Mathematical Problems in Engineering

Volume 2016 (2016), Article ID 2743251, 11 pages

http://dx.doi.org/10.1155/2016/2743251

## Exact Boundary Controllability for Free Traffic Flow with Lipschitz Continuous State

Department Mathematik, Friedrich-Alexander-Universität Erlangen-Nürnberg (FAU), Cauerstraße 11, 91058 Erlangen, Germany

Received 3 December 2015; Accepted 21 March 2016

Academic Editor: Chaudry Masood Khalique

Copyright © 2016 Martin Gugat. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

We consider traffic flow governed by the LWR model. We show that a Lipschitz continuous initial density with free-flow and sufficiently small Lipschitz constant can be controlled exactly to an arbitrary constant free-flow density in finite time by a piecewise linear boundary control function that controls the density at the inflow boundary if the outflow boundary is absorbing. Moreover, this can be done in such a way that the generated state is Lipschitz continuous. Since the target states need not be close to the initial state, our result is a global exact controllability result. The Lipschitz constant of the generated state can be made arbitrarily small if the Lipschitz constant of the initial density is sufficiently small and the control time is sufficiently long. This is motivated by the idea that finite or even small Lipschitz constants are desirable in traffic flow since they might help to decrease the speed variation and lead to safer traffic.

#### 1. Introduction

The Lighthill Whitham and Richards (LWR) model is a macroscopic model for traffic flow (see [1, 2]). In the model, the traffic flow is described by the solution of an initial boundary value problem with a hyperbolic partial differential equation (pde). The pde can be solved using the method of characteristics. In fact, since the pde is a scalar conservation law, for this model the characteristic curves are straight lines. A derivation of this macroscopic model as a limit of a microscopic model is given in [3].

In the control of traffic flow, for example, for the flow through a tunnel, it is desirable to have free-flow traffic in the tunnel where the density remains below a certain critical density that corresponds to maximal throughput in the system. In this way, congested flow in the tunnel is avoided. This can be achieved by controlling the traffic inflow into the tunnel; see [4, 5].

It is well known that the solutions of the LWR model can develop shocks in finite time. However, in order to decrease variations in speed among the cars and along the road solutions without shocks are desirable. In this paper we construct Lipschitz continuous controls that generate Lipschitz continuous states. This is motivated by the idea that the traffic safety is increased if the Lipschitz constant is decreased, since this might help to decrease the speed variation among vehicles on the road and as vehicles drive along the road, which in turn reduces the expected accident rate; see [6, 7]. Moreover, we hope that as the Lipschitz constant decreases, also the fuel efficiency of the traffic flow increases. Thus, the proposed controls point out possible improvements for the control of free-flow traffic that may help to improve the safety and fuel efficiency in the free-flow regime. However, in the example of tunnel traffic flow control there is a trade-off between decreasing the Lipschitz constant of the traffic flow density flow within the tunnel and possible longer waiting time of the drivers before they are allowed to enter the tunnel. The reason is that in order to generate Lipschitz continuous states we impose an upper bound on the rate of change of the controlled inflow traffic density. We want to emphasize that the method presented in this paper cannot dissolve existing congested flow within the tunnel.

Exact boundary controllability has been studied intensively for classical solution of quasilinear hyperbolic partial differential equations, in particular by Li and his group; see [8]. In this paper we want to consider solutions that are less regular, namely, solutions generated by Lipschitz continuous initial and boundary data. It turns out that the solutions can still be represented using the method of characteristics. Also in [9, 10] the exact controllability has been studied in the framework of classical solutions. Both studies focus on systems where the eigenvalues can change their sign. In [9] the St.-Venant system that is a model for the flow of water through channels is considered, whereas in [10] a general class of systems with vanishing characteristic speed is studied. In the LWR model the characteristic speed vanishes exactly at the critical density where the traffic flow changes from free flow to congested flow. Since our exact controllability is not restricted to desired states that are close to the initial state, we can consider it as a global exact controllability result within the free-flow regime. A result of this kind has also been presented in [11] for the St.-Venant system. In fact, the basic idea of the proof in [11] is similar to the proof presented here. However, in [11] classical solutions are considered.

In this paper we show how Lipschitz continuous control functions can be chosen in such a way that they generate Lipschitz continuous states and control the given initial density in finite time to any desired density below the critical density. While the density can be increased arbitrarily fast without generating shock, in order to decrease the density to a substantially smaller density some intermediate steps can be necessary. As an example, a piecewise linear controller is presented explicitly in Section 5 of the paper.

The paper has the following structure. In Section 2, the boundary control system that we want to consider is defined and a well-posedness result in the framework of solutions that are defined in the sense of characteristics is given. Based upon this result, in Section 3, we study the exact controllability of the system. We present a result on global exact boundary controllability to constant states with Lipschitz continuous solutions. To prove this result, we first show two results on local exact controllability. In Section 4 we present a sufficient condition for Lipschitz continuity of the generated states and provide Lipschitz constants explicitly. In Section 5, we summarize the constructed piecewise linear control functions in explicit form and describe the Lipschitz continuous states that they generate.

#### 2. The System

First we introduce some notation for the flux function.

##### 2.1. The Flux Function

Let a maximal density be given. Let a nonzero concave function that maps to with and be given.

The function defines the fundamental diagram (see [12]) and serves as the flux function in our traffic flow model. Define the critical density as the smallest point where the function attains its maximum value on . Assume that is continuously differentiable on . The concavity of implies that it is strictly increasing on . Moreover, is decreasing on .

Assume that is Lipschitz continuous on with the Lipschitz constant ; that is, for all , we have the Lipschitz inequality:

For define

*Remark 1. *Since in our analysis we only work in the uncongested phase, in fact only the assumptions on as a function defined on the interval are relevant for our analysis.

##### 2.2. The Initial Boundary Value Problem

Let denote the length of the considered road section. In particular, this could be the length of a tunnel. Let a Lipschitz continuous initial density with values in and the Lipschitz constant be given. Let a time be given. Let a Lipschitz continuous boundary density with the Lipschitz constant be given. Consider the system: Theorem 3 contains sufficient conditions for the existence of a solution of in the sense of the characteristic curves on the time interval . To avoid the generation of shocks at , it is assumed that and that and are compatible in the sense that . The sufficient conditions are stated in terms of the size of the Lipschitz constants for the initial density and the boundary density. Note that in Theorem 3 we do not control the growth of the Lipschitz constants during the evolution of the solution. The reason is that we only assume that the time is strictly smaller than the breaking time, which can be estimated quite accurately. The corresponding result for the Burgers equation has been stated in [13]. The size of the Lipschitz constants of the generated states is studied in Section 4. In Theorem 12 explicit representations for these Lipschitz constants are presented that are increasing functions of the Lipschitz constants of the initial density and the boundary density and also of the time . To make sure that the solution exists on a given possibly large time interval , the Lipschitz constants and must be sufficiently small. Note that the values of the boundary density and hence also the initial density at are assumed to be strictly below the critical density .

*Remark 2. *For , the boundary condition is equivalent to . Thus, the boundary condition can be implemented by controlling the influx of vehicles per second as in ramp metering (see, e.g., [14–16]).

For a given upper bound for the values of the boundary density and given values of the Lipschitz constants and , Theorem 3 guarantees the existence of the solution without shocks only on a possibly short time interval .

In the definition of and thus also in Theorem 3 the boundary conditions at are not considered. If congested flow occurs at , a shock wave might travel backwards into the region , thus interfering with the regular solution that is constructed in Theorem 3. To avoid this effect, throughout the paper we assume free outflow at on the whole time interval ; that is, we assume that is an absorbing boundary, where we do not impose boundary conditions.

Theorem 3 (quasilinear initial boundary value problem). *Let and be given. Assume that the initial density is Lipschitz continuous on with the Lipschitz constant and that for all one has .**Assume that the boundary density is Lipschitz continuous on with the Lipschitz constant and that for all one has .**Assume that the -compatibility conditions between and hold: that is, Assume that **Then system has a solution on in the sense of characteristics without shocks and rarefaction fans. In particular, the solution is continuous.*

*Proof of Theorem 3. *We consider a solution of in the sense of characteristics. We define the characteristic curves that intersect the -axis at the time ; we have for and For the characteristic curves that intersect the axis at a time we have for Then for each point there exists a characteristic curve such that or such that .

In order to obtain a well-defined solution in the sense of characteristics it is sufficient to show that is chosen in such a way that it is impossible that the characteristic curves intersect for .

Then at each point the solution is well defined since the value of is constant along the characteristic curves.

There are three possibilities of how two different characteristic curves can intersect at a point :(1)Two characteristic curves of type (6) intersect: the equation with , and implies and Due to our regularity assumptions on and , the function is Lipschitz continuous with the Lipschitz constant . Hence, (8) implies the inequality . Hence, if , an intersection of this type cannot occur for .(2)Two characteristic curves of type (7) intersect: the equation with , and implies Due to our regularity assumptions on and , the function is Lipschitz continuous with the Lipschitz constant . Thus, using the definition of in (2) we arrive at the inequality Thus, if , an intersection of this type is impossible for .(3)Characteristic curves of type (6) intersect characteristic curves of type (7): for and the equation implies Due to our compatibility assumption (4) we have Thus, we obtain the inequality If satisfies (5) an intersection of this kind cannot occur for .Thus, we have proved Theorem 3.

*Remark 4. *If satisfies the assumptions of Theorem 3, the compatibility assumption (4) holds, for all , and the control function is continuous and increasing on with values in , then the solution exists as a continuous solution in the sense of characteristics if .

The reason is that in this case for intersections of the characteristic curves of the second type, if we have ; hence, . Thus, there is no such intersection of the second type with and .

Moreover, since is increasing, for all , . Hence, for intersections of the characteristic curves of the third type, we have ; thus, in this case we have , so intersections of this type do not occur with .

*Example 5. *Let and and for define (see [12] and the introduction in [17]). Then and .

Let and for define the initial density: Figure 1 shows the graph of the initial density on the space interval .

Choose . We have , , and . Hence, For choose For define the control: Then the compatibility condition (4) holds and is Lipschitz continuous with the Lipschitz constant .

Choose . Then we have and hence due to (17) Theorem 3 implies that, for all (5) holds. In particular, the solution in the sense of characteristics exists without shocks and rarefaction fans on the time interval . If we choose as the right-hand side in (17), we have if and only if Hence, if (22) holds the solution exists at least until has attained the value . Figure 2 shows the characteristic curves corresponding to the generated state on for , , , and chosen as the right-hand side in (17). The horizontal axis is the space axis.