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Feedback Control of Scalar Conservation Laws with Application to Density Control in Freeways by Means of Variable Speed Limits

The paper provides results for the stabilization of a spatially uniform equilibrium profile for a scalar conservation law that arises in the study of traffic dynamics under variable speed limit control. Two different control problems are studied: the problem with free speed limits at the inlet and the problem with no speed limits at the inlet. Explicit formulas are provided for respective feedback laws that guarantee stabilization of the desired equilibrium profile. For the first problem, global asymptotic stabilization is achieved; while for the second problem, regional exponential stabilization is achieved. Moreover, the solutions for the corresponding closed-loop systems are guaranteed to be classical solutions, i.e., there are no shocks. The obtained results are illustrated by means of a numerical example.

In this work, we study a specific 1-D scalar conservation law, which is described by the following first-order Partial Differential Equation (PDE): where  is the state variable, u is the control input, 0 t  denotes time and 0 x  denotes spatial position. To the best of our knowledge, this is the first work that studies control problems related to the PDE (1). This particular 1-D scalar conservation law arises in the study of traffic dynamics in a freeway, where Variable Speed Limits (VSL) can be applied continuously (both in time and space) along the freeway. The PDE (1) is a variation of the standard first-order LWR model (see [21,26]) and was used in [12] for traffic networks with VSL which depend only on time. In the context of traffic dynamics, the state is the vehicle density  (in veh/km), and the control input u is the speed limit ratio ( 1 u  is the case where no speed limits are applied, and 0 u  is the case where no vehicle movement is allowed). Thus, the PDE (1) is accompanied by the constraint 01 u  .
It should be noted at this point that, although the PDE (1) has not been studied so far for control purposes, the effect of VSL on traffic flow and the exploitation of this effect for improved traffic flow efficiency have been studied mainly by use of discretized models (see [1,3,4,5,6,14,15,22,23]).
The objective of the present work is the stabilization of a spatially uniform equilibrium profile by means of a feedback law, and we consider two different variants of the related control problem: 1) The control problem where we are free to use any speed limit at the inlet ( 0 x  ); this implies that the inlet itself is considered controllable.
2) The control problem where we are not allowed to have speed limits at the inlet, i.e., ( ,0) 1 ut  ; thus, the inlet for this problem is not controllable, but internal control is still allowed.
The motivation for the study of the second control problem stems from the concern to avoid the creation of queues at the entrance of the freeway, which may be required in specific applications. The two control problems are qualitatively different, with the second problem being more demanding than the first one. The qualitative difference is mathematically expressed by the imposition of a boundary condition in the second control problem.
We provide explicit formulas for the feedback laws that guarantee exponential stabilization of the desired equilibrium profile. Moreover, the solutions of the closed-loop systems (whose existence is established in both cases by Banach's fixed point theorem) are classical solutions, i.e., continuously differentiable; therefore, there are no shocks. However, for the second control problem, we cannot achieve global stabilization (i.e., for all physically relevant initial conditions). The proposed feedback law for the second control problem achieves regional stabilization, and the region where stabilization can be achieved depends on the parameters of the controller. This is conform with physical intuition and traffic engineering experience, which has shown that extreme congestion phenomena cannot be handled without controlling the inlet flow (via VSL-induced mainstream metering as proposed in [4,6] or via ramp metering or both).
The structure of the present work is as follows: Section 2 is devoted to the presentation of the model and the two control problems related to the 1-D scalar conservation law (1). Section 3 provides the statements of the main results (Theorem 3.1 and Theorem 3.2), as well as a discussion thereof. The proofs of the main results are provided in Section 4. A simple illustrative example is presented in Section 5. Concluding remarks are provided in Section 6.
Notation. Throughout this paper, we adopt the following notation.

The LWR model with VSL
VSL affect the fundamental diagram (flow-density curve) in a nonlinear way (see Chapter 2 in [6] and also [9,13,16]). In general, for every freeway there exists a nonlinear function is the physical upper bound of density in the particular road and 0 max  q is the upper bound (capacity) of the flow of vehicles in the road, with , so that the flow q of vehicles at a point in freeway with density ] , 0 ( max    is given by the equation is the ratio of the imposed speed limit divided by the maximum mean speed that can be exhibited in the freeway without speed limit.
Assuming that the speed limit ratio, i.e., l , can be manipulated continuously (both in time and space), i.e., assuming that ) , ( x t l l  , we obtain the modified LWR model with VSL for We next use the following assumption.

(H1) There exists a continuous function
This assumption reflects the physical effect of VSL on the flow-density curve in a rather general way: the application of any speed limit , which is a special case of the class of functions proposed in [6] for the description of the effect of VSL to the flow-density curve. More specifically, using the following equations hold for all Under Assumption (H1) and definition (9), for each . Therefore, we may consider the simplified LWR model (1) Traffic flow theory allows us to use the following assumption for the function , defined by (9).

Remark 2.2: Assumption (H2) introduces the critical density
, which in traffic engineering is the density value that produces the maximum flow (capacity). The control objective is to construct a feedback law of the form

Two Control Problems
so that for any initial condition of the closed-loop system (1) with (10) and initial condition 5 The requirement (12) is important, because it implies that the VSL may be employed to address any appearing congestion problem, hence the freeway will practically operate without speed limits after an initial transient period, i.e. after the problem has been tackled; notice that this is a consequence of (4), (5), (6) and (9) is the constant involved in (H1)).

nd Problem: No Speed Limit at Inlet
One possible issue with the 1 st problem is that the inlet flow, which is equal to , may become small for a transient period. This may cause the creation of queues at the entrance of the freeway (or at upstream on-ramps, in case of ramp metering). One way to avoid the creation of queues is to require that no speed limit is applied at the entrance of the freeway, i.e., Moreover, in this case, the flow at the entrance of the freeway is assumed to be equal to the nominal flow, i.e., The control objective is to construct a feedback law of the form (10), satisfying (11) and , so that for a large number of initial conditions of the closed-loop system (1), (14) with (10) and initial condition , is unique and satisfies (12).
The 2 nd control problem described above is not expected to be solvable for arbitrary set points

Two possible solutions of the control problems
The following theorem guarantees that the 1 st problem is globally solvable, i.e., for all physically relevant initial conditions.
Then there exists a non-decreasing function , which satisfies the following estimates: The following result guarantees that the 2 nd control problem is solvable for a certain class of initial conditions (regional stabilization).

Theorem 3.2 (Regional Exponential Stabilization in the sup norm): Suppose that Assumption
the initial-boundary value problem (1) with (14), has a unique solution , which satisfies estimates (17), (18) as well as the following estimate:

Discussion of Main Results
We provide below a list of comments for the main results.
1) The solutions of both control problems guarantee asymptotic stability in the sup norm.
2) The 2nd control problem is more demanding than the 1st control problem. The reason that explains this difference between the two control problems is related to the inlet flow; while the inlet flow may be modulated in the 1st problem, it remains constant (and equal to () f   , i.e. typically very high) for the 2nd problem. This limits the range of possible VSL actions; for example, the speed limits in the 2nd problem cannot become very small, e.g. in order to dissolve a possible downstream congestion, because this could result in high accumulation of vehicles farther upstream due to the high inflow.
3) The difficulty in the solution of the 2 nd control problem is reflected in the fact that the control objective can be achieved only for a class of initial conditions, namely initial conditions for which (19) holds. Another fact that expresses the difficulty in the solution of the 2 nd problem is that the set point   cannot be equal to the critical density cr  , while there is no such constraint for the 1 st problem.

4)
Since condition (19) depends on the controller gains , 0  , there is a degree of freedom which can be used for the enlargement of the allowable set of initial conditions. However, the proof of Theorem 2 shows that the gains ,0  cannot be arbitrary (they must be sufficiently small) and that the gains affect the convergence rate (the proof of Theorem 3.2 shows that the constant c involved in (17) is equal to L   ). (21)

5) Inequality
In other words, for every , defined by (15). This interpretation allows the physical explantation of condition (19): condition (19) guarantees that there exists an appropriate weight term (not of the form (15)) so that the bottleneck occurs at the inlet 0 x  . Using the appropriate weight term, we construct the feedback law (20), which satisfies (13).

9) If
then the proof of Theorem 3.1 shows that the boundary condition (14) will also hold (although the boundary condition (14) is not imposed to the closed-loop system (1), (16)). However, as noted earlier, this fact does not imply that the inlet flow is constant. In this case (

Proofs of Main Results
We first provide the proof of Theorem 3.1.

Proof of Theorem 3.1:
Suppose that the initial value problem (1) Combining (22) and (15) we get for It follows from (23) and (1) that: Integrating (24) and using (22) and the initial condition where   For the derivation of (29), we have used definition (15) as well as the facts that , it follows from (28) Estimate (17) is a direct consequence of (30) in conjunction with (25) It follows from (35) that the following equations hold for We next show that for each   . Using (37) and the inequality Inequality (45) contradicts inequality (33). We conclude that (43) cannot hold. The proof is complete.  Figures 1 and 2 show the density profiles for the initial condition 22 0 ( ) Figures 3 and 4 show the corresponding speed limit ratios ( , ) u t x . Figures 1 and 2 show that the convergence rate of the closed-loop system (1) with (16) is faster than that of the closed-loop system (1), (14) with (20). Notice that we cannot increase the values of the controller gains , 0  to speed up the convergence rate of the closed-loop system (1), (14) with (20), because, as the proof of Theorem 3.2 showed, the controller gains ,0  must be sufficiently small. The speed limits are initially generally higher for the closed-loop system (1), (14) with (20) than those of the closed-loop system (1) with (16). However, the convergence rate of the speed limit 13 ratio to 1 is faster for the closed-loop system (1) with (16) than that of the closed-loop system (1), (14) with (20).

Illustrative Example
From a physical point of view, Figures 1-4 illustrate the control mechanism leading to the dissolution of the initial congestion in the freeway. Specifically, under both control laws, the speed limit control is shown to take low values upstream of the congestion, so as to limit the respective inflows; in the first case, this involves also some inlet reduction, which is not allowed for the second case. In terms of outflow, it may be seen that the convergence rate of the density at the outlet for the closed-loop system (1), (14) with (20) is significantly slower than the convergence rate of the density at the outlet for the closed-loop system (1) with (16); this happens because the feedback law (20) cannot affect the inlet flow rate, hence it tries to maintain large outflows by keeping the density at the outlet close to the critical desnity (hence maintaining a larger outflow) for a longer period of time. . Figures 5 and 6 exhibit exponential convergence of the sup norm of the state, as predicted by estimate (17). Again, as remarked above, Figures 5 and 6 show that the convergence rate of the closed-loop system (1) with (16) is faster than that the closed-loop system (1), (14) with (20).
As pointed out earlier, the feedback law (20) cannot be used for a set point equal to the critical density. This is not the case for the feedback law (16). Figure 7 shows the density profiles of the closed-loop system (1) with (16)  u t x ) profiles for the closed-loop system (1), (14) with (20).
The horizontal axis is

Concluding Remarks
The paper provides two different results for the stabilization of a spatially uniform equilibrium profile of a specific scalar conservation law that arises in the study of traffic dynamics under VSL. We have studied two different control problems: the problem with free speed limits at the inlet and the problem with no speed limits at the inlet. For the 1 st problem, global asymptotic stabilization was achieved; while for the 2 nd problem, regional exponential stabilization was achieved. The solutions of the closed-loop systems were classical solutions, i.e., there are no shocks. The obtained results were illustrated by means of a numerical example.
Future work will address the inhomogeneous case, i.e. consideration of a freeway stretch with spatially inhomogeneous flow-density relationships. Future research may also involve the study of the effect of discretization. In practice, the distributed control input ( , ) u t x must be kept constant for certain time and space intervals. In other words, the implementation of the feedback controllers (16) or (20) involves discretization both with respect to space and time. The effect of discretization may be important and should be investigated. Another research direction is the incorporation of VSL to 2 nd order models. In this way, the velocity dynamics, which are ignored by the 1 st order model (1), can be taken into account.