Stabilization of Second Order Nonlinear Equations with Variable Delay

# Stabilization of Second Order Nonlinear Equations with Variable Delay

Leonid Berezansky, Elena Braverman and Lev Idels
Dept. of Math, Ben-Gurion University of the Negev, Beer-Sheva 84105, Israel;
Dept. of Math & Stats, Univ. of Calgary, 2500 University Dr. NW, Calgary, AB, Canada T2N1N4;
Dept. of Math, Vancouver Island University (VIU), 900 Fifth St. Nanaimo, BC, Canada V9S5J5
Corresponding author. Email: maelena@ucalgary.ca
###### Abstract

For a wide class of second order nonlinear non-autonomous models, we illustrate that combining proportional state control with the feedback that is proportional to the derivative of the chaotic signal, allows to stabilize unstable motions of the system. The delays are variable, which leads to more flexible controls permitting delay perturbations; only delay bounds are significant for stabilization by a delayed control. The results are applied to the sunflower equation which has an infinite number of equilibrium points.

Keywords: non-autonomous second order delay differential equations; stabilization; proportional feedback; a controller with damping; derivative control; sunflower equation

It is well known that control of dynamical systems is a classical subject in engineering sciences. Time delayed feedback control is an efficient method for stabilizing unstable periodic orbits of chaotic systems which are described by second order delay differential equations, see Bo; Do; Fr1; Fr2; Kim; Liu; Reit; Gabor2; Gabor1; Wan. When introducing a control, we assume that the chosen equilibrium of an equation is unstable, and the controller will transform the unstable equation into an asymptotically or exponentially stable equation. Instability tests for some autonomous delay models of the second order could be found, for example, in Cah. Two basic proportional (adaptive) control models are widely used: standard feedback controllers with the controlling force proportional to the deviation of the system from the attractor, where is an equilibrium of the equation, and the delayed feedback control , see Boccal; Joh; Ko.

Proportional control fails if there exist rapid changes to the system that come from an external source, and to keep the system steady under an abrupt change, a derivative control was used in Bela; Reit; Vyh, i.e. , where, for example, or . In electronics, a simple operational amplifier differentiator circuit will generate the continuous feedback signal which is proportional to the time derivative of the voltage across the negative resistance, see Joh. A classical proportional control does not stabilize even linear ordinary differential equations; e.g. the equation with the control is not asymptotically stable for any , since any constant is a solution of this equation. The pure derivative control also does not stabilize all second order differential equations. For example, the equation with the control is not asymptotically stable for any control since any constant is a solution of this equation. Some interesting and novel results could be found in Ren; Ru; Si; Saberi2; Yan. For a linear non-autonomous model the effective multiple-derivative feedback controller was introduced in Saberi1, and a special transformation was used to transform neutral-type DDE into a retarded DDE. However, most of second order applied models are nonlinear, even the original pendulum equation. The main focus of the paper is the control of nonlinear delay equations, some real world models are considered in Examples 2.9,LABEL:ex3,LABEL:additional_example.

In the present paper we study a nonlinear second order delay differential equation

 ¨x(t)+l∑k=1fk(t,˙x(gk(t)))+m∑k=1sk(t,x(hk(t)))=u(t),  t≥t0, (1.1)

with the input or the controller , along with its linear version

 ¨x(t)+l∑k=1ak(t)˙x(gk(t))+m∑k=1bk(t)x(hk(t))=u(t),  t≥t0. (1.2)

Both equations (1.2) and (1.1) satisfy for any the initial condition

 x(t)=φ(t), ˙x(t)=ψ(t), t≤t0. (1.3)

We will assume that the initial value problem has a unique global solution on for all nonlinear equations considered in this paper, and the following conditions are satisfied:
(a1) are Lebesgue measurable and essentially bounded on functions, , , which allows to define essential eventual limits

 α=limsupt→∞l∑k=1|ak(t)|, β=limsupt→∞m∑k=1|bk(t)|; (1.4)

(a2) are Lebesgue measurable functions, , , , .

The paper is organized as follows. In Section 2 we design a stabilizing damping control for any linear non-autonomous equation (1.2). Under some additional condition on the functions and , such control also stabilizes equations of type (1.1). The results are based on stability tests recently obtained in BBD; BBI for second order non-autonomous differential equations. We also prove in Section 2 that a strong enough controlling force, depending on the derivative and the present (and past) positions, can globally stabilize an equilibrium of the controlled equation. In Section 3 classical proportional delayed feedback controller is applied to stabilize a certain class of second order delay equations with a single delay involved in the state term only. We develop tailored feedback controllers and justify their application both analytically and numerically.

\@xsect

We will use auxiliary results recently obtained in BBD; BBI.

###### Lemma 2.1.

(BBD, Corollary 3.2) If

 4b>a2,  2(a+√4b−a2)a√4b−a2α+4a√4b−a2β<1, (2.1)

where and are defined in (a1) by (1.4), then the zero solution of the equation

 ¨x(t)+a˙x(t)+bx(t)+l∑k=1ak(t)˙x(gk(t))+m∑k=1bk(t)x(hk(t))=0 (2.2)

is globally exponentially stable.

###### Lemma 2.2.

BBI Assume that the equation

 ¨x(t)+f(t,x(t),˙x(t))+s(t,x(t))+m∑k=1sk(t,x(t),x(hk(t)))=0 (2.3)

possesses a unique trivial equilibrium, where , , ,
, , .

If at least one of the conditions
1) ,        2)
holds, then zero is a global attractor for all solutions of equation (2.3).

We start with linear equations. Stabilization results for linear systems were recently obtained in Saberi1; Saberi2. Unlike Saberi1; Saberi2, the following theorem considers models with variable delays, however, the control is not delayed.

###### Theorem 2.3.

For any , and defined by (1.4) and

 λ>μ(λ):=(δ+√4−δ2)α+√(δ+√4−δ2)2α2+4√4−δ2βδδ√4−δ2, (2.4)

equation (1.2) with the control is exponentially stable.

###### Proof.

Equation (1.2) with the control

 ¨x(t)+l∑k=1ak(t)˙x(gk(t))+m∑k=1bk(t)x(hk(t))=−δλ˙x(t)−λ2x(t) (2.5)

has the form of (2.2) with and . Then the inequalities in (2.1) have the form

 4λ2>δ2λ2~{}~{}and~{}~{}2(δ+√4−δ2)δλ√4−δ2α+4δλ2√4−δ2β<1. (2.6)

The first inequality in (2.6) holds as , and the second one is equivalent to

 δλ2√4−δ2−2(δ+√4−δ2)αλ−4β>0. (2.7)

Condition (2.4) implies (2.7), which completes the proof. ∎

###### Corollary 2.4.

Let for some , where is defined in (2.4). Then for equation (1.2) with the control is exponentially stable.

For Theorem 2.3 yields the following result.

###### Corollary 2.5.

Eq. (1.2) with the control is exponentially stable if

 λ>√2(α+√α2+β). (2.8)
###### Remark 2.6.

For any equation (1.2) there exists such that condition (2.8) holds. Hence the stabilizing damping control exists for any equation of form (1.2).

###### Example 2.7.

For the equation

 ¨x(t)+(sint)˙x(g(t))+(cost)x(h(t))=0,  h(t)≤t, g(t)≤t, (2.9)

the upper bounds defined in (1.4) are . Hence, as long as in Corollary 2.5, equation (2.9) with the control is exponentially stable.

Let us proceed to nonlinear equation (1.1); its stabilization is the main object of the present paper. For simplicity we consider here nonlinear equations with the zero equilibrium, since the change of the variable transforms an equation with the equilibrium into an equation in with the zero equilibrium.

###### Theorem 2.8.

Suppose ,

 ∣∣∣fk(t,u)u∣∣∣≤ak(t), ∣∣∣sk(t,u)u∣∣∣≤bk(t),u≠0. (2.10)

Then for any , the zero equilibrium of (1.1) with the control

 ¨x(t)+l∑k=1fk(t,˙x(gk(t)))+m∑k=1sk(t,x(hk(t)))=−δλ˙x(t)−λ2x(t) (2.11)

is globally asymptotically stable, provided (2.4) holds with and defined in (1.4).

###### Proof.

Suppose is a fixed solution of equation  (2.11). Equation (2.11) can be rewritten as

 ¨x(t)+l∑k=1ak(t)˙x(gk(t))+m∑k=1bk(t)x(hk(t))=−δλ˙x(t)−λ2x(t),

where    Hence the function is a solution of the linear equation

 ¨y(t)+l∑k=1ak(t)˙y(gk(t))+m∑k=1bk(t)y(hk(t))=−δλ˙y(t)−λ2y(t), (2.12)

which is exponentially stable by Theorem 2.3. Thus for any solution of equation (2.12), and since is a solution of (2.12),

In particular, for condition (2.4) transforms into (2.8).

###### Example 2.9.

Consider the equation

 ¨x(t)+a(t)˙x(g(t))+b(t)sin(x(h(t)))=0,  h(t)≤t, g(t)≤t, (2.13)

with . Equation (2.13) generalizes the sunflower equation introduced by Israelson and Johnson in isa as a model for the geotropic circumnutations of Helianthus annuus; later it was studied in alf; liza; somolinos.

We have ; hence if condition (2.8) holds for and , then the zero equilibrium of equation (2.13) with the control in the right-hand side is globally exponentially stable. Equation (2.13) has an infinite number of equilibrium points , . To stabilize a fixed equilibrium we apply the controller .

For example, consider the sunflower equation

 ¨x(t)+˙x(t)+2sin(x(t−π))=0

with various initial conditions , where is constant for , which has chaotic solutions (see Fig. LABEL:figure2, left). Application of the controller , where and , for example, , stabilizes the otherwise unstable equilibrium , as illustrated in Fig. LABEL:figure2, right.

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