# Optimal Control of Particle Advection in  Couette and Poiseuille Flows

Authors: Teresa Grilo; Fernando Lobo Pereira; Sílvio Gama
Journal: Conference Papers in Mathematics
 (2013)

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## Abstract

We present the problem of minimum time control of a particle advected in  Couette and Poiseuille flows and solve it by using the Pontryagin maximum principle. This study is a first step of an effort aiming at the development of a mathematical framework for the control and optimization of dynamic control systems whose state variable is driven by interacting ODEs and PDEs which can be applied in the control of underwater gliders and mechanical fishes.


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## Body

## 1. Introduction

This paper represents a first step for the optimal control of dynamic systems whose state evolves through the interaction of ordinary differential equations and the partial differential equations, [1, 2], which will provide a sound basis for the design and control of new advanced engineering systems. In Figure 1, two representative examples of the class of applications are considered: (i) underwater gliders, that is, winged autonomous underwater vehicles (AUVs) which locomote by modulating their buoyancy and their attitude in its environment, and (ii) robotic fishes. Motion modeling of these two types of systems can be found in [3, 4] and [5], respectively.Underwater glider (a), robotic fish (b).
(a)
(b)In spite of the key roots of the Optimal Control Theory having been established in the sixties for control systems with dynamics given by ordinary differential equations, [6], its sophistication in multiple directions has been progressing unabated (see, among others, [7, 8]). However, there still remains a large gap in what concerns dynamic control systems driven by partial differential equations, [2], and it is largely inexistent for hybrid systems in the sense that the controlled dynamics involve both partial and ordinary differential equations.In this paper, we formulate and solve two optimal control problems. Each one of these problems corresponds to a particular solution of the incompressible Navier-Stokes equation in two spatial dimensions. These particular solutions are, respectively, the steady Couette and Poiseuille flows.The Couette flow is the steady laminar unidirectional and two-dimensional flow due to the relative motion of two infinite horizontal and parallel rigid plates [9]. The liquid between these two plates is driven by the viscous drag force originated by the uniform motion of the upper plate which moves in the x-direction with velocity v0 (the lower plate is at rest). In this case, the velocity of such a flow has a linear profile and is given by
(1)v(x,y)=(my,0),x∈ℝ,y∈[0,L]
with m=v0/L, the plates being L distance units apart (Figure 2(a)).Linear (a) and quadratic velocity field (b).
(a)
(b)The Poiseuille flow is the steady flow due to the presence of a pressure gradient between two fixed (i.e., with zero relative velocity) rigid plates [9]. In this case, a parabolic velocity profile is of the form
(2)v(x,y)=(a-aL2y2,0),x∈ℝ,y∈[-L,L],
where, now, L is half the distance between the upper and lower plates (Figure 2(b)).


## 2. Minimum Time Control Problem

Consider a flow in a channel of the widthL with a given velocity field and a particle placed in this flow. Let (0,b) be the initial position of the particle, with 0≤b≤L.The objective of this problem is to determine the control functionu(·)=(ux(·),uy(·)) to be applied to the particle so that it will move in the channel from the initial position to the end point (xf,b) in minimum time while subject to the flow field. Let X(t)=(x(t),y(t)) be the position of the particle at time t, the control problem can be formulated as follows:
(3)MinimizeTsubjecttoX˙(t)=F(X(t),u(t)),X(0)=(0,b),X(T)=(xf,b),∀t∈[0,T].y(t)∈[0,L],∥u(t)∥∞≤1,In the following two particular cases, for simplicity of notation, we will not indicate the timet as an independent variable of the other variables, although this is the case we are considering.


### 2.1. Couette Flow: Linear Velocity Profile

In the case of linear flow (with slopem=v0/L>0), the velocity field of the flow is given by v(x,y)=(y/m,0). So, the dynamics of this control system are given by F(X,u)=(y/m+ux,uy).The Pontryagin maximum principle, [6], allows us to determine the optimal control u*=(ux*,uy*) by using the maximization of Pontryagin’s  function H(X,P,u) (here, P=(px,py) is the adjoint variable satisfying -P˙=∇XH(X,P,u), ∇X being the gradient of H with respect to X) almost everywhere with respect to the Lebesgue measure (from here onwards, functions are specified in this sense), together with the satisfaction of the appropriate boundary conditions. So, being
(4)H(X,P,u)=px(ym+ux)+(py+γ)uy,
where γ is a certain function which reflects the activity of the state constraints of the variable y, it follows that
(5)-p˙x=0,-p˙y=pxm,
and, thus,
(6)px=Kx,py=Ky-Kxmt,
for some constants Kx,Ky>0. By taking into account that the position of the particle at time t is given by
(7)x(t)=bmt+∫0t‍ux(τ)dτ+1m∫0t‍(t-τ)uy(τ)dτ,y(t)=b+∫0t‍uy(τ)dτ,
we conclude by the maximization of Pontryagin’s function that ux*(t)=1 (Figure 2) and uy* is given by the following.(i)
Consider(8)uy*(t)={1,t∈[0,t*/2[,-1,t∈]t*/2,t*],for the case ofxf≤2(L-b). By substituting in the equations of the particle’s position, we conclude that the optimal time of (3) is given by
(9)t*=2((b+m)2+mxf-(b+m)).(ii)
Consider(10)uy*(t)={1,t∈[0,t1[,0,t∈]t1,t*-t1[,-1,t∈]t*-t1,t*],for the case ofxf>2(L-b), where
(11)t1=(b+m)2+2m(L-b)-(b+m),being now(12)t*=(2(b+m)2-2(b+m)(b+m)2+2m(L-b)0=+2m(L-b)+mxf(b+m)2+2m(L-b))×((b+m)2+2m(L-b))-1.The two cases in the definition ofuy* correspond to the situations in which the constraint is inactive and active, respectively.


### 2.2. Poiseuille Flow: Parabolic Velocity Profile

Similar arguments are applied to the case of flow with parabolic velocity flow (with vertex in(a,0)). Now, the dynamics of the control system is given by
(13)F(X,u)=(a-aL2y2+ux,uy),
and Pontryagin’s function is given by
(14)H(X,P,u)=px(a-aL2y2+ux)+(py+γ)uy.
It is easy to observe that the state constraint will be inactive along the optimal trajectory and that there is symmetry about the axis y=0. By using these observations in the application of the Pontryagin maximum principle, as well as the fact that the position of the particle is given by
(15)x(t)=at-aL2∫0t‍y2(τ)dτ+∫0t‍ux(τ)dτ,y(t)=b+∫0t‍uy(τ)dτ,
we conclude that ux*(t)=1 (Figure 2) and that uy* is defined by the following:(i)(16)uy*(t)={-1,t∈[0,t*/2[,1,t∈]t*/2,t*],ifxf≤2b, in this case, the minimum timet*being a root of the polynomial as follows:
(17)t*3-6bt*2+(12b2-12L2-12L2a)t*+12L2axf=0,(ii)(18)uy*(t)={-1,t∈[0,t1[,0,t∈]t1,t*-t1[,1,t∈]t*-t1,t*],ifxf>2b, being the optimal time as follows:
(19)t*=xf-(2a/L2)((2/3)t13-bt12)a+1-(a/L2)(b-t1)2,wheret1 is half of the value of the t* obtained in (17) with xf=2b.


## 2.1. Couette Flow: Linear Velocity Profile

In the case of linear flow (with slopem=v0/L>0), the velocity field of the flow is given by v(x,y)=(y/m,0). So, the dynamics of this control system are given by F(X,u)=(y/m+ux,uy).The Pontryagin maximum principle, [6], allows us to determine the optimal control u*=(ux*,uy*) by using the maximization of Pontryagin’s  function H(X,P,u) (here, P=(px,py) is the adjoint variable satisfying -P˙=∇XH(X,P,u), ∇X being the gradient of H with respect to X) almost everywhere with respect to the Lebesgue measure (from here onwards, functions are specified in this sense), together with the satisfaction of the appropriate boundary conditions. So, being
(4)H(X,P,u)=px(ym+ux)+(py+γ)uy,
where γ is a certain function which reflects the activity of the state constraints of the variable y, it follows that
(5)-p˙x=0,-p˙y=pxm,
and, thus,
(6)px=Kx,py=Ky-Kxmt,
for some constants Kx,Ky>0. By taking into account that the position of the particle at time t is given by
(7)x(t)=bmt+∫0t‍ux(τ)dτ+1m∫0t‍(t-τ)uy(τ)dτ,y(t)=b+∫0t‍uy(τ)dτ,
we conclude by the maximization of Pontryagin’s function that ux*(t)=1 (Figure 2) and uy* is given by the following.(i)
Consider(8)uy*(t)={1,t∈[0,t*/2[,-1,t∈]t*/2,t*],for the case ofxf≤2(L-b). By substituting in the equations of the particle’s position, we conclude that the optimal time of (3) is given by
(9)t*=2((b+m)2+mxf-(b+m)).(ii)
Consider(10)uy*(t)={1,t∈[0,t1[,0,t∈]t1,t*-t1[,-1,t∈]t*-t1,t*],for the case ofxf>2(L-b), where
(11)t1=(b+m)2+2m(L-b)-(b+m),being now(12)t*=(2(b+m)2-2(b+m)(b+m)2+2m(L-b)0=+2m(L-b)+mxf(b+m)2+2m(L-b))×((b+m)2+2m(L-b))-1.The two cases in the definition ofuy* correspond to the situations in which the constraint is inactive and active, respectively.


## 2.2. Poiseuille Flow: Parabolic Velocity Profile

Similar arguments are applied to the case of flow with parabolic velocity flow (with vertex in(a,0)). Now, the dynamics of the control system is given by
(13)F(X,u)=(a-aL2y2+ux,uy),
and Pontryagin’s function is given by
(14)H(X,P,u)=px(a-aL2y2+ux)+(py+γ)uy.
It is easy to observe that the state constraint will be inactive along the optimal trajectory and that there is symmetry about the axis y=0. By using these observations in the application of the Pontryagin maximum principle, as well as the fact that the position of the particle is given by
(15)x(t)=at-aL2∫0t‍y2(τ)dτ+∫0t‍ux(τ)dτ,y(t)=b+∫0t‍uy(τ)dτ,
we conclude that ux*(t)=1 (Figure 2) and that uy* is defined by the following:(i)(16)uy*(t)={-1,t∈[0,t*/2[,1,t∈]t*/2,t*],ifxf≤2b, in this case, the minimum timet*being a root of the polynomial as follows:
(17)t*3-6bt*2+(12b2-12L2-12L2a)t*+12L2axf=0,(ii)(18)uy*(t)={-1,t∈[0,t1[,0,t∈]t1,t*-t1[,1,t∈]t*-t1,t*],ifxf>2b, being the optimal time as follows:
(19)t*=xf-(2a/L2)((2/3)t13-bt12)a+1-(a/L2)(b-t1)2,wheret1 is half of the value of the t* obtained in (17) with xf=2b.


## 3. Conclusions and Future Work

The case studies discussed here are very simple and their difference concerns only the profile of the velocity of the fluid. Not only the dynamics of the control system are defined by a set of ODEs, but also the conditions resulting from the application of the Pontryagin maximum principle can be easily solved in an explicit way. The next step consists in deriving optimality conditions in the form of a maximum principle leading to the computation of the solution to optimal control problems for which the previous simplifications cannot be exploited. This study suggests that the optimality conditions to be developed will require an adjoint variable satisfying a mixed system with ODEs and PDEs, so that the optimal control can be obtained by maximizing an appropriated Pontryagin function, coupled with appropriate boundary conditions.



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*Source: 783510-2013-09-03.xml*