# Transition control of the Blasius boundary layer using passivity

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## Abstract

The control problem for linearised three-dimensional perturbations about a nominal laminar boundary layer over a flat plate (the Blasius profile) is considered. With a view to preventing the laminar to turbulent transition, appropriate inputs, outputs, and feedback controllers are synthesised that can be used to stabilise the system. The linearised Navier–Stokes equations are reduced to the Orr–Sommerfeld and Squire equations with wall-normal velocity actuation entering through the boundary conditions on the wall. An analysis of the work-energy balance is used to identify an appropriate sensor output that leads to a passive system for certain values of the streamwise and spanwise wavenumbers. Even when the system is unstable, it is demonstrated that strictly positive real feedback can stabilise this system using the special output.

## Keywords

Transition Control Boundary Layer Passivity## 1 Introduction

The viscous effects in unseparated flow over a body are concentrated in a thin layer adjacent to the body’s surface known as the boundary layer. The no-slip condition for the fluid velocity on this surface leads to a form of drag known as skin friction drag. The size of this force depends strongly on whether the flow is laminar or turbulent with laminar boundary layers producing less drag. Hence, there is a great deal of motivation to prevent transition between the two flow regimes from occurring.

Historically, the transition problem has been studied by linearising the Navier–Stokes equations about a nominal velocity profile consisting of a baseline laminar flow and addressing the stability of small perturbations [10, 17]. For boundary layer flows over a flat plate, this profile has typically been taken to be the two-dimensional Blasius solution [16]. If a two-dimensional spatial Fourier transform is taken of the linearised equations (corresponding to assuming spatially oscillating perturbations in the streamwise and spanwise directions), one arrives at the Orr–Sommerfeld equation describing the wall-normal velocity component and the Squire equation describing the wall-normal vorticity component. Transition can be studied by determining the eigenvalues of the Orr–Sommerfeld/Squire model governing the perturbations (Drazin and Reid 2001).

It has been noted by experimentalists that transition typically occurs at Reynolds numbers (based on distance along the plate) that are smaller than those predicted by linear eigenvalue theory [10]. It has been postulated that large transient growth in the flow perturbations can trigger transition via nonlinear mechanisms before the linear instability mechanism occurs. Transient growth in the Blasius boundary layer was studied by Butler and Farrell [7], who noted that at some Reynolds numbers, the worst case transient growth occurred at streamwise wavenumbers that were zero and nonzero spanwise wavenumbers.

Linear theory can be exploited using the associated models to develop feedback controllers which address stabilisation at those Reynolds numbers and wavenumbers where the instabilities would otherwise occur. Active feedback control requires the introduction of appropriate sensors and actuators and the design of feedback controllers. The history and use of linear state-space models based on the Orr–Sommerfeld/Squire system to design feedback controllers is described by Bewley [4].

Active control design has been investigated for several flows with emphasis on plane Poiseuille flow [5, 12] and the Blasius boundary layer [1, 2]. The Poiseuille flow corresponds to the fully developed flow in a channel between two parallel infinite plates and the Blasius boundary layer is the two-dimensional laminar flow over a semi-infinite flat plate. This paper will concentrate on the latter case. The use of linear models to develop linear controllers for what is generally considered to be a nonlinear phenomenon (transition to turbulence) has been championed by many authors who have argued that linear control systems based on linear models can deal with the initial linear amplification of disturbances, thus preventing the subsequent nonlinear transition behaviour [13].

The type and location of the sensors and actuators has a profound effect on the achievable stability, performance, and robustness of a feedback strategy. Sensor and actuator locations have been considered by Belson et al. [2] in the Blasius case. A very useful paradigm for robust feedback controller design is passivity-based control. Passive systems [9] are those that only store or dissipate energy. Strict passivity is a stronger property than passivity and corresponds to systems that only consume (dissipate) energy. The passivity theorem [9] states that the negative feedback interconnection of a passive system and a strictly passive system (with finite gain) is \(L_2\)-stable, that is, \(L_2\) (finite energy) inputs produce \(L_2\) (finite energy) outputs.

In a previous work [8], we studied the passivity property in two dimensions in the context of the Orr–Sommerfeld equation with actuation inputs implemented as wall-normal velocity. A dual output (based on energy analysis) was shown to be the second spatial derivative (normal direction) of the streamwise velocity perturbation at the wall. Although passivity of the model relating this input and output could not be demonstrated, it was shown that an appropriate closed-loop system could be made passive with a baseline Poiseuille flow but not with a Blasius flow. Passivity ideas have been employed by Sharma et al. [18] and Heins et al. [11] in the stabilisation of a Poiseuille flow (the fully developed flow in a channel between two parallel plates).

The paper is organised as follows. Section 2 defines the key notions involving passivity. In Sect. 3, we resume the passivity analysis from [8], this time in three spatial dimensions using both the Orr–Sommerfeld and Squire equations. In Sect. 4, spatial discretisation of the Orr–Sommerfeld and Squire equations is accomplished using Hermite cubic finite elements to describe the wall-normal velocity and vorticity components. This approach was originally used for the Orr–Sommerfeld equation with Poiseuille flows by Mamou and Khalid [14]. In Sect. 5, stabilisation using the passivity-based output is demonstrated at a Reynolds number and wavenumber pair that is open-loop unstable. Section 6 presents some concluding remarks.

## 2 Feedback controller design

### 2.1 Passivity and feedback design

Consider the feedback system shown in Fig. 1 where \({\varvec{d}}_1(t)\), \({\varvec{d}}_2(t)\), \({\varvec{m}}_1(t)\), and \({\varvec{m}}_2(t)\) are functions of time *t*. Generically, \({\varvec{m}} \in L_2\) if the \(L_2\)-norm satisfies \(||{\varvec{m}}||_2 {\mathop {=}\limits ^{\Delta }} \sqrt{\int _0^{\infty }{\varvec{m}}^\mathsf{T}(t) {\varvec{m}}(t)\,\mathrm {d}t} < \infty \) (the symbol \((\;)^\mathsf{T}\) denotes the matrix transpose and \((\;)^\mathsf{H}\) denotes the complex-conjugate transpose). We also have, \({\varvec{m}} \in L_{2e}\) (the extended \(L_2\)-space) if \(||{\varvec{m}}||_{2T} {\mathop {=}\limits ^{\Delta }} \sqrt{\int _0^T{\varvec{m}}^\mathsf{T}(t) {\varvec{m}}(t)\,\mathrm {d}t} < \infty \), \(0 \le T < \infty \). Note that \(L_2 \subset L_{2e}\). Consider a system \({\varvec{m}}(t) = ({\varvec{{{\mathcal {G}}}}}{\varvec{e}})(t)\) where the operator \({\varvec{{{\mathcal {G}}}}}: L_{2e} \rightarrow L_{2e}\) (possibly nonlinear and time-varying) maps the input \({\varvec{e}}\in L_{2e}\) into the output \({\varvec{m}}\in L_{2e}\). The gain of \({\varvec{{{\mathcal {G}}}}}\) (which is the induced norm on \(L_2\)) is defined to be \(||{\varvec{{{\mathcal {G}}}}}|| = \displaystyle {\sup _{\mathbf{0} \ne {\varvec{e}} \in L_2}} ||{\varvec{{{\mathcal {G}}}}}{\varvec{e}}||_2/||{\varvec{e}}||_2\).

If the system \({\varvec{{{\mathcal {G}}}}}\) is square (the number of inputs in \(\varvec{e}\) is equal to the number of outputs in \(\varvec{m}\)), the operator \({\varvec{{{\mathcal {G}}}}}\) is defined to be strictly passive if \(\int _0^T {\varvec{m}}^\mathsf{T}(t){\varvec{e}}(t)\,\mathrm {d}t = \int _0^T {\varvec{e}}^\mathsf{T}(t){\varvec{{{\mathcal {G}}}}}{\varvec{e}}(t)\,\mathrm {d}t \ge \delta + \epsilon \int _0^T {\varvec{e}}^\mathsf{T}(t){\varvec{e}}(t) \,\mathrm {d}t\), \(\forall {\varvec{e}} \in L_{2e}\), \(0\le T < \infty \), for some \(\epsilon >0\) and real constant \(\delta \) which may depend on the initial conditions of \({\varvec{{{\mathcal {G}}}}}\). If \(\epsilon =0\), the system is passive.

*s*denotes the complex-valued Laplace transform variable and \(\mathrm {i}=\sqrt{-1}\). Note that \(\varvec{G}(s)=\varvec{C}(s\varvec{I} -\varvec{A})^{-1}\varvec{B} + \varvec{D}\) is the transfer matrix corresponding to the state-space model in Eqs. (1) and (2). Here, \({\varvec{I}}\) is the identity matrix of appropriate dimension. If the system is minimal, i.e., it is controllable and observable, then \(L_2\)-stability of \({\varvec{{{\mathcal {G}}}}}\) (\({\varvec{e}} \in L_2\) implies that \({\varvec{m}}= {\varvec{{{\mathcal {G}}}}}{\varvec{e}} \in L_2\)) corresponds to the matrix \({{\varvec{A}}}\) having eigenvalues with negative real parts. For stable LTI systems, the gain can be shown (Vidyasagar, 1992) to be \(||{\varvec{{{\mathcal {G}}}}}|| = ||{\varvec{G}}(s)||_{\infty } = \displaystyle {\sup _{\omega \in {{\mathcal {R}}}}} {\bar{\sigma }}[{\varvec{G}}({\mathrm i}\omega )]\) where \({\bar{\sigma }}\) denote the largest singular value.

Passive LTI systems of this form correspond to the case where \({{\varvec{G}}}(s)\) is a positive real (PR) transfer function. When \({{\varvec{G}}}(s)\) is a proper real rational matrix function of *s*, it is positive real if no element of \({{\varvec{G}}}(s)\) has a pole in \({\mathfrak {R}e}\{s\} > 0\); \(\text{ He }[{{\varvec{G}}}({\mathrm i}\omega )] = (1/2)[{{\varvec{G}}}({\mathrm i}\omega ) + {{\varvec{G}}}^\mathsf{H}(\mathrm {i}\omega )]\ge \mathbf{0}\) for all real \(\omega \) with \(\mathrm {i}\omega \) not a pole of \({{\varvec{G}}}(s)\) (\(\text{ He }(\;\;)\) denotes the Hermitian part of a square matrix); and if \(\mathrm {i}\omega _0\) is a pole of any element of \({{\varvec{G}}}(s)\), it is at most a simple pole and the residue matrix \(\lim _{s\rightarrow \mathrm {i}\omega _0} (s-\mathrm {i}\omega _0) {{\varvec{G}}}(s)\) is non-negative definite Hermitian.

*s*, \({\varvec{K}}(s)\), is SPR if no element of \({{\varvec{K}}}(s)\) has a pole in \({\mathfrak {R}e}\{s\} \ge 0\); \(\text{ He }[{{\varvec{K}}}(\mathrm {i}\omega )] > \mathbf{0}\) for all real \(\omega \in (-\infty ,\infty )\); and \(\displaystyle {\lim _{\omega \rightarrow \infty }}\omega ^2\text{ He }[{{\varvec{K}}} (\mathrm {i}\omega )] > \mathbf{0}\). A system with transfer matrix \({\varvec{K}}(s) + \epsilon {\varvec{I}}\) is strictly passive with finite gain if \({\varvec{K}}(s)\) is SPR and \(\epsilon > 0\). The importance of passivity for feedback design lies in the passivity theorem [9], which addresses the feedback system shown in Fig. 1. One form of the passivity theorem states that if \({\varvec{{{\mathcal {G}}}}}\) is passive and \({\varvec{{{\mathcal {K}}}}}\) is strictly passive with finite gain, then \({\varvec{d}}_1, {\varvec{d}}_2 \in L_2\) implies that \({\varvec{e}}_1, \varvec{e}_2, {\varvec{m}}_1, {\varvec{m}}_2 \in L_2\).

It should be noted that although passivity of \({\varvec{{{\mathcal {G}}}}}\) and strict passivity of \({\varvec{{{\mathcal {K}}}}}\) are sufficient for \({\varvec{{{\mathcal {H}}}}}\) to be strictly passive, they are not necessary conditions. An interesting design problem for LTI systems is the following: given \({\varvec{G}}(s)\) (not necessarily passive), find \({\varvec{K}}(s)\) to render \({\varvec{H}}(s) = {\varvec{G}}(s)[{\varvec{I}} + {\varvec{K}}(s){\varvec{G}}(s)]^{-1}\) SPR [15] which would mean that the system in Fig. 2 is \(L_2\)-stable for passive \({\varvec{\Delta }}\).

### 2.2 Strictly positive real design

### 2.3 Model order reduction

### 2.4 Motivation for a passivity analysis and passivity-based control

This paper seeks to analyse the passivity properties of the linearised system relating the wall-normal velocity as actuation to an appropriate sensor output in the case of the flow perturbations acting on the Blasius boundary layer. Hence, the passivity analysis answers the question “what should be measured if the actuation is wall-normal blowing and suction?” Determining an output that leads to a passive system produces a system that is easy to stabilise since any strictly passive negative feedback controller leads to this result. Hence, a large family of stabilising controllers is available. Using a strictly passive feedback controller (or an SPR controller in the LTI case), one produces a closed-loop system whose stability is robust with respect to passive perturbations connected in a feedforward arrangement (which continues to produce a passive plant) or a negative feedback arrangement (which is stable if the nominal closed-loop system is strictly passive). As will be examined in a later section, the feedforward perturbation could correspond to the system nonlinearities. The feedback perturbation could correspond to unmodelled sensor and actuator dynamics.

## 3 Passivity analysis of the Orr–Sommerfeld/Squire equations

### 3.1 Blasius boundary layer

We consider a three-dimensional flow field occupying the region \((x,y,z) \in [0,\infty ]\times [a,b]\times [-\infty ,\infty ]\) with a base parallel laminar flow (*U*(*y*), 0, 0) and associated pressure field *P*(*x*, *y*, *z*, *t*). The Blasius boundary layer flow (Schlichting, 1979) is depicted in Fig. 3 and we shall take \(a=0\). Although \(b\rightarrow \infty \), a finite computational boundary for *b* will be employed as discussed below. The nominal laminar flow (*U*, *V*, 0) is known to be nonparallel (\(V\ne 0\)), but we shall make the approximation \(V=0\) and take *U*(*y*) to be the Blasius solution: \(U(y) = \mathrm {d}f(\eta )/\mathrm {d}\eta \) (this has been nondimensionalised using the free-stream velocity \(U_0\)) where \(\eta = y_d\sqrt{\rho U_0/(\mu x_d)}\) (\(x_{\text{ d }}\), \(y_{\text{ d }}\), and \(z_{\text{ d }}\) refer to dimensional coordinates) and \(f(\eta )\) is the solution of \(2\mathrm {d}^3f/\mathrm {d}\eta ^3 + (\mathrm {d}^2f/\mathrm {d}\eta ^2) f = 0\) with \(\mathrm {d}f(0)/\mathrm {d}\eta = f(0) =0\) and \(\mathrm {d}^2f(0)/\mathrm {d}\eta ^2 = 0.33205733622\) which yields the correct asymptotic boundary condition \(\mathrm {d}f(\eta )/\mathrm {d}\eta =1\) as \(\eta \rightarrow \infty \).

### 3.2 Orr–Sommerfeld/Squire equations

*u*(

*x*,

*y*,

*z*,

*t*),

*v*(

*x*,

*y*,

*z*,

*t*),

*w*(

*x*,

*y*,

*z*,

*t*), and

*p*(

*x*,

*y*,

*z*,

*t*) about the Blasius flow, the linearised incompressible Navier–Stokes equations [10] are

*H*. The boundary conditions are taken to be \(u(x,a,z,t) = u(x,b,z,t) = v(x,b,z,t) = w(x,a,z,t) = w(x,b,z,t)=0\) and the control variable is taken to be

*v*(

*x*,

*a*,

*z*,

*t*), which corresponds to wall-normal blowing and suction.

*v*and vorticity \(\zeta \):

*x*and

*z*directions, or alternatively letting

*u*and

*v*; note that the symbol \(\nu \) will not be used in this paper to refer to a fluid’s kinematic viscosity).

### 3.3 Passivity analysis

### 3.4 Measurements

*y*yields

### 3.5 Nonlinear passivity analysis

*x*, the nonlinear version of Eq. (10) with respect to

*y*, and the nonlinear version of Eq. (11) with respect to

*z*and evaluates them at the wall, one arrives at

In Fig. 4, a disturbance input \(d_{\text{ w }}(x,z,t)\) has been added in addition to the control input \(v_w(x,z,t)\) as well as a feedback perturbation \({\varvec{\Delta }}_{\text{ w }}:L_{2e,{\text{ w }}}\rightarrow L_{2e,{\text{ w }}}\). To present the fundamental stability result governing this setup, let us define the space of finite energy functions \(L_{2,{\text{ w }}} = \{ \mathbf{e}_{\text{ w }} \;|\; \lim _{T\rightarrow \infty } \langle \mathbf{e}_{\text{ w }},\mathbf{e}_{\text{ w }}\rangle _{{\text{ w }},T} < \infty \}\). Now apply Zames’s conic sector theorem [20] to this setup. Let us assume that the feedback systems \({\varvec{H}}_{\alpha \beta }(s) = {\varvec{G}}_{\alpha \beta } (s)[{\varvec{I}} + {\varvec{K}}_{\alpha \beta }(s){\varvec{G}}_{\alpha \beta }(s)]^{-1}\) are in the cone \([a_{\text{ c }},\infty ]\) for each \((\alpha ,\beta )\). Hence, the time and space domain operator \({\varvec{{{\mathcal {H}}}}}_{\text{ w }} \in \mathrm{cone}[a_{\text{ c }},\infty ]\) and we can state that \(d_{\text{ w }}\in L_{2,{\text{ w }}} \Rightarrow v_{yyy,{\text{ w }}}\in L_{2,{\text{ w }}}\;\mathrm{if} \; {\varvec{\Delta }}_{\text{ w }} \in \mathrm{cone}(0,-1/a_{\text{ c }})\).

## 4 Spatial discretisation using finite elements

*y*-domain [

*a*,

*b*] is broken into \(N_{\text{ e }}\) equally sized finite elements (width \(\ell \)) with the value of

*y*at the nodes (element boundaries) denoted by \(y_j=(j-1)\ell \), \(j=1,\ldots ,N_e+1\) where \(\ell = (b-a)/N_e\). Let us denote the value of \(\hat{v}\) and its derivative at the nodes by \(v_j(t) = \hat{v}(y_j,t)\) and \(v^{\prime }_j(t) = \hat{v}_y(y_j,t)\) with similar definitions for \(\zeta _j(t)\) and \(\zeta ^{\prime }_j(t)\). Within the

*j*th element, the following trial solutions are assumed:

Now, take the (real) control input to be \({\varvec{e}} = [{\mathfrak {R}e}\{v_1\}\;{\mathfrak {I}m}\{v_1\}]^\mathsf{T}\) and the (real) measurement output is taken to be \({\varvec{m}}(t) = {\varvec{m}}_{\mathrm{{os}}}(t) = [ {\mathfrak {R}e}\{\hat{v}_{yyy}(a,t)\}\; {\mathfrak {I}m}\{\hat{v}_{yyy}(a,t)\}]^\mathsf{T}\). If the (real) state vector is taken as \({\varvec{x}} = [ {\varvec{q}}_{\mathrm{{os}},r}^\mathsf{T}\;{\varvec{q}}_{\mathrm{{os}},i}^\mathsf{T}\; {\varvec{q}}_{\text{ sq },r}^\mathsf{T}\;{\varvec{q}}_{\text{ sq },i}^\mathsf{T}\; ]^\mathsf{T} - \mathrm{block diag}\{{\varvec{M}}_r^{-1}{\varvec{B}}_{2r}, {\varvec{M}}_r^{-1}{\varvec{B}}_{2r}\}\varvec{e}\) where \({\varvec{q}}_{\mathrm{{os}},r} = {\mathfrak {R}e}\{{\varvec{q}}_{\mathrm{{os}}}\}\), \({\varvec{q}}_{\mathrm{{os}},i} = {\mathfrak {I}m}\{{\varvec{q}}_{\mathrm{{os}}}\}\), \({\varvec{q}}_{\text{ sq },r} = {\mathfrak {R}e}\{{\varvec{q}}_{\text{ sq }}\}\), \({\varvec{q}}_{\text{ sq },i} = {\mathfrak {I}m}\{{\varvec{q}}_{\text{ sq }}\}\), then the methods of Damaren [8] can be used to derive a state-space model of the form in Eqs. (1) and (2).

## 5 Numerical example

Orr–Sommerfeld/Squire eigenvalues for Blasius case, \(Re= 800\), \(\alpha =0.25\), \(\beta = 0.2\)

Schmid and Henningson | Damaren \((N_e=240,b=24)\) | ||
---|---|---|---|

\({\mathfrak {R}e}\{\lambda /\alpha \}\) | \({\mathfrak {I}m}\{\lambda /\alpha \}\) | \({\mathfrak {R}e}\{\lambda /\alpha \}\) | \({\mathfrak {I}m}\{\lambda /\alpha \}\) |

Orr–Sommerfeld | |||

+ 0.00287572 | 0.39065421 | + 0.002826 | 0.390610 |

− 0.23434181 | 0.54772364 | − 0.234257 | 0.547720 |

− 0.31005379 | 0.33866341 | − 0.309990 | 0.338641 |

− 0.37872068 | 0.79181869 | − 0.378502 | 0.791717 |

− 0.40505900 | 0.65749147 | − 0.404910 | 0.657402 |

Squire | |||

− 0.13769021 | 0.23869653 | − 0.137679 | 0.238688 |

− 0.23747142 | 0.41904327 | − 0.237431 | 0.419016 |

− 0.31360121 | 0.57017612 | − 0.313527 | 0.570123 |

− 0.37342899 | 0.70889059 | − 0.373318 | 0.708801 |

− 0.41937502 | 0.84255313 | − 0.419228 | 0.842414 |

Let us return to the case \({\textit{Re}}=800\), \(\alpha = 0.25\), and \(\beta =0.20\) which is unstable (and hence nonpassive). Examining the eigenvalues of \({\varvec{A}}-{\varvec{B}}{\varvec{D}}^{-1}{\varvec{C}}\) reveals a zero at \(s= 0.2412\) and hence the system is unstable and nonminimum phase. However, as intimated in Eq. (28) et seq., energy dissipativeness may be possible using an SPR controller. To illustrate this, we employed the design procedure of Sec. 2.2 using a reduced-order modal model with the 16 most controllable/observable modes and \({\varvec{Q}} = {\varvec{C}}_\mathrm{r}^\mathsf{T}{\varvec{C}}_\mathrm{r}\) (it is only positive semidefinite in this case), \({\varvec{R}} = {\varvec{I}}\), and \({\varvec{Q}}_\mathrm{c} = 5 {\varvec{I}}\), where \({\varvec{C}}_\mathrm{r}\) is the output matrix of the reduced-order model. This leads to a stable closed-loop system. The eigenvalues of the Hermitian part of the closed-loop transfer matrix from \({\varvec{d}}_1\) to \({\varvec{m}}_1 = {\varvec{m}}_{\mathrm{{os}}}\) (\({\varvec{H}}(s) = {\varvec{G}}(s)[{\varvec{I}} + {\varvec{K}}(s){\varvec{G}}(s)]^{-1}\)) are depicted in Fig. 6. Although \({\varvec{K}}(s)\) is determined using a reduced-order model, the closed-loop stability calculation and the determination of \({\varvec{H}}(s)\) make use of the full-order model of \({\varvec{G}}(s)\). Given the negative excursion of the smallest eigenvalue, we conclude that \({\varvec{H}}(s)\) is not positive real since \(a_\mathrm{c} \doteq -0.108\). However, it does belong to the conic sector \([a_\mathrm{c},\infty ]\) and hence is guaranteed to be stable when placed in negative feedback with an uncertainty block \({\varvec{\Delta }}\) strictly belonging to the conic sector \((0,-1/a_\mathrm{c})\).

## 6 Conclusions

The important property of passivity has been examined in the case of the boundary-feedback controlled Orr–Sommerfeld/Squire equations. A study of the work-energy balance was used to select the appropriate sensed variables corresponding to wall-normal velocity actuation. This corresponded to the third derivative (wall-normal direction) of the wall-normal velocity, which it was demonstrated can be constructed from pressure measurements made along the wall. This choice of sensing and actuation was shown to lead to passivity when the streamwise wavenumber \(\alpha \) was equal to zero. This was validated by looking at the Nyquist plot for a typical case. Unfortunately, the Squire modes are not observable from this output which limited its applicability. However, it was shown that stabilisation could be achieved in an unstable case using a strictly positive real controller and this output. An analysis was presented that showed that a series of stabilising linear designs at each wavenumber pair could provide stability of the original nonlinear distributed parameter system.

## Notes

### Compliance with ethical standards

### Conflict of interest

The author declares that he has no conflict of interest.

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