Optimal Control of Particle Advection in Couette and Poiseuille Flows

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.

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.

(a)

(b)

Figure 1 

Underwater glider (a), robotic fish (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 𝑣𝑥,𝑦=𝑚𝑦,0,𝑥∈ℝ,𝑦∈[0,𝐿] (1) with 𝑚=𝑣₀/𝐿, the plates being 𝐿 distance units apart (Figure 2(a)).

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(b)

Figure 2 

Linear (a) and quadratic velocity field (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 𝑣=𝑎𝑥,𝑦𝑎−𝐿²𝑦²,0,𝑥∈ℝ,𝑦∈[−𝐿,𝐿], (2) 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 width 𝐿 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 function 𝑢(⋅)=(𝑢_𝑥(⋅),𝑢_𝑦(⋅)) to be applied to the particle so that it will move in the channel from the initial position to the end point (𝑥_𝑓,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: 𝑋̇(𝑡)(𝑋(𝑡)(𝑡,𝑋(0)=(),𝑋(𝑇)=𝑥Minimize𝑇subjectto=𝐹,𝑢))0,𝑏_𝑓[].𝑦(𝑡)∈[],‖𝑢(𝑡,𝑏,∀𝑡∈0,𝑇0,𝐿)‖_∞≤1, (3)

In the following two particular cases, for simplicity of notation, we will not indicate the time 𝑡 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 slope 𝑚=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+𝑢_𝑥,𝑢_𝑦).

The Pontryagin maximum principle, [6], allows us to determine the optimal control 𝑢^∗=(𝑢^∗_𝑥,𝑢^∗_𝑦) by using the maximization of Pontryagin’s function 𝐻(𝑋,𝑃,𝑢) (here, 𝑃=(𝑝_𝑥,𝑝_𝑦) is the adjoint variable satisfying −𝑃̇=∇_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 𝐻(𝑋,𝑃,𝑢)=𝑝_𝑥𝑦𝑚+𝑢_𝑥+𝑝_𝑦+𝛾𝑢_𝑦, (4) where 𝛾 is a certain function which reflects the activity of the state constraints of the variable 𝑦, it follows that −𝑝̇_𝑥−𝑝̇=0,_𝑦=𝑝_𝑥𝑚, (5) and, thus, 𝑝_𝑥=𝐾_𝑥,𝑝_𝑦=𝐾_𝑦−𝐾_𝑥𝑚𝑡, (6) for some constants 𝐾_𝑥,𝐾_𝑦>0. By taking into account that the position of the particle at time 𝑡 is given by 𝑥(𝑡)=𝑏𝑚𝑡+^𝑡₀𝑢_𝑥(𝜏)1𝑑𝜏+𝑚^𝑡₀(𝑡−𝜏)𝑢_𝑦(𝜏)𝑦(𝑡)𝑑𝜏,=𝑏+^𝑡₀𝑢_𝑦(𝜏)𝑑𝜏, (7) we conclude by the maximization of Pontryagin’s function that 𝑢^∗_𝑥(𝑡)=1 (Figure 2) and 𝑢^∗_𝑦 is given by the following.(i) Consider 𝑢^∗_𝑦(𝑡)=1,𝑡∈0,𝑡^∗,𝑡/2−1,𝑡∈^∗/2,𝑡^∗, (8) for the case of 𝑥_𝑓≤2(𝐿−𝑏). By substituting in the equations of the particle’s position, we conclude that the optimal time of (3) is given by 𝑡^∗=2(𝑏+𝑚)²+𝑚𝑥_𝑓−(𝑏+𝑚). (9)(ii)Consider 𝑢^∗_𝑦(𝑡)=1,𝑡∈0,𝑡₁,0,𝑡∈𝑡₁,𝑡^∗−𝑡₁,−1,𝑡∈𝑡^∗−𝑡₁,𝑡^∗, (10) for the case of 𝑥_𝑓>2(𝐿−𝑏), where 𝑡₁=(𝑏+𝑚)²+2𝑚(𝐿−𝑏)−(𝑏+𝑚), (11) being now 𝑡^∗=2(𝑏+𝑚)²−2(𝑏+𝑚)(𝑏+𝑚)²()()+2𝑚𝐿−𝑏+2𝑚𝐿−𝑏+𝑚𝑥_𝑓×(𝑏+𝑚)²+2𝑚(𝐿−𝑏)⁻¹. (12)

The two cases in the definition of 𝑢^∗_𝑦 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 𝐹()=𝑎𝑋,𝑢𝑎−𝐿²𝑦²+𝑢_𝑥,𝑢_𝑦, (13) and Pontryagin’s function is given by 𝐻(𝑋,𝑃,𝑢)=𝑝_𝑥𝑎𝑎−𝐿²𝑦²+𝑢_𝑥+𝑝_𝑦+𝛾𝑢_𝑦. (14) 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 𝑥(𝑡)𝑎=𝑎𝑡−𝐿^2𝑡₀𝑦²(𝜏)𝑑𝜏+^𝑡₀𝑢_𝑥(𝜏)𝑦(𝑡)𝑑𝜏,=𝑏+^𝑡₀𝑢_𝑦(𝜏)𝑑𝜏, (15) we conclude that 𝑢^∗_𝑥(𝑡)=1 (Figure 2) and that 𝑢^∗_𝑦 is defined by the following: (i) 𝑢^∗_𝑦(𝑡)=−1,𝑡∈0,𝑡^∗,𝑡/21,𝑡∈^∗/2,𝑡^∗, (16) if𝑥_𝑓≤2𝑏, in this case, the minimum time𝑡^∗being a root of the polynomial as follows: 𝑡^∗3−6𝑏𝑡^∗2+12𝑏²−12𝐿²−12𝐿²𝑎𝑡^∗+12𝐿²𝑎𝑥_𝑓=0, (17) (ii) 𝑢^∗_𝑦(𝑡)=−1,𝑡∈0,𝑡₁,0,𝑡∈𝑡₁,𝑡^∗−𝑡₁,1,𝑡∈𝑡^∗−𝑡₁,𝑡^∗, (18) if 𝑥_𝑓>2𝑏, being the optimal time as follows: 𝑡^∗=𝑥_𝑓−2𝑎/𝐿²(2/3)𝑡³₁−𝑏𝑡²₁𝑎+1−𝑎/𝐿²(𝑏−𝑡₁)², (19) where 𝑡₁ is half of the value of the 𝑡^∗ obtained in (17) with 𝑥_𝑓=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.

Acknowledgments

The first and the second authors gratefully acknowledge the financial support of the FCT funded R&D projects PEST-OE/EEI/UI0147/2011 and PTDC/EEA-CRO/104901/2008. The work of the third author was supported, in part, by FCT through the CMUP—Centro de Matemática da Universidade do Porto.