Abstract
In this chapter, we introduce HamiltonJacobi PDEs. These PDEs are related to conservation laws and their solutions are the antiderivative (in space) of the Entropy solutions of the corresponding conservation law, given that some assumptions are satisfied.
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6.1 Introduction
In this chapter, we introduce HamiltonJacobi PDEs. These PDEs are related to conservation laws and their solutions are the antiderivative (in space) of the Entropy solutions of the corresponding conservation law, given that some assumptions are satisfied.
Roughly, a HamiltonJacobi PDE reads for the Cauchy problem on \(\mathbb {R}\) for a given flux function \(f:\mathbb {R}\rightarrow \mathbb {R}\) as
Assuming that the solutions are significantly smooth, we can differentiate the PDE with respect to the spatial variable and obtain
Setting ∂ _{x} q ≡ u for a given function \(u:(0,T)\times \mathbb {R}\rightarrow \mathbb {R}\) one ends up with
the conservation law in u with flux function f and initial datum \( q_{0}^{\prime }(x),x\in \mathbb {R}\). Having a closer look into the theory of HamiltonJacobi PDEs, one major advantage is that the solutions to these equations remain Lipschitz continuous in case the initial datum is Lipschitz continuous (and in the case one is in the bounded domain case, also the boundary datum).
Even more, there is an explicit solution formula taking advantage of the Legendre Fenchel dual to write down the solution at every given timespace point as an convex optimization problem. This is of great importance as it not only allows to evaluate solutions without a predetermined numerical grid, the specific solution structure enables it also to study details and behavior of solutions.
These loosely described approaches will be made rigorous in the next section where we introduce the concept of strong and generalized solutions.
6.2 Strong Solutions
We start with the problem setup and some assumptions on the flux function which contain convexity. The same results also hold for concave functions under minor modifications.
Definition 27 (The HamiltonJacobi PDE with Initial Datum)
Given a flux function \(f:\mathbb {R}\rightarrow \mathbb {R}\) and initial datum \(q_{0}:\mathbb {R}\rightarrow \mathbb {R}\) we call the following initial value problem
a HamiltonJacobi PDE, and \(q:(0,T)\times \mathbb {R}\rightarrow \mathbb {R}\) its solution.
To guarantee the existence and uniqueness of solutions, we require some assumptions on initial datum and flux function. This is detailed in the following
Assumption 1 (Flux Function and Initial Datum)
We assume that

1.
\(q_{0}\in W^{1,\infty }_{\mathit{\text{loc}}}(\mathbb {R}):\ q_{0}^{\prime }\in L^{\infty }(\mathbb {R})\)

2.
\(f\in W^{1,\infty }_{\mathit{\text{loc}}}(\mathbb {R})\) strictly convex satisfying \(\lim _{x\rightarrow \pm \infty } \tfrac {f(x)}{x}=\infty \).
For the explicit solution formula for every fixed spacetime point we require the convex conjugate of the function which we define as follows.
Definition 28 (Convex Conjugate or the Legendre Fenchel Transform)
Suppose that f satisfies item 2 in assumption 1, i.e., is in particular strictly convex and let \(I\subseteq \mathbb {R}\) be a closed interval. Then, we define the Legendre Fenchel transform f ^{∗} of f on I as
where the domain for the Legendre transform f ^{∗} on I is defined as
Remark 13 (Convex Conjugate)
Note that due to item 2, \(\operatorname {Dom}(f^{*})\) is an interval in \(\mathbb {R}\) or entire \(\mathbb {R}\). Furthermore, f ^{∗} is also a convex function.
Given these definitions and assumptions we can state the main Theorem 33:
Theorem 33 (Existence/Uniqueness of Strong Solutions and the LaxHopf Formula)
Let Assumption 1 hold. Then, the initial value problem in Definition 27 admits a unique strong solution \(q\in W^{1,\infty }_{\mathit{\text{loc}}}((0,T)\times \mathbb {R}), \partial _{2}q\in L^{\infty }((0,T)\times \mathbb {R})\) and the solution at timespace point \((t,x)\in (0,T)\times \mathbb {R}\) can be posed as
with f ^{∗} as in Definition 28 . The latter identity is called LaxHopf formula.
Proof
The proof can be found in [126, Chapter 3.3]. □
Remark 14 (Interpretations of the Solution Formula and More)
As pointed out this solution formula for q requires it to solve a minimization problem at every given point in spacetime. However, as f ^{∗} is strictly convex and q _{0} grows at most linearly due to the assumption on \(q_{0}^{\prime }\in L^{\infty }(\mathbb {R})\), this minimization problem always possesses one and only one minimal point for every given \((t,x)\in (0,T)\times \mathbb {R}\).
The following theorem makes the connection between the solution of the HamiltonJacobi equation and the Entropy solution of the corresponding conservation law. It thus details what had been described in Sect. 6.1. For the general theory of scalar conservation laws and types of solutions, particularly Entropy solutions, we refer the reader to [51].
Theorem 34 (Relation to the Corresponding Conservation Law)
Given Assumption 1 , the spatial derivative of the solution stated in Eq.(6.3) in Theorem 33 is the unique Entropy solution of the conservation law
Proof
The proof can be found in many text books, we again refer to [126, Chapter 3.4, Chapter 11]. □
This result is the key why instead of solving a conservation law, one might solve the corresponding HamiltonJacobi equation. One advantage of the solution is that it is Lipschitz continuous. Due to the gain in regularity one can apply methods on control and optimization which one could not so easily apply on the level of conservation laws.
In the following, we will also look into the case where boundary datum is prescribed, i.e., when we are on a bounded or semibounded domain.
Before addressing the named question we look in detail into the bounded domain case:
6.2.1 The Bounded Domain Case
It is worth mentioning some results for the bounded domain cases:
Theorem 35 (LaxHopf Formula on \(\mathbb {R}_{>0}\))
Let \(T\in \mathbb {R}_{>0}\) and f satisfy Assumption 1 . Let the initial value \(u_{0} \in L^{\infty }(\mathbb {R}_{>0})\) be given and assume that u _{b} ∈ L ^{∞}((0, T)). Then, there exists a unique strong Lipschitz continuous solution to the following HamiltonJacobi equation \(q:(0,T)\times \mathbb {R}_{\geq 0}\rightarrow \mathbb {R}\) with Neumann boundary datum
The solution q can be represented by means of a three dimensional restricted minimization problem, for \((t,x)\in \mathbb {R}_{\geq 0}\times [0,T]\)
Furthermore, the spatial derivative of q, i.e.,
is the entropy solution of the conservation law
satisfying the boundary condition in the sense of [ 29 ]
with
the possibly attained boundary datum.
Proof
The proof can be found in [184]. □
Remark 15 (Some Remarks to the Boundary Datum and the Solution Formula)
The Neumann boundary datum on the level of HamiltonJacobi equations can be interpreted as a Dirichlet boundary datum for the corresponding conservation law. This is also why the boundary datum can only hold in the sense of [29] and also needs to be projected into what the flux function f can “handle.”
Finally, from the solution formula in Theorem 35 one can see that the solution is computed in two parts. The first part, i.e., \(min_{y\in \mathbb {R}_{\geq 0}} \Big \{tf^{*}\big (\tfrac {xy}{t}\big )+q_{0}(y) \Big \}\) is the solution which directly propagates the initial datum (compare Eq. (6.3)). The second part of the minimization is responsible for the interaction of boundary datum and initial datum.
A further result is available for two sided boundary datum, however, in this case the solution formula is much more involved as the two boundaries can influence each other over time. We refer to [185].
6.3 Generalized Solutions
In this section we will introduce more general solutions, the socalled BarronJensen/Frankowska solutions [30, 134] for HamiltonJacobi equations. Then, the spatial derivative might not even exist in a strong sense so that an interpretation on the level of conservation laws as in the previous section is not possible anymore. However, from an applied point of view the generalized solutions for the HamiltonJacobi equations have other desirable features. For example, they enable the reconciliation of data points incompatible with conservation laws, as often measured in experimental data [74]. In addition, the interpretation of these solutions goes back to the 1968 seminal article of Karl Moskowitz [234], which gives a practitioner interpretation to these solutions in terms of vehicle counts (which can be directly measured with loop detectors). The basic idea follows [21, 72,73,74] and related work. As this work is mainly concerned with traffic flow applications, the flux functions chosen here are concave. In addition, for the conservation law in \(q:(0,T)\times X\rightarrow \mathbb {R}\) for t ∈ [0, T] and \(x\in X\subset \mathbb {R}\) a bounded interval
we redefine the corresponding HamiltonJacobi equation as
so that for smooth solutions we obtain
We make this precise requiring the following
Assumption 2 (Assumption on the Flux Function)
Given a maximal density \(u_{\max }\in \mathbb {R}_{>0}\) we assume that the flux function f considered is concave, and Lipschitz, i.e., \(f\in W^{1,\infty }((0,u_{\max }))\) . As it will be important for the following Legendre Fenchel transform in Definition 29 to have f defined on \(\mathbb {R}\) we extend it by the following procedure. Define
we extend f by \(\tilde {f}:\mathbb {R}\rightarrow \mathbb {R}\) as follows
and have that also \(\tilde {f}\in W^{1,\infty }_{\mathit{\text{loc}}}(\mathbb {R})\) is concave. The extension is also illustrated in Fig. 6.1 . We also define
The reformulation of HamiltonJacobi equations as in Eq. (6.5) also necessitates the redefinition of the Legendre transform as
Definition 29 (Legendre Fenchel Transform in Traffic Flow Modeling)
For a given \(u_{\max }\in \mathbb {R}_{>0}\) let a concave flux function \(f\in W^{1,\infty }([0,u_{\max }])\) be given and recall its extension \(\tilde {f}\) as in Assumption 2, the Legendre Fenchel transform reads for \(u\in \operatorname {Dom}(f^{*})\) as
For specific flux functions, as, for instance, for triangular flux functions, one can compute the dual function explicitly. We first define a triangular flux function as follows and then compute its dual.
Definition 30 (Triangular Flux Function)
We call \(f\in W^{1,\infty }([0,u_{\max }])\) for a given \(u_{\max }\in \mathbb {R}_{>0}\) a triangular flux function if it satisfies
where the model parameters are the free flow speed \(v\in \mathbb {R}_{>0}\), the critical density \(u_{\text{c}}\in [0,u_{\max })\), the congestion speed \(w\in \mathbb {R}_{<0}\) and the maximal density \(u_{\max }\), satisfying \(v u_{\text{c}}=w (u_{\text{c}}u_{\max })\). The flux function is illustrated in Fig. 6.2.
To illustrate the complexity of the computations we give in the following an explicit expression for the Legendre Fenchel transform of f:
Remark 16 (Computation of the Dual Function of the Triangular Flux)
Let the flux as in Eq. (6.10) be given, and recall the transform in Definition 29 we compute its dual as
where we have used \(vu_{\text{c}}=w(u_{\text{c}}u_{\max })\) so that we obtain as effective domain \(\operatorname {Dom}(f^{*})=[v,w]\).
Assumption 3 (Notation for the Bounded Domain Case)
The spatial interval on which we will consider the previously introduced Eq.(6.4) will be the interval \((A,B)\subset \mathbb {R}\) with \(A,B\in \mathbb {R},\ A<B\) . For notational convenience, we will sometimes write
We then start with a basic definition of value conditions, one key ingredient for the generalized solution:
Definition 31 (Value Condition)
For \(T\in \mathbb {R}_{>0}\) a value condition \(\mathfrak {c}:\operatorname {Dom}(\mathfrak {c})\subseteq [0,T] \times X\rightarrow \mathbb {R}\) is a lower semicontinuous function. We extend \(\mathfrak {c}\) by \(\tilde {\mathfrak {c}}:[0,T]\times X\) on the entire spacetime horizon by defining
For reasons of convenience we replace in the following \(\tilde {\mathfrak {c}}\) again by \(\mathfrak {c}\) which is now defined on [0, T] × X but admits as effective domain on the original \(\operatorname {Dom}(\mathfrak {c})\).
With this definition we can present the main theorem of this section, the generalized LaxHopf formula based on viability theory [22]. This viability epi solution to the HamiltonJacobi equation is obtained in epigraphical form from the computation of a viability kernel under an auxiliary dynamics (related to characteristics). The corresponding result is given in the form of a epigraphical set; the lower envelope of this set is defined as the solution. Once established, the results of this procedure can be fully characterized as follows (details outside of the scope of this book, see, for example, [36]).
Theorem 36 (Generalized LaxHopf Formula)
The viability episolution \(m_{\mathfrak {c}}\) for \(\mathfrak {c}\) as in Definition 31 can be expressed as:
Proof
See [74]. □
The following Proposition 11 enables it to compute solutions for different input datum (boundary datum, initial datum, datum on trajectories) separately, and then stick all of them together to one solution.
Proposition 11 (Infmorphism Property)
For i ∈ I where I is a finite set, let \({\mathfrak {c}}_{i}:\operatorname {Dom}\subset (0,T)\times \mathbb {R}\rightarrow \mathbb {R}\) be as in Definition 31 . Then, defining
a property known as infmorphism property holds:
with \(m_{\mathfrak {c}_{i}}\) being the solution for the value condition \(\mathfrak {c}_{i}\) as in Theorem 36.
Proof
The proof can be found in [74]. □
6.3.1 Piecewise Affine Initial and Boundary Datum
In this section, we will investigate the generalized LaxHopf formula for specific initial and boundary datum, namely for piecewise affine datum. The restriction to this type of initial and boundary datum offers to compute the solution efficiently as a convex optimization problem, or—even more—state the solution explicitly.
On the level of conservation laws this means that we approximate initial and boundary datum piecewise constant. As piecewise constant functions are dense in \(L^{1}(\mathbb {R}),\ L^{1}(X)\) respectively, we can approximate every initial and boundary flux datum as precise as we want.
6.3.2 Piecewise Affine Initial Datum
Defining piecewise affine initial datum, we recall Eq. (6.6) so that we know that m _{x}(0, x) = −u _{0}(x) and as \(u_{0}(x)\in [0,u_{\max }]\ \forall x\in X \text{ a.e.},\) we define as follows
Definition 32 (Affine Initial Conditions)
We consider for \(T\in \mathbb {R}_{>0}\) the following affine initial condition \(m_{0}\!:\operatorname {Dom}(m_{0})\subset [0,T]\times X\rightarrow \mathbb {R}\):
where \( \underline {\alpha },\overline {\alpha }\in X\) with \( \underline {\alpha }\leq \overline {\alpha }\) and \(a\in [u_{\max },0],b\in \mathbb {R}\). The effective domain of m _{0} is
Lemma 5 (LaxHopf Formula Affine Initial Conditions in Definition 32)
The generalized solution to the HamiltonJacobi equation with initial datum as in Definition 32 can be expressed for (t, x) ∈ (0, T) × X as
The solution can actually be stated explicitly for triangular flux functions:
Lemma 6 (Explicit Solution Formula for Definition 32 and Triangular Flux)
For triangular flux function as in Definition 30 the solution formula in Lemma 5 can be stated as
Proof
According to Definition 32 and \(\operatorname {Dom}(f^{*})=[v,w]\) we have for (t, x) ∈ (0, T) × X so that \([\frac {Ax}{\nu ^{\flat }},+\infty )\cap \left [ t\overline {\gamma },t \underline {\gamma }\right ]\neq \emptyset \) (recall that a ≤ 0)
so that we obtain indeed
which can be reformulated to holds exactly the proposed solution formula. □
6.3.3 Piecewise Affine Left Hand Side Boundary Datum
As the boundary datum for the HamiltonJacobi equation is timeintegrated flow datum, we define as follows:
Definition 33 (Affine Left Hand Side Boundary Datum)
We consider the following upstream boundary datum γ : [0, T] × X:
where \(c\in [0,f_{\max }],d\in \mathbb {R}\) and \( \underline {\gamma },\overline {\gamma }\in [0,T]\) with \(\overline {\gamma } \underline {\gamma }\geq 0.\) The effective domain of γ is
Lemma 7 (LaxHopf Formula for Affine Left Hand Side b.c.)
The generalized solution to the HamiltonJacobi equation with left hand side boundary datum as in Definition 34 can be expressed for (t, x) ∈ [0, T] × X ∖{A} as
In case of a triangular flux function, we present the explicit solution formula in the following Lemma 8.
Lemma 8 (Explicit Solution Formula for Lemma 7 and Triangular Flux)
For triangular flux function as in Definition 30 the solution formula in Lemma 7 can be stated for \(t\in [ \underline {\gamma },T]\) as
Proof
According to Definition 32, we have for \((t,x)\in (0,T)\times X, \ [\frac {Ax}{\nu ^{\flat }},+\infty )\cap \left [ t\overline {\gamma },t \underline {\gamma }\right ]\neq \emptyset \) (recall that \(c\in [0,f_{\max }]\))
recall that ν ^{♭} = v
and also \(c\leq f_{\max }=f(u_{c})=u_{\text{c}}v\)
from which the conclusion follows. □
Definition 34 (Affine Right Hand Side Boundary Datum)
We consider the following right hand side boundary datum β : [0, T] × X:
where \(e\in [0,f_{\max }],\ f\in \mathbb {R}\) are given as well as \(\overline {\beta }, \underline {\beta }\in [0,T]\) with \(\overline {\beta } \underline {\beta }\geq 0\). The effective domain of β is therefore
Lemma 9 (LaxHopf Formula for Affine Right Hand Side b.c.)
The generalized solution to the HamiltonJacobi equation with right hand side boundary datum as in Definition 34 can be expressed for (t, x) ∈ (0, T) × X ∖{B} as
As in Lemma 8, we will give the solution explicitly for triangular flux function and left hand side boundary datum:
Lemma 10 (Explicit Solution Formula for Lemma 9 and Triangular Flux)
For triangular flux function as in Definition 30 the solution formula in Lemma 9 can be stated for \(t\in [ \underline {\beta },T]\) as
Proof
According to Definition 32, we have for \((t,x)\in (0,T)\times X, \ [\tfrac {Bx}{\nu ^{\sharp }},+\infty ) \\ \cap \left [t\overline {\beta },t \underline {\beta } \right ]\neq \emptyset \) (recall that \(e\in [0,f_{\max }]\))
recall that ν ^{♯} = −w
and also \(c\leq f_{\max }=f(u_{c})=u_{\text{c}}v\)
from which the conclusion follows. □
Having stated the solution for any piecewise affine initial datum and boundary datum, we can now move to present a solution formula for unification of such piecewise affine initial and boundary datum.
Definition 35 (The Piecewise Affine Data)
Let the sets \(I,J,K\subset \mathbb {N}_{\geq 1}\) be the index sets for the piecewise affine initial, left, and right hand side data. We assume that I, J, K < ∞. Then, we consider for (i, j, k) ∈ I × J × K the initial, left, and right hand side boundary datum
with corresponding effective domain and \(\boldsymbol {a}\in [u_{\max },0]^{I},\boldsymbol {b}\in \mathbb {R}^{I}, \underline {\boldsymbol {\alpha }},\overline {\boldsymbol {\alpha }}\in X^{I}\) with \( \underline {\boldsymbol {\alpha }}\leq \overline {\boldsymbol {\alpha }}\), \(\boldsymbol {c}\in [0,f_{\max }]^{J},\boldsymbol {d}\in \mathbb {R}^{J}, \underline {\boldsymbol {\gamma }},\overline {\boldsymbol {\gamma }}\in [0,T]^{J}\) with \( \underline {\boldsymbol {\gamma }}\leq \overline {\boldsymbol {\gamma }},\) \(\boldsymbol {e}\in [0,f_{\max }]^{K},\boldsymbol {f}\in \mathbb {R}^{K}, \underline {\boldsymbol {\beta }},\overline {\boldsymbol {\beta }}\in [0,T]^{K}\) with \( \underline {\boldsymbol {\beta }}\leq \overline {\boldsymbol {\beta }}\) the corresponding vectors collecting in their entries the different components of the set of piecewise affine linear initial and boundary data.
By the foregoing Theorems we have for every (i, j, k) ∈ I × J × K the existence of a generalized solution to the corresponding HamiltonJacobi equation, just incorporating the specific initial, upstream, downstream, internal data. Each solution is denoted by
This brings us to the following Theorem, which incorporates all these conditions into one single solution.
Theorem 37 (Solution to the HamiltonJacobi Equation for Piecewise Affine Data)
There exists a solution of the HamiltonJacobi Equation, incorporating the finite sequences of initial, boundary, and internal data, which we call \(c:[0,T]\times X\rightarrow \mathbb {R}\) . Then, the solution can be determined for (t, x) ∈ [0, T] × X by the formula
where the involved functions are introduced in Definition 6.15.
Proof
The proof is a direct consequence of the infmorphism property as stated in Proposition 11 in Eq. (6.13) and Definition 35. □
Theorem 38 (Convexity of the Solutions When Changing the Parameters of Initial and Boundary Datum in Definition 35)
For any (t, x) ∈ [0, T] × X the solution m(t, x) as defined in Eq.(6.16) is convex with regard to the input variables of initial and boundary datum, i.e., a, b, c, d, e, f in the corresponding dimension.
Proof
We only sketch the proof. Due to the specific construction of the solution being the minimum of the minimum of different functions the solution is convex if each of the contributing solutions in Eq. (6.16) is convex. We start with two initial conditions defined on the same domain, so take according to Definition 32 for given \( \underline {\alpha },\overline {\alpha }\in X: \underline {\alpha }\leq \overline {\alpha }\) as initial datum \(m_{0}(t,x)=ax+b,\ \tilde {m}_{0}(t,x)=\tilde {a}x+\tilde {b}\) satisfying the constraints in Definition 32 and in particular \((a,\tilde {a})\in [u_{\max },0]^{2}\). Then, the corresponding solutions are given in Lemma 6 and applying the definition of convexity, we obtain for λ ∈ (0, 1) and \(t\in [0,T],\ x\in [\max \{A, \underline {\alpha }+wt\},\min \{B,\overline {\alpha }+wt\}]\)
where the inequality in the previous calculations follows from the fact that

\(\big (a \underline {\alpha }\geq a\overline {\alpha }\big )\ \wedge \ \big (\tilde {a} \underline {\alpha }\geq \tilde {a}\overline {\alpha }\big )\) as \(a,\tilde {a}\leq 0\)

\(\{a\in [u_{\max },0]:\ \lambda a+(1\lambda )\tilde {a}\geq u_{\text{c}}\} \subseteq \{a\in [u_{\max },0]:\ \lambda a\geq u_{\text{c}}\}\)

\(\{\tilde {a}\in [u_{\max },0]:\ \lambda a+(1\lambda )\tilde {a}\geq u_{\text{c}}\} \subseteq \{\tilde {a}\in [u_{\max },0]:\ (1\lambda ) \tilde {a}\geq u_{\text{c}}\}\)
for every λ ∈ [0, 1]. Similar calculations for the boundary datum using the solution formulae in Lemma 8, Lemma 10 give the convexity with regard to the input parameters of the boundary datum. □
Remark 17 (Greenshields Flux Function)
A similar explicit solution formula can be computed for other flux functions like the quadratic Greenshields [159] flux function \(f(x)=vx(u_{\max }x),\ x\in [0,u_{\max }].\) In this case the Legendre Fenchel dual f ^{∗} will be quadratic as well so that one obtains a quadratic scalar optimization problem with constraints.
As previously stated Theorem 37 gives us a solution formula for any initial and boundary datum where one can also decouple the computation of each part of initial datum and boundary datum. However, for physical relevant solutions (i.e., that there exists the spatial derivative of the solution almost everywhere) one needs to prescribe additional constraints, the socalled compatibility constraints assuring that initial datum and boundary datum fit to each other. For instance, at the space point (0, 0) where initial and boundary datum meet, the datum needs to be continuous over this corner. In addition, assume that we have two parts of piecewise affine linear initial datum as parametrized in Definition 35 so that \(\overline {\alpha }_{1}= \underline {\alpha }_{2}\). Then, the corresponding datum does not need to satisfy a continuity assumptions, which would be \(a_{1}\overline {\alpha }_{1}+b_{1}=a_{2}\overline {\alpha }_{1}+b_{2}\). Same does not necessarily hold true for the remaining initial and boundary conditions and is the reason why one needs additional compatibility conditions.
6.3.4 Compatibility Conditions
To make sure that compatibility is satisfied so that we indeed obtain solutions of the underlying LWR PDE, we state the following
Theorem 39 (Compatibility Condition)
For the sequence of initial and boundary data as in Definition 35 the data is compatible iff
it holds
where \(m_{\mathfrak {c}}\) is the solution operator for an initial or boundary condition \(\mathfrak {c}\) as discussed in Sect. 6.3.2.
Proof
See [74]. □
The compatibility condition might not look very applicable as it states compatibility by just computing the solution and checking then. However, in the case where we can compute the solution explicitly as we have shown in the previous Sect. 6.3.2 for piecewise affine linear datum, these inequalities can be checked directly. Even more, as the datum is piecewise affine linear Eq. (6.17) does not need to be checked for all (t, x) ∈ T × X but only at the boundaries of the domain of each datum and at possible intersections of the corresponding solutions.
6.4 Optimization with HamiltonJacobi Equations
In this section, we show for an easy example how the introduced theory and framework in Sect. 6.3 can be used to formulate optimal control problems in a very efficient way as convex (or even linear) optimization problems.
Problem 2 (Problem Considered)
Assume we have at a specific time \(T\in \mathbb {R}_{>0}\) measured the road density as \(u_{T}\in L^{\infty }(X;[0,u_{\max }])\) and the downstream flow \(f_{\text{B}}\in L^{\infty }((0,T);[0,f_{\max }])\). Can we infer the initial state \(u_{0}\in L^{\infty }(\mathbb {R})\) and upstream flow f(u(⋅, A)) ∈ L ^{∞}((0, T))?
We will address this problem by means of an optimal control problem. As we will take advantage of the previously developed HamiltonJacobi theory and generalized solutions (Sect. 6.3), we need to reformulate f _{B} and u _{T} as boundary values and end values for the HamiltonJacobi equations. Using the relation in Eqs. (6.6) and (6.5) we obtain for the HamiltonJacobi equation the following:
Remark 18 (Reformulation in Terms of HamiltonJacobi Equations)
The downstream datum and end datum in Problem 2 read for the HamiltonJacobi equations for (t, x) ∈ [0, T] × (A, B) as
Note that the choice of integral bounds makes the end term compatible with the boundary term in the way that h _{T}(B) = 0 = h _{B}(T). This is necessary as the datum we would like to track should be Lipschitz continuous. Although, h _{T} and h _{B} are Lipschitz on their corresponding space/ time domain, they will not satisfy compatibility at the spacetime point (T, B).
Now, we are able to formulate the corresponding optimal control problem in terms of HamiltonJacobi equations as in Definition 36:
Definition 36
The Optimal Control Problem Considered For \((\nu ,\sigma )\in \mathbb {R}_{>0}^{2}\) we consider the following constrained minimization problem
where m is the solution of the HamiltonJacobi equation m for initial datum m _{0} and left hand side (upstream) boundary datum h _{A} as stated in Theorem 36 and Proposition 11 for the corresponding value functions.
We will not go into details whether a minimum exists in the chosen functional setup but directly approach the problem in a simplified version: Restricting the flux to a triangular flux as in Definition 30 and the initial and boundary datum to a piecewise affine datum, we can simplify the problem as follows:
Definition 37 (A FiniteDimensional Optimization Problem for Triangular Flux Function and Piecewise Affine Linear Initial and Left Hand Side Boundary Datum)
Chose for the left hand side boundary datum and initial datum as in Eq. (6.15) the finite set I and J with corresponding \(\overline {\boldsymbol {\alpha }}, \underline {\boldsymbol {\alpha }}\in X^{I}\) so that \(\cup _{i\in I} [ \underline {\alpha }_{i},\overline {\alpha }_{i}]=\overline {X}\) and \(\overline {\boldsymbol {\gamma }}, \underline {\boldsymbol {\gamma }}\in [0,T]^{J}\) so that \(\cup _{j\in J} [ \underline {\gamma }_{j},\overline {\gamma }_{j}]=[0,T]\) a finitedimensional approximation to Definition 36 reads as
As this minimization problem will have results which are not of interest as the corresponding initial and boundary values are not attained we can add a penalization for b, d and compatibility constraints. We then obtain
Theorem 40 (A Convex Optimization Problem)
Adding to the optimization problem in Definition 37 the corresponding compatibility constraints Theorem 39 , we still obtain a convex optimization problem in the optimization variables
Proof
This is a direct consequence of the structure of the compatibility constraints in Theorem 39, and the convexity of the solution in Theorem 38. □
Of course, the previously outlined optimization problem could easily be generalized to more complex situations and the underlying convexity structure could still be applied. In addition, as an explicit solution formula is available, the computation of the minimization problems is fast. Under specific circumstances and more manipulations one can even obtain linear or quadratic and/or mixedinteger programs [61, 74, 221].
6.5 Bibliographical Notes
For general theory and viscosity solutions of HamiltonJacobi equation we refer the reader to [94, 95] where the authors consider timedependent and independent HamiltonJacobi equations with a Hamiltonian which can also be explicitly space and timedependent with Dirichlet boundary conditions and as Cauchy problem. They introduced a notation of solutions, i.e., viscosity solutions for which existence and uniqueness of solutions and stability properties can be obtained (compare also [28]). For a rather comprehensive presentation on optimal control, the related HamiltonJacobiBellman equations (as optimality conditions) and the named viscosity solutions we refer the reader to [27, 71]. In the book [225] general solutions of HamiltonJacobi equations are discussed, in [30] and [134] semicontinuous viscosity solutions. For viability theory we refer to [22, 36] and for HamiltonJacobi equations with inequality constraints to [21].
Applications of HamiltonJacobi equations and related theory for transportation can be found in [72,73,74] which are one of the main sources for the latter chapter.
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Bayen, A., Monache, M.L.D., Garavello, M., Goatin, P., Piccoli, B. (2022). Control Problems for HamiltonJacobi Equations Coauthored by Alexander Keimer . In: Control Problems for Conservation Laws with Traffic Applications. Progress in Nonlinear Differential Equations and Their Applications(), vol 99. Birkhäuser, Cham. https://doi.org/10.1007/9783030930158_6
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