4.

The main result of the paper states that, under conditions (1.2) and (1.5), for any $\gamma\in(0,1)$, we can find a $\delta>0$ such that after applying control (1.3) for $z_{0}\in(K-u,K+u)$ followed by control (1.4), for $z_{\tau}\in(K-\delta,K+\delta)$, we have $\lim\limits_{n\to\infty}z_{n}=K$ with probability greater than $1-\gamma$.

Calculations in Sections 7 show that the radius $\delta$ of the interval for the initial value $x_{0}$ can be extremely small. To enhance this situation, in Section 3.3, we suggest the combined method, where the first control pushes the solution into a larger interval $(K-\beta,K+\beta)$, where $(K-\delta,K+\delta)\subset(K-\beta,K+\beta)\subset(K-u,K+u)$, and the solution is returned to $(K-\beta,K+\beta)$ each time it gets out. Applying an alternative stabilization by noise approach, see Lemma 2.3, we prove that the solution still reaches $(K-\delta,K+\delta)$ in a.s. finite time. Thus we consider a control with switching. Since the moment when the solution enters the designated interval is random, the switching happens at random moments, which brings some extra complications and makes proofs quite technical (see Section 7).

The main differences between the results of our paper and some previous research, for example, [23], can be outlined as follows.

We combine stabilization by noise with other control methods. However, at the final stage we have only stabilization by a state-dependent noise. 2. 2.

In general we don’t assume that $q$ is small. However, the larger is $q$, the smaller the initial neighbourhood should be. We note that in numerical runs wider neighbourhoods are applied than rigorously predicted theoretically, the estimates are sufficient only. Also, $\kappa$ in (1.2) can be any positive number, while $\kappa=1$ necessarily in [23]. 3. 3.

In [23], switching between two equilibrium points is controlled: the noise brings the zero equilibrium which loses stability, to become stable again, in a small neighbourhood of the equation parameter. In our examples, we consider mostly positive equilibrium points which lose stability (giving rise to stable cycles and eventually chaos), and become stable under controls with stochastic perturbations. The maps which are stabilized can even be chaotic.

The paper is organized as follows. After describing all relevant definitions, assumptions and notations for equation (1.1) in Section 2, we state the main results of the paper justifying the possibility to stabilize a map by noise in Section 3. However, such stabilization is stipulated by the choice of the initial point in the close proximity of the initial point to the unstable equilibrium, especially for chaotic maps. To alleviate this requirement, Section 3.3 considers a combination of another method applied at some finite and initially evaluated number of steps, and the noise control. This allows to start MNC from a significantly wider interval, in fact practically from any positive value. In Section 4 we consider several types of equations, in particular Ricker and logistic, as well as another type for which none of the results developed in [23] can be applied. In numerical simulations, we use either uniformly or Bernoulli distributed noise. Finally, Section 5 contains a summary and a discussion, and describes possible developments of the present research. Most of involved proofs of the main results are given in Appendix A. We introduce Directed Walks Control (DWC) as an additional method, all the details are postponed till Appendix B.

2 Preliminaries

Let $(\Omega,{\mathcal{F}},(\mathcal{F}_{n})_{n\in\mathbb{N}},{\mathbb{P}})$ be a complete filtered probability space. In the paper we deal with three sequences of random variables $(\xi_{n})_{n\in\mathbb{N}}$, $(\zeta_{n})_{n\in\mathbb{N}}$, and $(\chi_{n})_{n\in\mathbb{N}}$, each of them satisfying the following assumption.

Assumption 2.0

Each sequence $(\zeta_{n})_{n\in\mathbb{N}}$, $(\chi_{n})_{n\in\mathbb{N}}$ and $(\xi_{n})_{n\in\mathbb{N}}$ consists of identically distributed random variables, and $\zeta_{n}$, $\chi_{n}$ and $\xi_{n}$ are mutually independent random variables satisfying $|\zeta_{n}|\leq 1$, $|\chi_{n}|\leq 1$, $|\xi_{n}|\leq 1$, $n\in\mathbb{N}$.

We use the standard abbreviation “a.s.” for the wordings “almost sure” or “almost surely” with respect to the fixed probability measure $\mathbb{P}$ throughout the text. A detailed discussion of stochastic concepts and notation may be found, for example, in [28].

Everywhere below, for each $t\in[0,\infty)$, we denote by $[t]$ the integer part of $t$, $\mathbb{N}_{0}:=\mathbb{N}\cup\{0\}$, and, for $\delta,u\in(0,\infty)$, $0<\delta<u$,

[TABLE]

In the paper we consider identically distributed random variables, $\xi_{n}$. Sometimes, when we deal with their probabilities, $\mathbb{P}\{\xi_{n}>a\}$, or expectations, $\mathbb{E}\ln(1+q-\sigma\xi_{n})$, the index $n$ could be omitted.

2.1 Kolmogorov’s Law of Large Numbers and two more lemmas

In this section we formulate the Kolmogorov’s Law of Large Numbers, see i.e. [28], and two lemmas. The proof of Lemma 2.2 is straightforward and is thus omitted. The proof of Lemma 2.3 can be found, e.g in [11] or [14].

Theorem 2.1** ((Kolmogorov)).**

Let $(v_{n})_{n\in\ {\mathbb{N}}}$ be the sequence of independent random variables with $\theta_{n}^{2}=Var(v_{n})<\infty$. Let $S_{n}=v_{1}+\dots+v_{n}$ and an increasing sequence of $b_{n}>0$ be such that $b_{n}\uparrow+\infty$ as $n\to\infty$ and

[TABLE]

*Then a.s., $\displaystyle\frac{S_{n}-{\bf E}S_{n}}{b_{n}}\rightarrow 0$. *

Lemma 2.2**.**

*Let $(e_{n})_{n\in\ \mathbb{N}}$, $e_{n}\in{\mathbb{R}}$ be an increasing sequence, $\displaystyle\lim_{n\to\infty}e_{n}=+\infty$, and $(b_{n})_{n\in\ \mathbb{N}}$ be a sequence satisfying $\displaystyle\sum_{j=0}^{\infty}b_{j}<\infty$. Then $\displaystyle\lim_{n\to\infty}\frac{1}{e_{n}}\sum_{j=0}^{n}b_{j}e_{j}=0$. *

Lemma 2.3**.**

*(see [11, 14])

Let $(\xi_{n})_{n\in\ \mathbb{N}}$ be a sequence of independent identically distributed random variables such that $\mathbb{P}\left\{\xi_{n}\in(a,b]\right\}=p_{1}\in(0,1)$ for some interval $(a,b]$, $a<b$, and each $n\in\mathbb{N}_{0}$. Then for each nonrandom $\bar{J}\in{\mathbb{N}}$, the probability*

[TABLE]

2.2 Transformation of the equation

In this section we formulate the main assumptions about $\rm f$, and transform the equation

[TABLE]

moving the equilibrium $K$ to 0, as well as introduce some notations and comments on the MNC (Multiplicative Noise Control) method.

Assumption 2.3

Let $\mathrm{f}:[0,\infty)\to[0,\infty)$ be a continuous function which has a unique positive fixed point $K>0$, $(z-K)({\rm f}(z)-K)<0,~{}z\neq K$,

[TABLE]

whenever $z\in I_{u,K}$, and for some fixed positive $u,q,C,\kappa$,

[TABLE]

Denote

[TABLE]

When $\rm f$ satisfies Assumptions 2.3, transformation (2.5) moves the equilibrium $K$ to zero. For the function $f$ defined by (2.5), we have

[TABLE]

where $\phi$ satisfies (2.4). So now the equation

[TABLE]

has the unstable zero equilibrium. To stabilize the equilibrium, we introduce the noise term $\sigma x_{n}\xi_{n}$ into the right-hand-side of (2.7). This leads us to the stochastic difference equation

[TABLE]

For the sequence $(\xi_{n})_{n\in\mathbb{N}}$ satisfying Assumption 2.0 we set

[TABLE]

and note that $\Theta_{n}$ are also mutually independent and identically distributed random variables. By (2.6), for $x_{n}\in I_{u}$, we have

[TABLE]

so, when $x_{n}\in I_{u}$, equation (2.8) can be written as

[TABLE]

For $\Theta_{n}$ defined as in (2.9) we set

[TABLE]

and note that $v_{n}$ are also mutually independent and identically distributed. Also, both $\Theta_{n}$ and $v_{n}$, are bounded:

[TABLE]

In this paper, our main assumption about the values of the coefficient $q$ and of the noise intensity $\sigma$ is the following.

Assumption 2.3

Let Assumption 2.0 hold and $\Theta_{n}$ be defined as in (2.9). There exists $\lambda>0$ such that

[TABLE]

Remark 2.4**.**

As was mentioned in Introduction, Assumption 2.3 is fulfilled for many common independent identically distributed $\xi_{n}$ when $q$ and $\sigma$ are small enough and $\sigma^{2}>2q$. In Section 4, we derive $\mathbf{E}v_{n}$ for two particular distributions of $\xi_{n}$, uniform continuous and Bernoulli. Obtained formulae show that Assumption 2.3 is fulfilled for not so small $q$ and $\sigma$. Actually in the case of Bernoulli distribution for each $q>0$ we can find $\sigma$ such that Assumption 2.3 holds. Computer simulations of these cases are also provided in Section 4.

Whenever (2.13) holds, the application of the Kolmogorov’s Law of Large Numbers, i.e. Theorem 2.1 with $v_{i}:=\ln|\Theta_{i}|$, $b_{i}:=i$, gives that, a.s.,

[TABLE]

Relation (2.14) implies that $\forall\varepsilon\in(0,\lambda)$, there exists a random $\mathcal{N}_{1}=\mathcal{N}_{1}(\omega,\varepsilon)$ such that $\forall n\geq\mathcal{N}_{1}$ we have, a.s.

[TABLE]

3 Main results

In this section we present three results about a.s. convergence of a solution $z$ to the equilibrium after application of various stochastic controls.

The first result refers to application of only MNC, when either a solution starts at $I_{\delta,K}$ or an arbitrary control method brings it into $I_{\delta,K}$ at some a.s. finite random moment $\tau$.

The second result shows that DWC method can actually bring the solution from $I_{u,K}$ to $I_{\delta,K}$ using a.s. finite number of steps $\tau$.

The third result is a combination of DWC and MNC methods, when instead of a small neighbourhood $I_{\delta,K}$ we start applying MNC when a solution reaches a much bigger interval $I_{\beta,K}$.

3.1 MNC only

Consider an a.s. finite random variable $\tau$ which takes non-negative integer values and satisfies the following assumption.

Assumption 3.0

Assume that the random variable $\tau:\Omega\to\mathbb{N}_{0}$ is a.s. finite and independent of $\xi_{i}$ for all $i\in\mathbb{N}$.

Assumption 3.0 implies that $\tau$ is also independent of $\Theta_{i}$, defined by (2.9), and $v_{i}=\ln|\Theta_{i}|$, for all $i\in\mathbb{N}$.

Assume that a solution either starts at $I_{\delta,K}$ or an arbitrary control method brings it into $I_{\delta,K}$. Recall that MNC has the form

[TABLE]

Theorem 3.1**.**

Let $\gamma\in(0,1)$, Assumptions 2.0, 2.3 and 2.3 hold and an a.s. finite random moment $\tau$ satisfy Assumption 3.0. Then there exist positive constants $\varsigma$, $\eta$ and $\delta$, such that for any solution $z_{n}$ to (3.1) we have

1. (i)

there exists a set $\Omega_{\gamma}$, $\mathbb{P}\Omega_{\gamma}>1-\gamma$, such that, on $\Omega_{\gamma}$,

1. (a)

$|z_{m}-K|\leq\eta e^{-\varsigma(m-\tau)},\quad\mbox{for all}\quad m\geq\tau$, 2. (b)

$z_{m}\to K,\quad\text{as}\quad m\to\infty;$** 2. (ii)

there exists a set $\Omega^{[1]}_{\gamma}$, $\mathbb{P}\Omega^{[1]}_{\gamma}>1-\gamma$, and a nonrandom $n_{0}\in\mathbb{N}$, such that, on $\Omega^{[1]}_{\gamma}$,

[TABLE]

The proof of Theorem 3.1 is given in Appendix A, Section 7.5.

3.2 DWC reaches $I_{\delta,K}$ in a.s. finite time

Now consider the case when a solution $z$ with $z_{0}\in I_{u,K}$, is brought to $I_{\delta,K}$ by DWC method, introduced in (1.3) (see details in the Appendix B). Let $\tau$ be the moment when the solution $z_{n}$ for the first time reaches the interval $I_{\delta,K}$,

[TABLE]

Assume that for constants $C$, $u$ and $\kappa$ from the Assumption 2.3 the following condition holds

[TABLE]

Theorem 3.2**.**

Let $\gamma\in(0,1)$, Assumptions 2.0, 2.3, 2.3 and condition (3.3) hold. Then there exist positive constants $\varsigma$, $\eta$, $\delta$ (the same as in Theorem (3.1)), and parameters of DWC (1.3), such that the moment $\tau$ defined in (3.2) satisfies Assumption 3.0, and therefore the statement of Theorem 3.1 holds.

The proof of Theorem 3.2 is given in Appendix B, Section 8.2.

3.3 A combined method

Let assumptions of Theorem 3.1 hold and $u>\delta$ be the same as in Theorem 3.1.

In this section we combine DWC (1.3) with MNC (3.1), in order to be able to start applying MNC (3.1) in a bigger interval $I_{\beta,K}$, $u>\beta\gg\delta$ (i.e. with $\beta$ instead of $\delta$). The size of $I_{\beta,K}$ should be such that for each $x_{n}\in I_{\beta,K}$, under further application of $MNC$ at every step, we have

1. (i)

$x_{n+1}\in I_{u,K}$; 2. (ii)

there are $\iota\in(0,1)$ and $\bar{s}\in\mathbb{N}$ such that $\xi_{n+i}\in(1-\iota,1]$, $i=1,\dots,\bar{s}$ implies $x_{n+\bar{s}}\in I_{\delta,K}$.

The combined method can be described as follows. For $x_{0}\in I_{u,K}$ we apply DWC method (1.3) (see Section 8 for details) which takes at most $\bar{s}_{1}$ steps to reach $I_{\beta,K}$. As soon the solution is in $I_{\beta}$, we apply MNC method (3.1) (see Section 7 for details). However, at this stage, a solution can get out of $I_{\beta,K}$, but only into $I_{u,K}$ by (i) above. If this happens we apply again the DWC method to get the solution back inside $I_{\beta,K}$ using at most $\bar{s}_{1}$ steps. Note that, once $x\in I_{u,K}$, also $x_{n+1}\in I_{u,K}$. To prove that after a.s. finite numbers of steps the solution gets into $I_{\delta,K}$, we apply Lemma 2.3. As soon as $x_{n}\in I_{\delta,K}$, by the results in Section 7, a solution becomes asymptotically stable and no more DWC is necessary.

In addition to Assumptions 2.0, 2.3 and 2.3 we suppose that

[TABLE]

[TABLE]

Remark 3.3**.**

*Condition (3.5) holds in any of the two cases

(a) $\xi_{n}$ are Bernoulli distributed, $\mathbb{P}\{\xi=1\}=0.5$, , $\mathbb{P}\{\xi=-1\}=0.5$;

(b) $\xi_{n}$ are continuous random variables with a positive density on $[-1,1]$.*

In case (a) we have $p_{\iota}=0.5$ since $\mathbb{P}\{\xi>1-\iota\}=\mathbb{P}\{\xi=1\}=\frac{1}{2}.$

Condition (3.4) holds, in particular, if $\sigma<2$ and $q<\frac{\sigma^{2}}{2}$, since $q<\frac{\sigma^{2}}{2}<\sigma$.

Define

[TABLE]

By (3.5) we have $1>1-\iota>\frac{q}{\sigma},$ which implies $|\underline{\Theta}_{\iota}|<1$.

Theorem 3.4**.**

Let $\gamma\in(0,1)$, Assumptions 2.0, 2.3, 2.3, conditions (3.3), (3.4) and (3.5) hold. Let $\underline{\Theta}_{\iota}$ and $\bar{\Theta}$ be defined as in (3.6) and $\beta$ satisfy the inequality

[TABLE]

Then there exist positive constants $\varsigma$, $\eta$, $\delta$ (the same as in Theorem (3.1)), and parameters of DWC (1.3), such that a method which combines application of DWC to $I_{u,K}$ followed by application of MNC to $I_{\beta,K}$ brings solution to $I_{\delta,K}$ at the moment $\tau$, which satisfies Assumption 3.0, and therefore the statement of Theorem 3.1 holds.

The proof of Theorem 3.4 is given in Appendix B, Section 8.3.

4 Examples and computer simulations

In this section we consider two types of functions $\rm f$, logistic and Ricker, as well as an additional example of an unbounded $\rm f$, and two types of random variables $\xi$, continuous uniformly distributed on $[-1,1]$ random variable and Bernoulli distributed random variable

[TABLE]

For $\xi$ defined by (4.1) by direct calculations we obtain

[TABLE]

From (4.2) we conclude that $\lambda>0$ holds if and only if the following estimation is fulfilled:

[TABLE]

Estimation (4.3) allows parameters $q$ and $\sigma$ to be quite large. So even for the values of the equation parameters when the non-controlled equation is chaotic, and the derivative $|{\rm f}^{\prime}(K)|=1+q$ with substantial $q>0$, the equilibrium $K$ can be stabilized with the help of a noise of type (4.1). However, such a stabilization comes at a price: the interval $I_{\delta,K}$ becomes incredibly small. In this situation it is reasonable to apply the combined method, see Section 3.3, Section 8 from Appendix B, and examples below.

In some of the examples we use parameters of MNC and DWC methods, which are defined and discussed in Appendices.

Remark 4.1**.**

Condition (4.3) immediately implies (3.4).

If $\xi$ is continuously uniformly distributed on $[-1,1]$, direct calculations lead to

[TABLE]

From (4.4) we conclude that, for big $\sigma$, the first term in the brackets of the right-hand side is dominating, even though the second term can be negative, see examples and simulations below. So only when the derivative ${\rm f}^{\prime}(K)=-(1+q)$ is close to -1 (like $q\leq 0.3$), stabilization with the help of a continuous uniformly distributed noise could be achieved.

In the rest of the section we provide examples with computer simulations which illustrate our theoretical results and also show that, for smaller $q$, stabilization is possible with MNC only.

4.1 The logistic map.

Consider the truncated logistic equation

[TABLE]

which has a nonzero equilibrium at $K=1-\frac{1}{r}$, and the coefficients in (2.3) are: $C=r$, $q=r-3$, $\kappa=1$. We can show that $u\leq\frac{1}{4r}$ and coefficients in (8.1) are

[TABLE]

A similar to (4.5) truncation is applied to MNC

[TABLE]

For $\iota$ satisfying (3.5), applying (3.7), we can show that $\beta$ can be calculated as

[TABLE]

4.1.1 Logistic map: stabilization by noise only

In estimation of the proximity of $x_{0}$ to the equilibrium required to apply MNC and achieve stabilization, very small $\delta$ are obtained. However, in simulations much wider neighbourhoods allow such a control. First, we consider stabilization by noise (MNC) only. We use (4.6), otherwise, we do not employ any method to keep a solution in a vicinity of the equilibrium. The initial value is also not assumed to be in a small neighbourhood of $K$.

We apply continuous uniformly distributed on $[-1,1]$ noise in Examples 4.2,4.3.

Example 4.2**.**

Let $r=3.1$, then $q=0.1$, $K=0.677$. Applying (4.4) with $\sigma=0.4$, we get $\lambda\approx-0.0723<0$, for $\sigma=0.6$ we get $\lambda\approx-0.0405<0$ as illustrated on Fig. 1, upper left for $\sigma=0.4$ and upper right for $\sigma=0.6$. In both cases there is no stabilization, as predicted.

If $\sigma=0.8$ then $\lambda\approx 0.012>0$, for $\sigma=0.9$, $\lambda\approx 0.051>0$, see Fig. 1, lower left for $\sigma=0.8$ and lower, right for $\sigma=0.9$. In both cases we observe stabilization by MNC, slower for $\sigma=0.8$ and faster for $\sigma=0.9$.

Next, we illustrate the interplay between $q$ and $\sigma$ in MNC: the larger is $q$, the bigger $\sigma$ is required for stabilization.

Example 4.3**.**

Let $r=3.3$, then $K=0.7$, $q=0.3$. For $\sigma=0.8$ and $\sigma=1.2$ corresponding $\lambda$’s are negative ($\approx-0.19$ and $\approx-0.05$, respectively), so MNC does not stabilize the equation, see Fig. 2, the upper row. However, compared to $\sigma=0.8$, see Fig. 2, upper left, for $\sigma=1.2$, upper right, the solution is more concentrated around $K=0.7$.

For $\sigma=1.3$, we get $\lambda\approx 0.044>0$, and MNC is stabilizing, see Fig. 2, lower left. For $\sigma=1.4$, $\lambda\approx 0.1245>0$ guarantees convergence (Fig. 2, lower right).

Next, we proceed to the logistic map with a Bernoulli distributed $\xi$.

Example 4.4**.**

Let $r=3.3$, then $q=0.3$, $K=0.7$. Substituting $\sigma=0.6$ in (4.3) we conclude that stabilization is not guaranteed, see Fig. 3, left. However, for $\sigma=0.85$ MNC provides stabilization, as illustrated in Fig. 3, right.

4.1.2 The logistic map: stabilization with the combined method

Finally, we consider the case when a combination of DWM and MNC is required to achieve stabilization.

Example 4.5**.**

For $r=4$ we have $q=1$. We can show that the parameters $\sigma=2.1$, $\lambda=0.4457$, $u=0.05$, $\beta=0.00617$ satisfy (4.3), (2.13), (4.7).

In order to estimate $\delta$ by (7.20) we need to have $\bar{N}$ and $M$, see (7.2), (7.11), (7.18). For $\gamma=0.9$ using Chebyshev’s inequality we get $\bar{N}=272$, by the normal approximation we have $\bar{N}=74$. The number $M=\bar{\Theta}^{\bar{N}-1}=4.1^{\bar{N}-1}$ is big even for $\bar{N}=74$, which implies that $\delta$ is very small. So in this case it is reasonable to apply the combined method.

In numerical implementations, we choose $\sigma=2.1$, $u=0.125$, $\beta=0.015$. For the parameters of DWM, we use $a=1.25$, $\displaystyle\alpha_{j}=2.2a^{-(j+1)}u$, see Fig. 4. Similarly to Section 4.1.1, we illustrate by simulations that stabilization is possible even in the case when the procedure is more general than theoretically described. In Section 4.1.1, we stabilized a positive equilibrium with noise only. Here, we implement a combined algorithm with $u$ and $\beta$ larger than theoretically predicted, which is illustrated in Fig. 4.

4.2 The Ricker map

Consider the Ricker equation

[TABLE]

which has a positive equilibrium at $K=1$, $f^{\prime}(1)=1-r$, $q=r-2$.

Example 4.6**.**

Let $r=2.3$, then $q=0.3$. First, consider $\sigma=1.2$ with $\xi$ continuous uniformly distributed in $[-1,1]$. Applying (4.4) we get $\lambda=-0.05<0$, the same inequality is valid for $\sigma=1$. Fig. 5 illustrates that we do not observe stabilization for $\sigma=1$ (upper left) and $\sigma=1.2$ (upper right). For $\sigma=1.22$ and $\sigma=1.24$, we also get similar inequalities: $\lambda=-0.037<0$ and $\lambda=-0.023<0$, respectively. However, on Fig. 5 for $\sigma=1.22$ (lower left) and $\sigma=1.24$ (lower right) we observe stabilization, which does not contradict to our theory, as the results are just sufficient.

Next, we proceed to a larger $r$, for which the value of $\delta$ is extremely small and just MNC, starting from a bigger than $I_{\delta}$ interval, cannot stabilize a solution. Similarly to the logistic case, we consider the Bernoulli distribution.

For $r\geq 3$ and $u$ small enough, we have,

[TABLE]

[TABLE]

So we can set $\displaystyle\underline{L}:=e^{-ru}(r(1+u)-1),\quad 1+\bar{q}:=e^{ru}(r(1-u)-1)$ and find such $u$ that $\underline{L}>\bar{q}.$ We claim that, for $r=3$, it is possible to take $u=0.1$. Indeed, $1+\bar{q}=2.278$, $\underline{L}=1.71$ and $\bar{q}=2.278-1=1.278<1.71=\underline{L}.$

So on $I_{0.1,\,1}$ estimate (8.1) takes place with $\bar{q}=1.278$, $\underline{L}=1.71,$ and we can find $\beta$ for construction of the DWC algorithm. However, in numerical runs, we illustrate that a bigger $u$ can work as well.

Example 4.7**.**

Let $r=3$, $\sigma=2.1$, with noise (4.1), then condition (4.3) holds. We take $u=0.125$ and $\beta=0.015$. For the parameters of DWC, we use $a=1.25$, $\displaystyle\alpha_{j}=3a^{-(j+1)}u$. In Fig. 6, we observe stabilization. Note that the map is chaotic.

4.3 An example with $\kappa<1$

Finally, we consider an unbounded function for which, to the best of our knowledge, none of the previous results allows to prove the possibility of stabilization by noise.

Example 4.8**.**

Consider the equation

[TABLE]

which involves $\rm f$ satisfying (1.2) with $q=0.5$, $K=1$, $\kappa=0.5$. Thus, $\rm f$ is not Lipschitz at $K=1$, and all the results [23] fail for this case.

We use either uniform continuous or Bernoulli distribution in (4.9) with $\sigma=1.8$, see Fig. 7, left and right, for the initial value $x_{0}=1+10^{-11}$ and $x_{0}=1+10^{-8}$, respectively.

5 Discussion

We have proved that, when the equilibrium is unstable which includes the scenarios of stable cycles with high amplitude or even chaos, we still can stabilize equations, for example, by using the Bernoulli type of noises (4.1), if the initial value is in a neighbourhood $I_{\delta,K}$ of the positive equilibrium $K$. In simulations, we also considered a continuous uniformly distributed noise. However, the stronger the instability, the bigger the noise coefficient we need to take, which implies that the interval $I_{\delta,K}$ of the initial values becomes very small. To avoid the necessity to bring a solution to $I_{\delta,K}$, we apply a combined method and start a solution from a bigger interval $I_{\beta,K}$ pushing it back into $I_{\beta,K}$ each time when it gets out to $I_{u,K}$. We have justified that with the initially fixed probability, the combined method will deliver the solution into $I_{\delta,K}$ in a finite number of steps. After that moment we apply multiplicative noise control which eventually brings a solution to $K$.

The results of the paper can be summarized as follows.

We developed a two-step method, where at the final stage stabilization is achieved only by applying a multiplicative noise. However, numerical simulations illustrate that, for a wide range of problems, where there is a stable two-cycle in a non-controlled equation, stabilization by noise is possible, without any additional preliminary methods. Assumption 2.3 is absolutely crucial for the possibility to stabilize by noise, and the stronger is instability (measured, for example, by the distance of the map parameter from the bifurcation point where stability is lost), a larger noise amplitude should be chosen to guarantee stabilization. 2. 2.

In some works, e.g. [23], stabilization is considered in the case when stability switches between the two equilibrium points, and the value of the bifurcation parameter is quite close to the bifurcation point, in particular, $q=0.05$. First, we consider stabilization of an arbitrary point, allowing a period-doubling bifurcation. Second, stabilization was achieved quite far from the bifurcation point where the considered equilibrium point lost stability: $q=0.1,0.3$ and even $q=1$. In Examples 4.5 and 4.7, the non-controlled equations were chaotic. In order to stabilize, we successfully implemented the combined method.

Note that Assumption 2.3 assumes existence of exactly one positive equilibrium (which is satisfied for Ricker and logistic maps). However, the results of the present paper can be applied to the case of several positive equilibrium points, once, using a deterministic control, we bring a solution in a neighbourhood of a chosen equilibrium $K_{1}$ and set up a stabilizing stochastic perturbation dependent on $x_{n}-K_{1}$. Then, for any $x_{0}>0$, a solution converges to $K_{1}$.

Two other most substantial extensions of the present work are listed below.

• •

In the present paper, we stabilized scalar first order difference equations. It would be interesting to consider stabilization of higher order equations and systems by noise. It is natural to expect that a solution should be in a neighbourhood of the equilibrium, in order to achieve stabilization. Thus, some deterministic or stochastic method bringing a solution into a target domain, should precede stabilization by noise.

• •

So far we only stabilized an equilibrium. An interesting question is whether it is possible to stabilize an unstable cycle. It would probably be even more interesting to prove that, under certain conditions, a stochastic perturbation reduces a cycle amplitude, and construct relevant estimations.

6 Acknowledgment

The authors thank the referees for valuable comments which greatly contributed to the presentation of the paper. Both authors are grateful to the American Institute of Mathematics SQuaRE program during which this research was initiated. The first author was supported by the NSERC grant RGPIN-2015-05976.

7 Appendix A

In this section we derive exponential estimates of MNC solutions and prove Theorem 3.1.

We consider the case when the solution has already reached the target interval $I_{\delta,K}=(K-\delta,K+\delta)$ which happened at some random moment $\tau$. For simplicity of calculations, in this section we assume that the equilibrium $K$ equals zero, so instead of $z_{n}$ we denote the solution by $x_{n}$.

We assume everywhere below that $\mathbb{P}\{\tau<\infty\}=1$, as we prove it for the Directed Walk Control in Appendix B. Once this happens, we apply MNC (Multiplicative Noise Control) which is based on the application of Kolmogorov’s Law of Large Numbers (see Sections 2.1 and 2.2). So we consider equation (2.10) with the initial value $x_{\tau}\in I_{\delta}$, where $\tau$ is the first moment when $x$ enters $I_{\delta}$. The fact that, in general, $\tau$ is a random variable, brings some extra technical difficulties.

In Section 7.1 we introduce two classes of sets, which will be helpful in the proofs. The first class consists of sets, for each of which the value of $\tau$ is constant. Construction of the second class is based on Kolmogorov’s Law of Large Numbers and on the number for which a desired estimate on the sums of $\ln|\Theta_{i}|$, where $\Theta$ are defined in (2.9), holds with a given probability. Then we evaluate probabilities of certain intersections and unions of those sets.

In Section 7.2 we define several parameters, in particular $\delta$, the radius of the initial interval which guarantees local asymptotic stability. In Section 7.3, after obtaining a general estimate for solutions of the equation with MNC, we derive an exponential estimate for the solution with a certain value at a step $\tau$ treated as an initial value. In Sections 7.4 we discuss the case $\mathbb{P}\{\tau=0\}=1$, which means that, with probability 1, the solution starts from the interval $I_{\delta}$. And, finally, in Sections 7.5 we give the proof of Theorem 3.1.

7.1 Random moment $\tau$ and classes of probability sets

7.1.1 Definition of $\Omega_{\tau k}$

For a random variable $\tau$ satisfying Assumption 3.0 and for each $k\in\mathbb{N}_{0}$, we set

[TABLE]

Since $\Omega_{\tau k}$ are mutually exclusive, we also have $\sum_{i=0}^{\infty}\mathbb{P}\Omega_{\tau i}=1.$

7.1.2 Definition of $\Omega_{\gamma k}$

Fix $\gamma\in(0,1)$ and, based on (2.15), find a nonrandom number $\bar{N}=\bar{N}(\gamma,\varepsilon)$ such that

[TABLE]

For each $k\in\mathbb{N}_{0}$, we set

[TABLE]

In general, $\Omega_{\gamma k}\neq\Omega_{\gamma j}$ for $k\neq j$, but since $v_{n}$ are identically distributed, we have $\mathbb{P}\left\{\Omega_{\gamma k}\right\}=\mathbb{P}\left\{\Omega_{\gamma j}\right\}$, $\forall k,j\in\mathbb{N}_{0}$, where $\mathbb{P}\left\{\Omega_{\gamma 0}\right\}$ is estimated in (7.2). Then, for all $k\in\mathbb{N}_{0}$,

[TABLE]

By (7.3), for all $n\geq\bar{N}$ and all $k\in\mathbb{N}_{0}$ we have on $\Omega_{\gamma k}$,

[TABLE]

7.1.3 Estimation of some probabilities

Let $\{\Omega_{\tau k}\}_{k\in\mathbb{N}}$, $\{\Omega_{\gamma k}\}_{k\in\mathbb{N}}$ be defined by (7.1) and (7.3), respectively. Due to the fact that $\tau$ is independent of $v_{i}$, we have for nonzero left-hand side,

[TABLE]

Note that, for $k\neq i$, the sets $\Omega_{\tau k}\cap\Omega_{\gamma k}$ and $\Omega_{\tau i}\cap\Omega_{\gamma i}$ are mutually exclusive since $\Omega_{\tau k}\cap\Omega_{\tau i}=\emptyset$, $i\neq j$. Denote, for $s\in\mathbb{N}$,

[TABLE]

Then, applying (7.1), we get

[TABLE]

The proof of the following lemma is standard and thus is omitted.

Lemma 7.1**.**

Let $\gamma\in(0,1)$ and $\Omega_{\tau,\gamma,s}$ be defined as in (7.6). Then there exists $n_{0}\in\mathbb{N}_{0}$ such that

[TABLE]

7.1.4 Definition and estimation of $\mathcal{B}_{n,k}$ and $\mathcal{A}_{k}$ for $n\leq\bar{N}$

For each $n\in\mathbb{N}$, $k\in\mathbb{N}_{0}$, based on the linear part of equation (2.10), we construct non-negative random variables $\mathcal{B}_{n,k}$ and $\mathcal{A}_{k}$ by

[TABLE]

where $\bar{N}>1$ is such that (7.2) and (7.4) hold. By Assumption 2.0 and (2.12), random variables $\Theta_{n}$ and $v_{n}$ are bounded by $\bar{\Theta}$ and $\bar{v}$, respectively. If we set

[TABLE]

then $M>1$, and for all $\omega\in\Omega$, $n\in\mathbb{N}$, $k\in\mathbb{N}_{0}$,

[TABLE]

Note that for each $k\in\mathbb{N}_{0}$ by (7.3) and (7.5), we have for $n\geq\bar{N}>1$ on $\Omega_{\gamma k}$,

[TABLE]

7.2 More definitions and relations

Let $\lambda$ be defined as in (2.13) and let $C,\kappa$ be from Assumption 2.3. Choose some $\varsigma>0$ and $\varepsilon>0$ such that

[TABLE]

Recall that (7.2) and (7.4) hold for $\bar{N}$. Then (7.5) implies that for $n\geq\bar{N}$, $\omega\in\Omega_{\gamma k}$ and each $k\in\mathbb{N}_{0}$, we have for $\mathcal{B}_{n,k}$ defined as in (7.9),

[TABLE]

By (7.14), we get $-\lambda+\varsigma+\varepsilon<0$, so

[TABLE]

For $n\geq\bar{N}$ we have on $\Omega_{k}$, for each $k\in\mathbb{N}_{0}$,

[TABLE]

Since $\kappa\varsigma-2\varepsilon>0$ and $-\lambda+\varsigma+\varepsilon<0$, we can apply Lemma 2.2 with $b_{j}:=e^{-(\kappa\varsigma-2\varepsilon)j}$, $e_{j}:=~{}e^{-(-\lambda+\varsigma+\varepsilon)(j+1)}$, and choose a nonrandom $\bar{N}_{1}\in\mathbb{N}$ such that, for $n\geq\bar{N}_{1}$,

[TABLE]

where $C$ is from (2.4) in Assumption 2.3. Let

[TABLE]

$u$ and $\kappa$ be also from Assumption 2.3, and $M$ be defined as in (7.11). Choose

[TABLE]

and

[TABLE]

which obviously satisfies $\delta<1$ and $\delta<\frac{\eta}{2}$.

7.3 Exponential estimates of solutions.

In this section we derive exponential estimates of solutions to equations with different initial values.

7.3.1 General estimation of solutions

Let $\tau$ satisfy Assumption 3.0, i.e. $\tau$ be a positive integer-valued a.s. finite random variable which is independent of all $\xi_{i}$, and let $\nu$ be some a.s. finite positive random variable. Fix some $k\in\mathbb{N}_{0}$ and consider the following modification of equation (2.10)

[TABLE]

Everywhere further we assume that $\displaystyle\prod_{i}^{j}=1$ if $i>j$. The following lemma is obtained by induction.

Lemma 7.2**.**

Let $k\in\mathbb{N}_{0}$, condition (2.4) hold, $\Theta_{i}$ be defined as in (2.9), and $\Omega^{*}\subseteq\Omega$. Assume that $y_{n}$ is a solution to (7.21) with a random initial value $\nu$ such that $\nu\in I_{u}$ on $\Omega^{*}$, and, for some $n\in\mathbb{N}_{0}$, on $\Omega^{*}$, $y_{i}\in I_{u}$, $i=1,2,\dots,n$. Then, on $\Omega^{*}$,

[TABLE]

7.3.2 Exponential estimates of solutions of auxiliary equations

Next, we proceed to exponential estimations for solutions of (7.21).

Lemma 7.3**.**

Let $\gamma\in(0,1)$ and $k\in\mathbb{N}_{0}$. Let Assumptions 2.0, 2.3, 2.3 and 3.0 hold. Let the sets $\Omega_{\tau k}$, $\Omega_{\gamma k}$ and numbers $\varsigma$, $\eta$, $\delta$, be defined by (7.1), (7.3), (7.14), (7.19) and (7.20), respectively. Let $y_{n}$ be a solution to (7.21) with the initial value $y_{0}=\nu$, where $\nu$ is a random variable such that $\nu\in I_{\delta}$ on $\Omega_{\tau k}$. Then, on $\Omega_{\gamma k}\cap\Omega_{\tau k}$,

[TABLE]

Proof 7.4**.**

Since the sequence $(\xi_{n})_{n\in\mathbb{N}}$ satisfies Assumptions 2.0, the same holds for sequences $(\xi_{k+n})_{n\in\mathbb{N}}$ with any $k\in\mathbb{N}_{0}$.

We prove (7.23) by induction. Note that (2.12) and (7.11) imply that, for each $k\in\mathbb{N}_{0}$, $|\Theta_{k}|\leq\bar{\Theta}\leq M.$ Since $|y_{0}|\leq\delta<u<1$ on $\Omega_{\gamma k}\cap\Omega_{\tau k}$, we can apply (2.4), recall (2.9) and get for $n=1$, on $\Omega_{\gamma k}\cap\Omega_{\tau k}$,

[TABLE]

since $\delta\leq(M+C)^{-1}e^{-\varsigma}\eta$.

Now assume that, for some $n\in\mathbb{N}$, (7.23) holds on $\Omega_{\gamma k}\cap\Omega_{\tau k}$ for all $i\leq n$, and prove that it holds on $\Omega_{\gamma k}\cap\Omega_{\tau k}$ for $n+1$. Then we can apply estimate (7.22) from Lemma 7.2 with $\Omega^{*}=\Omega_{\gamma k}\cap\Omega_{\tau k}$, recall the definition of random variable $\mathcal{B}_{n,k}$ in (7.9) and conclude that, on $\Omega_{\gamma k}\cap\Omega_{\tau k}$,

[TABLE]

Next, we prove that $T^{[k]}_{i}(n+1)\leq\frac{1}{2}\eta e^{-\varsigma n}$ on $\Omega_{\gamma k}\cap\Omega_{\tau k}$, for each $i=1,2$. For $\bar{N}_{2}$ defined as in (7.18) we consider two cases: $n<\bar{N}_{2}$ and $n\geq\bar{N}_{2}$.

Estimation of $T^{[k]}_{1}(n+1)$. For $n<\bar{N}_{2}$, since $|y_{0}|\leq\delta<\frac{1}{2}\eta M^{-1}e^{-\varsigma\bar{N}_{2}}$ and by (7.10), (7.12), we have on $\Omega_{\gamma k}\cap\Omega_{\tau k}$

[TABLE]

For $n\geq\bar{N}_{2}$, since $\delta<\frac{1}{2}\eta$ and by (7.13), (7.15), (7.16),

[TABLE]

Estimation of $T^{[k]}_{2}(n+1)$. For $n<\bar{N}_{2}$, by (7.9), (7.10), (7.12), we have on $\Omega_{\gamma k}\cap\Omega_{\tau k}$,

[TABLE]

By substituting estimate (7.19) of $\eta$ into (7.24) we arrive at

[TABLE]

For $n\geq\bar{N}_{2}$, applying (7.16) and (7.17), we get

[TABLE]

which completes the proof.

7.4 Some remarks and the case $\tau\equiv 0$.

Remark 7.5**.**

By construction of estimate (7.23) and the choice of $\eta$, on $\Omega_{\gamma k}\cap\Omega_{\tau k}$, a solution $y_{n}$ to equation (7.21), which starts in $I_{\delta}$ will stay in $I_{u}$ for all $n\in{\mathbb{N}}$. This implies that, on $\Omega_{\gamma k}\cap\Omega_{\tau k}$, $y_{n}$ is also a solution to the equation

[TABLE]

Note that $y_{n}$ does not necessarily belong to $I_{\delta}$ for all $n$. However, after at most $\bar{n}$ steps, $y_{n}$ returns to $I_{\delta}$ and remains there. The number $\bar{n}$ can be calculated as the minimum $n$ for which $\eta e^{-\varsigma n}<\delta$. In other words,

[TABLE]

If the initial value $\nu$ belongs to the interval $I_{\delta}$ with probability 1, i.e. $\mathbb{P}\left\{\nu\in I_{\delta}\right\}=1$, we have $\mathbb{P}\left\{\tau=0\right\}=\mathbb{P}\left\{\Omega_{\tau 0}\right\}=1$ and $\mathbb{P}\left\{\Omega_{\tau k}\right\}=0$ for all $k>0$. So

[TABLE]

and then, by (7.6) and (7.7), we have

[TABLE]

Thus we arrive at the corollary which can be treated as the main result for $\nu\in I_{\delta}$.

Corollary 7.6**.**

Let $\gamma\in(0,1)$, Assumptions 2.0, 2.3 and 2.3 hold, $\varsigma$, $\eta$, $\delta$, be defined by (7.14), (7.19), (7.20), respectively, and $x_{n}$ be a solution to (2.10) with $x_{0}=\nu$, where $\mathbb{P}\left\{\nu\in I_{\delta}\right\}=1$. Then, with a probability greater than $1-\gamma$,

[TABLE]

Remark 7.7**.**

If $\mathbb{P}\left\{\nu\in I_{\delta}\right\}=1$, estimate (7.8) holds with $n_{0}=0$.

7.5 Proof of Theorem 3.1

Set, as in (2.5), $x_{n}=z_{n}-K$. For an arbitrary $k\in\mathbb{N}_{0}$, we have $\tau(\omega)=k$ on $\Omega_{\tau k}$, by definition, therefore, $x_{\tau}(\omega)=x_{k}(\omega)$ whenever $\omega\in\Omega_{\tau k}$.

Denoting $y_{n}^{[k]}:=x_{k+n}$, we consider on $\Omega_{\tau k}$ the following problem

[TABLE]

see (2.5). By Remark 7.5 (and Lemma 7.3) we conclude that, on $\Omega_{\tau k}$, $y^{[k]}_{n}$ is also a solution to (7.21) with $\nu=x_{k}$. Lemma 7.3 and Remark 7.5 imply the estimates

[TABLE]

which hold on $\Omega_{\tau k}\cap\Omega_{\gamma k}$, for each $n\in\mathbb{N}_{0}$. Set, for each $n\in\mathbb{N}_{0}$,

[TABLE]

then (7.27) yields that, for each $n\in\mathbb{N}_{0}$,

[TABLE]

Substituting $m:=n+\tau$ into (7.28) and applying (7.7), we arrive at part (i),(a). Part (i),(b) is a corollary of (i),(a).

Choose $n_{0}$ and set $\Omega^{[1]}:=\Omega_{\tau,\gamma,n_{0}}$, as in Lemma 7.1. Then

[TABLE]

From (7.28) we can conclude that, whenever $m\geq n_{0}$, with a probability exceeding $1-\gamma$, we have the estimate

[TABLE]

which concludes the proof.

8 Appendix B

In this section we discuss Directed Walks Control with changing parameters and present proofs of Theorems 3.2 and 3.4.

8.1 Directed Walks Control with changing parameters

Assume that $z_{n}\in I_{u,K}$, and define the control that pushes the next $z_{n+1}$ towards  $K$ by applying (1.3). To the best of our knowledge, this method has not been introduced earlier. Obviously, $\alpha_{j}$ should depend on the closeness of $x$ to $K$, to avoid the situation when $|x_{n+1}-K|>|x_{n}-K|$. In a sequence of $\alpha=\alpha_{j}$ in (1.3), we assume that $\alpha_{j}=a^{-1}\alpha_{j-1}$, $j\in{\mathbb{N}}$, where $a>1$, i.e. $(\alpha_{j})$, $j=0,1,\dots,k-1$, is a geometrically decaying finite positive sequence.

In (1.3), $\zeta_{n}$ and $\chi_{n}$ satisfy Assumption 2.0, $\ell,\ell_{c}>0$ are some parameters, which, together with a number $k\in\mathbb{N}$, finite sequences of numbers $\alpha_{j}$ and intervals $J^{(j)}$, will be described below.

We recall that the purpose of subsequent controls with $\alpha_{j}$ is to bring a solution, which is originally in the interval $I_{u,K}=J^{(0)}$, to the interval $I_{\delta,K}=J^{(k)}=I^{(1)}$.

Assume that there exist $\bar{q}>0$, $\underline{L}>0$, and $u>0$ such that

[TABLE]

So far (8.1) is the only limitation on $\rm f$ and $u$-neighborhood of $K$. This is equivalent to $\bar{q}<\underline{L}\leq 1+\bar{q}$, where $1+\bar{q}$ and $\underline{L}$ are the upper and the lower bounds of $|{\rm f}(z)-K|/|z-K|$, respectively. In particular, (8.1) is satisfied if ${\rm f}$ is continuously differentiable and the variation of $|{\rm f^{\prime}}(x)|$ in $I_{u,K}$ is less than one. Lemma 8.1 presents another sufficient condition.

Lemma 8.1**.**

Let function $\rm f$ satisfy Assumption 2.3 and condition (3.3) hold. Then condition (8.1) holds with $\underline{L}:=1+q-Cu^{\kappa}$, $\bar{q}:=q+Cu^{\kappa}$.

Proof 8.2**.**

Note first that, by (3.3), we have

[TABLE]

So $\underline{L}$ is well defined and satisfies the first condition in (8.1). To check the second one we note that, under Assumption 2.3, we have

[TABLE]

and thus

[TABLE]

Remark 8.3**.**

Let us note, first, that (8.1) is the only condition required for successful implementation of DWC method, and, second, (8.1) is less restrictive than Assumption 2.3: for example, ${\rm f}(z)=1+1.4(z-1)-0.2|z-1|$ satisfies (8.1) with $K=1$, $\bar{q}=0.6$, $\underline{L}=1.2$, while Assumption 2.3 is obviously not satisfied.

Remark 8.4**.**

If $I_{u,K}$ includes an attractive interval of $\rm f$, the non-controlled sequence enters $I_{u,K}$ at a certain step: $z_{n}\in I_{u,K}$ for some $n\in{\mathbb{N}}$, where $n$ depends on $z_{0}$. If, for example, $\rm f$ is a function with an attractive interval $[z_{1},z_{2}]$ and $K\in(z_{1},z_{2})$ (this includes the case of a stable 2-cycle $\{z_{1},z_{2}\}$, $z_{1}\in(0,K)$, $z_{2}\in(K,\infty)$ ), and we choose $u\geq\max\{K-z_{1},z_{2}-K\}$ then, after a finite number of steps, a non-controlled sequence $(z_{n})$ enters $I_{u,K}$.

Based on (8.1), choose some

[TABLE]

Define

[TABLE]

where $[t]$ is the maximal integer not exceeding $t$, and a sequence of shrinking neighborhoods of $K$

[TABLE]

Thus $k_{\delta,u}$ is the number of the last interval in the sequence $(I^{(j)})$ which strictly includes $I_{\delta,K}$, so $I^{(k_{\delta,u}+1)}\subseteq I_{\delta,K}\subset I^{(k_{\delta,u})}$. Hence for the nested intervals in (1.3) we have

[TABLE]

Let us define a bound for $\ell_{c}$ and a constant $B$ satisfying

[TABLE]

This allows us to determine bounds for $\ell$ as follows:

[TABLE]

Lemma 8.5**.**

Let condition (8.1) hold, $a$ and $\varepsilon_{0}$ be defined as in (8.2). Then $\ell_{c}$, $B$ and $\rho$ are positive numbers, well defined by (8.6) and (8.7), respectively.

Proof 8.6**.**

Let us note that by (8.6), $\ell_{c}\in(0,1)$. Further, $B$ is well defined in (8.6) if

[TABLE]

which is equivalent to

[TABLE]

and is satisfied due to the definition of $\varepsilon_{0}$ in (8.2). The fact that all other parameters in (8.6)-(8.7) are well defined follows from the inequalities

[TABLE]

For each $j=0,1,\dots,k_{\delta,u}+1$ we define

[TABLE]

where $a$ and $B$ are from (8.2) and (8.6), respectively.

The following theorem proves that suggested DWC method (1.3), after a certain number of steps, brings a solution with $z_{0}\in I_{u,K}$ to $I_{\delta,K}$.

Theorem 8.7**.**

Let Assumption 2.0 and conditions (8.1), (8.6) hold, $\delta\in(0,u)$ and $\mu$ be defined as in (8.7). Let $z_{n}$ be a solution to (1.3) with the initial value $z_{0}\in I_{u,K}$, where $J^{(j)}$, $k$ and $\alpha_{j}$ satisfy (8.4), (8.5) and (8.8).

Then there exists a random moment $\tau$ such that, for any $\omega\in\Omega$,

[TABLE]

Proof 8.8**.**

If, for some $n\in\mathbb{N}$ and $j=0,1,\dots,k_{u,\delta}$, we have $z_{n}\in I^{(j)}\setminus I^{(j+1)}$, then

[TABLE]

*We have, by (8.6)-(8.8), $\displaystyle\frac{\left|\alpha_{j}(1+\ell_{c}\chi_{n+1})\right|}{|{\rm f}(z_{n})-K|}\leq\frac{\underline{L}a^{-(j+2)}u(1+\ell_{c})B}{\underline{L}a^{-(j+1)}u}=\frac{(1+\ell_{c})B}{a}<1$

and $\displaystyle\frac{\left|\alpha_{j}(1+\ell_{c}\chi_{n+1})\right|}{|{\rm f}(z_{n})-K|}\geq\frac{\underline{L}a^{-(j+2)}u(1-\ell_{c})B}{(1+\bar{q})a^{-j}u}=\frac{\underline{L}(1-\ell_{c})B}{a^{2}(1+\bar{q})}$.*

*Note that $\displaystyle\frac{\underline{L}(1-\ell_{c})B}{a^{2}(1+\bar{q})}<\frac{(1-\ell_{c})B}{a^{2}}\leq\frac{(1-\ell_{c})}{a(1+\ell_{c})}<1$. Then,

$\displaystyle\left|1-\frac{\alpha_{j}(1+\ell_{c}\chi_{n+1})}{|{\rm f}(z_{n})-K|}\right|=1-\frac{\alpha_{j}(1+\ell_{c}\chi_{n+1})}{|{\rm f}(z_{n})-K|}<1-\frac{\underline{L}(1-\ell_{c})B}{a^{2}(1+\bar{q})}$.*

By (8.6) and the above estimates we have

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So $z_{n+1}\in I^{(j)}$. If $z_{n+1}\in I^{(j)}\setminus I^{(j+1)}$, then we repeat the above calculations. Since $\displaystyle(1-\mu)^{\bar{m}}a^{-j}u<a^{-j-1}u$ for

[TABLE]

after at most $\bar{m}$ steps we have $z_{n+\bar{m}}\in I^{(j+1)}$. This holds for each $j\leq k_{u,\delta}$. Therefore, a.s., after at most

[TABLE]

steps the solution $z_{n}$ will be in $I_{\delta,K}$.

Remark 8.9**.**

Theorem 8.7 implies that $n_{j}$ when switching between $(j-1)$-th and $j$-th control in (1.3) occurs, are a.s. finite random variables, and $n_{j}-n_{j-1}\leq\bar{m}$ with $\bar{m}$ defined by (8.10).

*So, started in $J^{(0)}$ solution $z_{n}$ of equation (1.3), for the first time reaches the interval $J^{(k)}\equiv I_{\delta,K}$ at a random time $\tau=\tau(u,\delta)$ with an upper estimate as in (8.9). After that moment, the new control by noise only (MNC) starts working. This control was discussed above, in particular, in Appendix A. *

8.2 Proof of Theorem 3.2

Theorem 3.2 is a corollary of Theorems 8.7 and 3.1 and Lemma 8.1.

8.3 Proof of Theorem 3.4

Proof 8.10**.**

For simplicity of calculations, we assume $K=0$.

For $x_{n}\in I_{\beta}$, we have by Assumption 2.3

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Denoting $\displaystyle\beta_{1}:=\sup\{\beta>0:\bar{\Theta}\beta+C\beta^{1+\kappa}<u\}$, we conclude that

[TABLE]

Remark 8.11**.**

Note that $\beta_{1}$ is easily found in the case $\kappa=1$. Also, assuming that $\beta<1$ (which is not a restriction!) we can take $\displaystyle\beta_{1}<\frac{u}{\bar{\Theta}+C}$.

Now we look for $\beta_{2}$ and a nonrandom number of steps $\bar{s}$ such that

[TABLE]

Assume that $x_{n}\in I_{\beta_{2}}$ and $\xi_{n+1}\in(1-\iota,1]$, then

[TABLE]

Since $|\underline{\Theta}_{\iota}|<1$, we can take

[TABLE]

choose

[TABLE]

assuming without loss of generality that $\beta>\delta$, and set

[TABLE]

Then, for $i\leq\bar{s}$ on $\Omega_{\iota,n}:=\{\omega\in\Omega:\xi_{n+i}\in(1-\iota,1],\,i=1,2,\dots,\bar{s}\}$,   $\displaystyle|x_{n+i}|\leq H_{\iota}^{i}\beta$.

So, $x_{n+\tau_{1}}\in I_{\delta}$, on $\Omega_{\iota,n}$ after at most $\bar{s}$ steps, where the random moment $\tau_{1}\leq\bar{s}$.

Also, by the DWC method (see Section 8.1), we need at most $\bar{s}_{1}$ steps to reach $I_{\beta}$ from $I_{u}$. Let $\bar{J}=\bar{s}+\bar{s}_{1},\,\,(a,b]=\left(1-\iota,1\right].$ By condition (3.5) we have $\mathbb{P}\left\{\xi\in\left(1-\iota,1\right]\right\}=p_{\iota}>0.$ Applying now Lemma 2.3, we conclude that there exists an a.s. finite random moment $\mathcal{N}$ such that, a.s., $\xi_{\mathcal{N}+i}\in\left(1-\iota,1\right]$, $i=1,\dots,\bar{J}$. Consider a solution $x_{\mathcal{N}}$ at the moment $\mathcal{N}$. It can be either $x_{\mathcal{N}}\in I_{u}\setminus I_{\beta}$ or $x_{\mathcal{N}}\in I_{\beta}$.

If $x_{\mathcal{N}}\in I_{\beta}$ and since $\xi_{\mathcal{N}+i}\in\left(1-\iota,1\right]$ for $i=1,\dots,\bar{s}$, we have

[TABLE]

For $x_{\mathcal{N}}\in I_{u}\setminus I_{\beta}$, applying the DWC method, which needs not more than $\bar{s}_{1}$ steps to reach $I_{\beta}$, we have $x_{\mathcal{N}+\tau_{2}}\in I_{\beta}$, $\tau_{2}\leq\bar{s}_{1}$. Since, a.s.,

[TABLE]

we have, a.s.,

[TABLE]