### A. PDE Network Model

In this paper, we use the PDE model of air traffic flow initiated in [4], and later extended in [40]. This approach models jetways as paths composed of a series of line segments known as links, which range between 30 and 200 mi in length. In previous work, 13 742 trajectories across the NAS were reduced to build a network of 1598 links [40], a subset of which is used in this paper.

We represent each link *k* on a path as a segment [0,*L*] and denote by *u*(*x*,*t*) the number of aircraft between distances zero and *x* at time *t*. In particular, *u*(0,*t*) = 0 and *u*(*L*,*t*) is the total number of aircraft in the path modeled by [0, *L*] at time *t*. In this paper, we assume a time-varying velocity profile *v*(*x*,*t*) > 0, which represents the aggregate velocity of aircraft flow at position *x* and time *t*. Applying the conservation of mass to a control volume comprised between positions *x* and *x*+*h*, and letting *h* tend to zero, the following relation between the spatial and temporal derivatives of *u*(*x*,*t*) is obtained [4]
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$$\cases{{\partial u(x,t)\over\partial t}+v(x,t){\partial u(x,t)\over\partial x}\cr\quad=v(0,t){\partial u(0,t)\over\partial x}&$(x,t)\in(0,L)\times(0,T]$\cr u(x,0)=u_{t_{0}}(x)&$x\in[0,L]$\cr u(0,t)=0&$t\in[0,t]$}\eqno{\hbox{(1)}}$$where the term *v*(0,*t*)(∂ *u*(0,*t*)/∂ *x*) represents a prescribed rate at which aircraft enter the link (at *x* = 0). We introduce the density of aircraft ρ(*x*,*t*) as the weak derivative of *u*(*x*,*t*) with respect to *x*ρ(*x*,*t*) = (∂ *u*(*x*,*t*)/∂ *x*), so that the evolution of aircraft density is a solution of the partial differential equation
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$$\cases{{\partial\rho(x,t)\over\partial t}+v(x,t){\partial\rho(x,t)\over\partial x}\cr\quad+{\partial v(x,t)\over\partial x}\rho(x,t)=0&$(x,t)\in(0,L)\times(0,T]$\cr\rho(x,0)=\rho_{t_{0}}(x)&$x\in[0,L]$\cr\rho(0,t)=\rho_{x_{0}}(t)&$t\in[0,T]$}\eqno{\hbox{(2)}}$$with initial and boundary conditions ρ_{to}(*x*) and ρ_{x0}(*t*). This PDE is a linear advection equation with positive velocity and a source term (∂ *v*(*x*,*t*)/∂ *x*)ρ(*x*,*t*). Clearly, the two systems (1) and (2) are equivalent and model the same physical phenomenon. Note that while (2) is linear in the state ρ(*x*,*t*), if *v*(*x*,*t*) is also introduced as an unknown parameter to be determined by the optimization program, (2) becomes nonlinear when considered as a constraint.

This model of air traffic flow on a link can be extended to a network of links. We let denote the set of all links in the network. For each link , we associate the set with the set of all links in the network in which flow merges into link *k*. We represent the portion of flow exiting link that enters link *k* by β_{m,k}, where 0≤β_{m,k}≤ 1. Because flow exiting *m* may split onto multiple links including *k*, we require that for a fixed *m*, . That is, the flow exiting from link *m* and diverging to all other links *i* must be conserved. For the simple network shown in Fig. 1, we have , and . The system of partial differential equations on a general network can be written as shown in the equation at the bottom of the next page.
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$$\cases{{\partial\rho_{k}(x,t)\over\partial t}+v_{k}(x,t){\partial\rho_{k}(x,t)\over\partial x}+{\partial v_{k}(x,t)\over\partial x}\rho_{k}(x,t)=0&$(x,t)\in(0,L_{k})\times(0,T],k\in\BBK$\cr\rho_{k}(x,0)=\rho_{t_{0},k}(x)&$x\in[0,L_{k}],k\in\BBK$\cr\rho_{k}(0,t)=\rho_{x_{0},k}(t)+{\sum_{m\in\BBM_{k}}\beta_{m,k}\rho_{m}(L_{m},t)v(L_{m},t)\over v_{k}(0,t)}&$t\in[0,T],k\in\BBK$}\eqno{\hbox{(3)}}$$ The well-posedeness of (3), and the existence and uniqueness of the solution on the network, is proved in [40].

### B. Optimal Control Problem

With the PDE model defined above, we now pose the problems of maximizing the network throughput and minimizing delays in the en route and arrival airspace as optimal control problems. In other words, we seek to find the globally optimal velocity profiles *v*_{k}(*x*,*t*) with respect to some network performance metric, for all links in the system. The novelty of this work is that unlike previous treatments of these optimal control problems, we do not solve for the velocity profile directly. Instead, we introduce a new formulation in which the velocity is computed from the optimal density and flux solutions. The most important contribution of this paper is that with this change of variables, the PDE network optimal control problem can be posed as a convex optimization problem, subsequently enabling globally optimal solutions which can be computed using efficient and well-developed LP or QP techniques.

*1) Constraints*

Following standard optimal control terminology, we encode the dynamics of the system in the form of constraints [8]. The key constraint in the optimization program is thus the constitutive model equation given in the form of PDEs of the ρ_{k}(*x*,*t*) functions. This also includes a prescribed initial density distribution ρ_{t0,k}(*x*), which represents the aircraft initially airborne. The conditions at the boundaries of each link ρ_{k}(0,*t*) are defined by aircraft entering the network from international flights or lower altitude traffic ρ_{x0,k}(*t*) and aircraft entering from other links in the network. We impose upper and lower bounds on the velocity profile on each link *v*_{k,min}(*x*,*t*)≤ *v*_{k}(*x*,*t*)≤ *v*_{k,max}(*x*,*t*) to keep traffic flow consistent with the physical capabilities of the aircraft in the NAS.

An upper bound on the density on each link ρ_{k}(*x*,*t*)≤ρ_{k,max}(*x*,*t*) is added for two reasons. First, the FAA has established a minimum horizontal separation distance between aircraft in the en route environment of five nautical miles. By imposing an upper bound on the density, we capture the essence of this requirement in an aggregated form, following the work in the field of Eulerian models [3], [4], [23], [24], [31], [32], [34], [36], [40]. Secondly, if air traffic controllers are used to dispatch the optimal control strategy to the pilots, the integral of the upper bound on the density corresponds to the maximal number of aircraft in a sector that the air traffic controller can legally handle. We also require the density to be nonnegative to be physically meaningful.

*2) Transformation of the Problem Into a Convex Optimization Program*

If these constraints are imposed along with the network PDE model, the optimization program for a single junction becomes
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$$\eqalignno{&{\bf min}:\quad J\cr&\quad{\bf st}:\quad 0\leq\rho_{k}(x,t)\leq\rho_{k,\max}(x,t)\cr&\qquad\qquad\quad(x,t)\in[0,L_{k}]\times[0,T],k\in\BBK\cr&\qquad\qquad v_{k,\min}(x,t)\leq v_{k}(x,t)\leq v_{k,\max}(x,t)\cr&\qquad\qquad\quad(x,t)\in[0,L_{k}]\times[0,T],k\in\BBK\cr&\qquad\qquad \ (3).&\hbox{(4)}}$$

The goal of the optimization problem is to find the optimal *v*_{k}(⋅,⋅) such that the objective function *J* is minimized. The output of this program is thus an optimal speed control policy to be applied by the air traffic controller.

The principle difficulty with solving (4) is that the PDE is a nonlinear constraint in the optimization variables, as was mentioned earlier. Thus, even if linear discretization schemes are used, the resulting constraints will be nonlinear. In the past, standard PDE optimization techniques such as adjoint methods have been used to solve these types of problems [4], [16], [31], [32], [35]. As explained earlier, the benefit of these methods is their generality. The drawbacks are the difficulty to implement them (using BFGS routines [41]), the lack of guarantees of numerical convergence and subsequent degree of suboptimality, and, in practice, the high computational cost of the resulting algorithms. In particular, recent work on second-order methods [32] displayed improved convergence to suboptimal values of first-order methods, which also underlines the lack of global certificates of optimality for this general class of methods (the performance of the second order method is so superior to the performance of the first-order method that it emphasizes the suboptimality of the latter very strikingly).

We now propose a change of decision variables that makes the previous constraints in (4) linear
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$$q_{k}(x,t)=\rho_{k}(x,t)v_{k}(x,t)\eqno{\hbox{(5)}}$$where *q*_{k}(*x*,*t*) is known as the flux function and can be interpreted as the amount of aircraft that flow through a point *x* on the link per unit time at time *t*. The nonlinear constraint in terms of decision variables ρ_{k}(*x*,*t*) and *v*_{k}(*x*,*t*) can be transformed into a linear constraint in terms of decision variables ρ_{k}(*x*,*t*) and *q*_{k}(*x*,*t*). The control variable *v*_{k}(*x*,*t*) is completely absent in the resulting formulation, and so it must be computed from the optimal solutions for density and flux obtained by solving the following equivalent continuous optimization problem:
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$$\eqalignno{{\bf min}:&\quad J\cr{\bf st}:&\quad 0\leq\rho_{k}(x,t)\leq\rho_{\max}(x,t)\cr&\qquad(x,t)\in[0,L_{k}]\times[0,T],k\in\BBK\cr&\quad v_{k,\min}(x,t)\rho_{k}(x,t)\leq q_{k}(x,t)\cr&\qquad\leq v_{k,\max}(x,t)\rho_{k}(x,t)\cr&\qquad(x,t)\in[0,L_{k}]\times[0,T],k\in\BBK\cr&\quad{\partial\rho_{k}(x,t)\over\partial t}+{\partial q_{k}(x,t)\over\partial x}=0\cr&\qquad(x,t)\in(0,L_{k})\times(0,T],k\in\BBK\cr&\quad\rho_{k}(x,0)-\rho_{t_{0},k}(x)=0\cr&\qquad x\in[0,L_{k}],k\in\BBK\cr&\quad q_{k}(0,t)-q_{x_{0},k}(t)\cr&\qquad-\sum_{m\in\BBM_{k}}\beta_{m,k}q_{m}(L_{m},t)=0\cr&\qquad t\in[0,T],k\in\BBK . &\hbox{(6)}}$$Note also that the velocity constraint changed from bounds on *v*_{k}(*x*,*t*) in (4) to a linear constraint on *q*_{k}(*x*,*t*) and ρ_{k}(*x*,*t*) in (6). In this latter form, any linear discretization scheme will yield a discrete convex formulation that can be solved using either LP or QP techniques given an appropriate choice of the objective function and its discretization, or general convex optimization techniques if the objective function *J* is a general convex function.

*3) Objective Functions*

In order to pose convex objective functions, we introduce a single airport α in the set of all airports in the network. If we wish to denote the number of arrivals of aircraft at airport α through time *t* by η_{α}(*t*), we can maximize the total number of arrivals by forming the following objective function: **max**: . If we let a link terminating at airport α be denoted by *k*_{α}, then *k*_{α} belongs to the subset of links terminating at one of the airports in the set of all airports defined by . We note that the integral represents the cumulative arrivals η_{α}(*t*) at airport α at time *t*, and so this objective can be implemented in terms of the flux as
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$${\bf min}:\quad J=-\sum_{k_{\alpha}\in\BBK_{\BBA}}\int\limits_{0}^{T}q_{k_{\alpha}}\left(L_{k_{\alpha}},t\right)\,dt\eqno{\hbox{(7)}}$$where **max**:*J* = −**min**:−*J* has been used to obtain a convex minimization program in standard form. Note that the throughput objective function is linear in the flux function *q*. With a proper (linear) discretization in space and time, it will lead to a linear program when subject to linear constraints. This objective function is used in Sections III and IV to highlight the correct numerical implementation of the convex optimization problem.

We introduce a second control objective: to minimize delays by matching the desired flight plans as closely as possible. This is accomplished by first computing the desired density and flux distributions ρ_{k,des}(*x*,*t*) and *q*_{k,des}(*x*,*t*) corresponding to the desired flight plans of all aircraft in the network. In practice, this is a useful formulation when some unforeseen event, such as inclement weather, reduces the capacity of the network and forces deviations from the desired schedule
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$$\displaylines{{\bf min}:J=\sum_{k\in\BBK}\int\limits_{0}^{T}\int\limits_{0}^{L_{k}}\left(\left(q_{k}(x_{k},t)-q_{k,{\rm des}}(x_{k},t)\right)^{2}\right.\hfill\cr\hfill\left.+\,\left(\rho_{k}(x_{k},t)-\rho_{k,{\rm des}}(x_{k},t)\right)^{2}\right)\,dx_{k}\,dt.\quad\hbox{(8)}}$$This objective will be implemented as a quadratic program when subject to linear constraints and is demonstrated on a benchmark problem implemented in Section V.

To emphasize the usefulness of the convex framework, we note that a third control objective can also be implemented, which is to match a desired arrival schedule at all airports as closely as possible [31]. If we define the desired number of aircraft that have arrived at airport α by time *t* as η_{α,des}(*t*), the objective formulation is **min**: . In terms of the flux function, this objective can be written as
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$$\displaylines{{\bf min}:\quad J=\sum_{k_{\alpha}\in\BBK}\int\limits_{0}^{T}\left(\int\limits_{0}^{t}q_{k_{\alpha}}\left(L_{k_{\alpha}},s\right)\,ds\right.\hfill\cr\hfill\left.-\,\int\limits_{0}^{t}q_{k_{\alpha},{\rm des}}\left(L_{k_{\alpha}},s\right)ds\right)^{2}dt.\quad\hbox{(9)}}$$Again, this objective can be implemented as a quadratic program when subjected to linear constraints.

For the objective functions involving scheduling, such as maximizing the throughput of the network (7), a choice of *T*≤ 24 h is natural. The optimal control strategies can be computed as soon as the flight plans of each aircraft are established for the day. In the event of a perturbation that creates deviations from a prescribed flight plan (8) or arrival schedule (9), *T* may range between 6 and 12 h, or a time at which the perturbation no longer effects the flight schedule. Although it is computationally feasible, we do not consider *T*≥ 24h simply because other control strategies such as grounding flights become more appropriate. The framework presented in this paper assumes scheduled flights will eventually take place.