# Boundary Control for a Certain Class of Reaction-Advection-Diffusion System

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Function Spaces

## 3. Reaction-Advection-Diffusion (R-A-D) System

## 4. Target System

**Lemma**

**1**

**Poincaré Inequality**). [12] For any $w\in {H}^{1}(0,1)$ (Sobolev space) the following inequalities hold

**Proof.**

## 5. Backstepping Transformation

## 6. Integral Equation Solution

## 7. Neumann Stabilizing Controller

## 8. MATLAB${}^{\circledR}$ Code

`MATLAB`${}^{\circledR}$[19]. To this end, we use the function

`pdepe`. This function is able to solve initial-boundary value problems for parabolic and elliptic PDEs in the one space variable x and time t. This function solves PDEs of the form

`MATLAB`${}^{\circledR}$, the syntax for the

`pdepe`function is as follows

wheresol=pdepe(m,pdefun,icfun,bcfun,xlinspace,tlinspace);

`m`is defined in (131),

`pdefun`is a function that defines the components of the PDE to be solved,

`icfun`is a function that defines the initial condition,

`bcfun`is a function that defines the boundary conditions,

`xlinspace`is a vector with elements $[{x}_{0},{x}_{1},\dots ,{x}_{n}]$ concerning with n specific points (defined by the user) for which a solution is required for every value of

`tlinspace`, also a vector with elements $[{t}_{0},{t}_{1},\dots ,{t}_{f}]$ for f specific points (also defined by the user).

`a`represents the left boundary condition and

`b`represents the right boundary condition, the solution components must satisfy a boundary condition of the form

`pdefun`function is declared as it is shown below.

`function[c,f,s]=pdefun(x,t,u,DuDx)`

`global lambda; global b;`

`c=1;`

`f=DuDx;`

`s=b*f + lambda*u;`

`icfun`is declared as follows.

`function u0=icfun(x)`

`global initial`

`global i`

`u0 = initial(i);`

`i=i+1;`

`bcfun`function is declared following the form given by (133).

`cntrllr`is for the control law (actuation signal).

`function[pl,ql,pr,qr] = bcfun(xl,ul,xr,ur,t)`

`global cntrllr; global b;`

`pl=b/2*ul;`

`ql=1;`

`pr=-cntrllr;`

`qr=1;`

`pdepe`function. The main code that handles these last functions is shown next.

`clear all;`

`global cntrllr; cntrllr=0;`

`global lambda; lambda=30;`

`global b; b=10;`

`global lambda0; lambda0=lambda-b^2/4;`

`m=0;`

`x0=0; xf=1; xn=40;`

`x=linspace(x0,xf,xn+1);`

`t0=0; step=0.001; tf=step; tn=3;`

`zf=20; tt=zf*step; ttn=zf*tn;`

`t=linspace(t0,tf,tn);`

`T=linspace(t0,tt,ttn+1);`

`tt`is for the simulation time and

`ttn`is for the total number of simulation data, i.e., the sum of all solution slices. The initial conditions are defined through the following code.

`global initial; initial=5*(1-2*sin(3*pi*x/2));`

`u=initial;`

`global i; i=1;`

`pdepe`function returns the solution in a 3-D array called

`sol`, where

`sol(i,j,k)`approximates the k-th component of the solution ${u}_{k}$ evaluated at $t\left(i\right)$ and $x\left(j\right)$. The size of

`sol`is

`ttn`-by-

`xn`-by-

`initial`and the command to extract the solution of the PDE for the defined size is as follows.

`u = sol(:,:,1);`

`ND=5;`

`tg=zeros(1,ttn/ND+1);`

`ug=zeros(ttn/ND+1,xn+1);`

`ug(1,:)=initial;`

`for`cycle is implemented to compute the whole solution of the PDE system. The code shown at the bottom is used to plot the solution.

`for z=1:zf`

`sol=pdepe(m,@pdefun,@icfun,@bcfun,x,t);`

`u=[u; sol(:,:,1)];`

`root = sqrt(lambda0*(1-x.^2));`

`bss1 = besseli(1,root);`

`bss2 = besseli(2,root);`

`coc1 = (3/2)*bss1./root;`

`coc2 = bss2./(1-x.^2);`

`coc1(end) = coc1(end-1)*.89;`

`coc2(end) = coc2(end-1)*.89;`

`coc = coc1 + coc2;`

`int = coc.*exp(b/2*(x-1)).*u(end,:);`

`integ = trapz(x,int);`

`u_1 = u(end,end);`

`cntrllr = -((lambda0+b+1)/2)*u_1 - lambda0*integ;`

`initial = u(end,:);`

`i=1;`

`t0=t0+step;`

`tf=tf+step;`

`t=linspace(t0,tf,tn);`

`end`

`for h=1:ttn/ND`

`tg(1,h+1) = T(1,ND*h+1);`

`ug(h+1,:) = u(ND*h+1,:);`

`end`

`figure(1);`

`surf(x,tg,ug);`

`view(45,30);`

`title(’Reaction-Advection-Diffusion Equation ’,’FontSize’,12’);`

`xlabel(’Distance $x$’,’Interpreter’,’Latex’,’FontSize’,12’);`

`ylabel(’Time $t$’,’Interpreter’,’Latex’,’FontSize’,12’);`

`zlabel(’Solution $u(x,t)$’,’Interpreter’,’Latex’,’FontSize’,12’);`

`view(55,30)`

`figure(2);`

`plot(tg,ug(:,end)’);`

`xlabel(’Time $t$’,’Interpreter’,’Latex’,’FontSize’,15’);`

`ylabel(’Solution $u(1,t)$’,’Interpreter’,’Latex’,’FontSize’,15’);`

`grid on;`

`pdefun`,

`icfun`and

`bcfun`functions, respectively. The main code is saved as BCRAD.m which contains the remaining MATLAB${}^{\circledR}$ code in the order presented above. The BCRAD.m file must be compiled and then executed to get the numerical solution.

## 9. Simulation Results

## 10. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

DPSs | Distributed Parameters Systems |

ISS | Input-to-State Stability |

ODEs | Ordinary Differential Equations |

PDEs | Partial Differential Equations |

P(I)DEs | Partial Integro-Differential Equations |

R-A-D | Reaction-Advection-Diffusion |

R-D | Reaction-Diffusion |

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**MDPI and ACS Style**

Cruz-Quintero, E.; Jurado, F.
Boundary Control for a Certain Class of Reaction-Advection-Diffusion System. *Mathematics* **2020**, *8*, 1854.
https://doi.org/10.3390/math8111854

**AMA Style**

Cruz-Quintero E, Jurado F.
Boundary Control for a Certain Class of Reaction-Advection-Diffusion System. *Mathematics*. 2020; 8(11):1854.
https://doi.org/10.3390/math8111854

**Chicago/Turabian Style**

Cruz-Quintero, Eduardo, and Francisco Jurado.
2020. "Boundary Control for a Certain Class of Reaction-Advection-Diffusion System" *Mathematics* 8, no. 11: 1854.
https://doi.org/10.3390/math8111854