## Abstract

A general form of a distributed feedback control algorithm based on the speed–gradient method is developed. The goal of the control is to achieve nonlinear wave localization. It is shown by example of the sine-Gordon equation that the generation and further stable propagation of a localized wave solution of a single nonlinear partial differential equation may be obtained independently of the initial conditions. The developed algorithm is extended to coupled nonlinear partial differential equations to obtain consistent localized wave solutions at rather arbitrary initial conditions.

This article is part of the themed issue ‘Horizons of cybernetical physics’.

## 1. Introduction

Localized nonlinear waves with permanent shape and velocity (bell-shaped or kink-shaped) are important features of nonlinear dynamic processes. First, these waves transfer considerable energy. On the other hand, they carry information about features of the medium (wave guide) where they propagate. This information may be obtained by measuring the amplitude and velocity of the wave. Stable propagation of nonlinear localized waves of permanent shape is provided by some balances resulting from the features of the system, in particular the balance between the nonlinear and dispersive features of the medium. Correspondingly, variations in the physical properties of the medium result in the failure of localization. In particular, visco-elasticity destroys the balance between nonlinearity and dispersion for strain wave propagation in an elastic wave guide. Also, wave propagation is governed by the solution of a partial differential equation that should satisfy certain initial and boundary conditions. However, in real problems it is difficult to provide consistent initial and boundary conditions. This is more important for equations with time derivatives of the second order. Even if a nonlinear equation possesses the exact solution in the form of a localized wave of permanent shape, the wave does not definitely arise at any initial and boundary conditions. Therefore, a tool is needed that recovers the balance or consistency between the conditions to support the stable propagation of a localized nonlinear wave. It seems that such a possibility may be realized by application of control methods.

There are a variety of control methods of nonlinear waves such as the sensitivity of the wave solutions to the type of nonlinear equation, the initial and boundary conditions, the aim of the control, etc. Thus, nonlinear reaction–diffusion systems were studied in [1–3], nonlinear modulation equations were considered in [4–8] and the sine-Gordon equation solutions were subjected to control in [9–11]. The control algorithm may be obtained using conservation laws: in most of the previous papers on this subject control problems were studied using asymptotic procedures that restrict consideration by small variations. Also, an important point is the presence of dissipative terms in the equations considered. Another way is to use the speed–gradient method [12]. In most of the previous papers, control problems were studied for the special structure of a controlled physical system, allowing for arbitrary change in the system dynamics by the controlling action. Up to now, there has been no universal algorithm for the control of nonlinear waves, in particular localized waves.

In this paper, we develop a universal distributed control algorithm that may be used for the generation and stable propagation of localized nonlinear waves of various shapes. The plan of the paper is as follows. Section 2 is devoted to a discussion about the aim of the control of localized waves. Various aims are discussed and compared. Section 3 concerns the development of the control algorithm. Section 4 considers an example of the control application for a single nonlinear equation as well as an extension of the algorithm to coupled nonlinear equations.

## 2. Goal of nonlinear wave control

Introducing a control into a physical system ensures that the system has the desired processes; this is known as the control aim. One can find a variety of control aims in the literature; for example, controlling the position in reaction–diffusion systems [1]. Also, the control of the shape of the plane wave in reaction–diffusion systems is established in [2]. Similarly, the excitation and stable propagation of an envelope wave solution [6, 9] or multi-phase sine-Gordon solutions [5] may be considered. The desired control behaviour of a periodic wave in reaction–diffusion systems was studied in [3], while control of the wave maximum to avoid blow-up has been considered in [7, 8]. For coupled equations, the generation of an optical mode has been suggested as the goal of the control in [13]. Finally, the desired energy of a wave may be the aim of the control [14]. Therefore, the key point in the development of a control algorithm is the formalization of the control goal in the form of a certain goal function.

Consider, for example, the sine-Gordon equation,
2.1 which possesses, in particular, an exact travelling wave solution in the form of a kink or an antikink,
2.2 where *V* is the phase velocity and is the initial phase. The minus sign in equation (2.2) corresponds to the case of the antikink that satisfies the boundary conditions,
2.3 The initial conditions should be consistent,
2.4 However,
2.5 at , destroys the consistency, and the stable antikink propagation is not realized due to the growing oscillations on the wavefront [15].

To prevent the development of oscillations, one can try to achieve the value of the energy corresponding to the antikink. However, as shown in [14], this does not result in serious deviations in the wave behaviour. One cannot control either the shape or the velocity of the wave. Our idea relates to the simultaneous control of the shape and velocity, thus the control has two aims.

## 3. Choice of control method

Here, feedback control is employed. The development of the control algorithm is carried out on the basis of the speed–gradient method [12], which was previously used for oscillation control problems, see, for example, [12, 16]. The feedback methods for nonlinear waves were developed in [1, 2, 7, 8, 10, 11], while feed-forward (non-feedback) controlling actions were considered in [4–6, 9]. The main feature of our method is the simultaneous inclusion of the desired shape and velocity of the wave in the feedback.

The next feature of the control algorithm concerns its distribution. Non-distributed control algorithms, where the control function has finite dimension and is applied to certain points of the system (typically, to its boundary conditions), were developed in [3, 9–11, 13], where the control function does not depend on the coordinate. In contrast, distributive control has infinite dimensions and may be different for every spatial coordinate in the system [1, 2, 7, 8]. Our non-distributive algorithm does not allow us to localize the nonlinear wave solution to the sine-Gordon equation [14, 17]. Therefore, it seems that localized waves may be efficiently controlled by a distributive algorithm.

One can control boundary conditions to achieve a significant difference in the behaviour of the wave [11, 13, 18]. However, this seems to be problematic for the control of localized waves. Another way is to control the value of one of the coefficients of the equations or to include control terms [1–3, 7–9].

Let us illustrate this with the example of the sine-Gordon equation. Our previous results [15, 17] show that the control of the coefficient allows us to provide stable propagation of those waves whose shape is described by analytical travelling wave solutions to the equation. However, waves with another shape are not generated. This is why modification of the algorithm from [15, 17] is needed. Let us introduce the control function into the sine-Gordon equation (2.1) in the form of the additional term,
3.1 Let us define the aim of the control as the tendency of the solution of equation (3.1) to the desired wave solution, . Let us introduce a *distributed* error of the shape of the wave as
3.2

Then, one obtains 3.3

Let us introduce the auxiliary error function as follows: 3.4 where are the parameters of the algorithm.

Then, the functional *Q* may be defined as
3.5 An auxiliary control goal is to diminish the functional (3.5). However, this does not depend *explicitly* on the control function *u* while its first temporal derivative does,
3.6 where … denotes terms that do not depend on *u*. Then, the distributed control function *u* is assumed to diminish ,
3.7 where is the parameter of the algorithm. Without loss of generality one may assume that .

The control may be switched on/off forcibly,
3.8 where *H* is the unit-step function, is the time for switch on and is the time for switch off. The switch on/off factors in equation (3.8) may be used together or alone.

## 4. Control of localized nonlinear waves

### (a) Single equation

As noted before, the control of solutions to the sine-Gordon equation has been studied in our previous papers [15, 17]. However, it was found that only particular localized waves may be supported by the algorithm used in [15, 17]. Now, we first consider the improved algorithm developed in §3 to achieve the target function ,
4.1 which does not correspond to any analytical solution to equation (2.1). Here *k*, , , *W*, *A* and *B* are the free parameters. We chose their values to obtain the shape of the wave in the form of an antikink with a hole and a bump; see the dashed line profile in figure 1.

The initial conditions are taken in the form (2.4); these are consistent with the exact antikink solution, but not with the desired solution (4.1). The boundary conditions are described by equation (2.3); they are consistent with equation (4.1).

For simulations, the values of the parameters are chosen as
while the parameters of the control are
4.2 To demonstrate the role of the control, the switch-on control is used in the expression for *u* (3.8) with . The switch-off may also be used to show that after switching off the control the wave (4.1) cannot stably propagate. However, only the delayed switch-on is used in figure 1.

Numerical simulations of equation (3.1), shown in figure 1, describe the process of transformation of the initial conditions (2.4) under the control which is switched on at . One can see that propagation of the antikink (2.2) happens before the control is switched on at . Then, transformation to the desired wave (4.1) happens in a short time. Both the velocity of (4.1) and its shape are changed in the numerical solution with the hole before the wavefront and the hump behind it. Further stable propagation of the wave (4.1) is observed. With stable propagation, the control term is very small according to its definition, and equation (3.1) is almost the same as the original sine-Gordon equation without control. However, using the switch-off component in the control function and introducing the switch-off time demonstrates the failure of the stable propagation of the wave (4.1) after forcibly switching off the control.

The choice of the values of the parameters of the control (4.2) is not unique, and other values give rise to the same stable wave propagation. However, one has to note the importance of taking into account the control of both the shape and the velocity: for only the velocity of the wave (4.1) may be achieved, not its shape for any value of *γ*.

### (b) Coupled equations

Consider nonlinear coupled equations,
4.3 and
4.4 which describe highly nonlinear dynamic processes in diatomic crystals [19]. Here *v* is the longitudinal macro-strain and *u* is the micro-displacement characterizing the movement of defects of the crystalline structure.

The exact travelling wave solution represents coupled functions *u*, *v* [19],
4.5 and
4.6 where and the parameters are
One has to note that the initial position of both waves is defined by the same parameter
, and their phase velocity *V* is the same. Numerical simulations in [20–22] revealed significant violation of the localization of the waves in the case when the positions of the initial pulses for *u* and *v* do not coincide or when their initial velocities are different. To prevent delocalization caused by imperfections in the initial conditions, a distributed control algorithm is developed using the same procedure as for single equations. Let us begin by introducing control function in only one equation (4.3),
4.7 while equation (4.4) does not contain any control term. The function *w* will control only the behaviour of *v*. The initial conditions are
4.8 and
4.9 where
4.10 Typical shapes of the waves are shown in figure 2. Let us choose the goal function in the form of the exact solution for *v* (4.5), but with an initial phase defined as ,
4.11 In particular, may be chosen equal to or . The function is defined as the first temporal derivative of equation (4.11). Then,
4.12
4.13
4.14 and the function *w* is defined using the same procedure as that used before. Finally, we obtain
4.15 where .

The combined profiles of the waves for various times and without control are shown in figure 3. One can see an additional secondary localized wave *u* arising due to inconsistencies in the initial conditions shown in figure 2. This large localized wave is accompanied by a small secondary localized wave *v*. At the same time, there are combined localized waves *u* and *v* that result from initially separated waves (4.8) and (4.9).

The combination of the profiles in the presence of the control (4.11) at corresponding to the initial phase of *v* in equation (4.8) is shown in figure 4. Now the secondary wave *u* is partly delocalized and modulated. Comparison of the profiles at
and in figure 4 shows the partial delocalization because this secondary wave does not completely disappear as time goes on, but it decreases between figure 4*a* and figure 4*b*. Also now there is no secondary localized wave *v*.

The choice of in the aim function (4.11) corresponds to the initial phase for *u* in equation (4.9). One can see in figure 5 that a combination of the maxima of *u* and *v* is even more perfect than in figure 4. The profile of *v* is smooth and corresponds well to the goal function. Moreover, no secondary localized waves arise behind the primary localized waves *u* and *v*.

## 5. Conclusion

This new distributed feedback control algorithm allows the generation and further stable propagation of rather arbitrary localized nonlinear waves in the processes governed by single equations. Extension of the algorithm to coupled equations does not result in perfect results for both functions. A possible reason for this lies in the application of the control to only one variable that is chosen for physical reasons. The algorithm appears to be universal—it has the same functional form for different equations. The control should be distributive to support localization of the wave. It is important to follow two goals, achievement of the desired shape and velocity, simultaneously.

## Author's contributions

A.P. carried out the theory. B.A. performed the numerical simulations. Both authors read and approved the manuscript.

## Competing interests

The authors declare that they have no competing interests.

## Funding

This work was performed in IPME RAS and was supported by the Russian Science Foundation, project 14-29-00142.

## Footnotes

One contribution of 15 to a theme issue ‘Horizons of cybernetical physics’.

- Accepted October 9, 2016.

- © 2017 The Author(s)

Published by the Royal Society. All rights reserved.