Hyperbolic Partial Differential Equations in Optimization and Control

Transport phenomena are ubiquitous in many physical, biological, and societal processes: applications range from fluid dynamics for air and spacecrafts, swarming dynamics of animal cohorts, data or product flows in large-scale computing or production systems [7], and description of emergent phenomena in collective dynamics [11]. Typically, these problems exhibit a temporal and spatial scale, a finite speed of information propagation, and highly nonlinear behavior. Mathematically, these phenomena allow for a unified description by general hyperbolic partial differential equations (PDEs). Some classical examples date back to 1757 when Leonhard Euler published Principes généraux du mouvement des fluides [5], known today as the Euler equations of fluid dynamics:

\[{\partial}_{t}\rho+{\nabla}_{\mathbf{x}}\cdot\rho\mathbf{u}=0,\enspace{\partial}_t(\rho\mathbf{u})+{\nabla}_{\mathbf{x}}\cdot(\rho{\mathbf{u}}\otimes{\mathbf{u}})+{\nabla}_{\mathbf{x}}\;p=0,\enspace{\partial}_tE+{\nabla}_{\mathbf{x}}\cdot(E+p){\mathbf{u}}=0.\]

The Euler equation \((1)\) in conservative form solves for the density \(\rho,\) velocity \(\mathbf{u},\) and total energy \(E\) as functions of space \(\mathbf{x}\) and time \(t\). The ideal pressure\(, p={\rho}e({\gamma}-1),\) together with the ratio of the specific heats \(\gamma,\) and the internal energy \(e,\) is related to the total energy by \(E= {\rho}e+\frac{1}{2}||\mathbf{u}|^2.\)

Since the Euler equations were published, tremendous progress has been made in both analytical and numerical aspects of hyperbolic PDEs, including results on well-posedness and the development of efficient, high-order methods on arbitrary grids. Due to its prominence within a wide range of applications, a strong demand for methods that emphasize optimization, optimal control, and controllability and stabilization, has emerged. Although researchers in different areas of expertise often approach these problems with a different focus and varying methods, all applied mathematicians share a substantial body of common knowledge and background on transport phenomena. Thus, the interdisciplinary nature of this research and analysis attracted many researchers interested in the novel fundamental theoretical results, as well as more recent results on corresponding numerical schemes.

Despite immense progress, the two approaches that drive both the theoretical and numerical developments of hyperbolic PDEs are still disjoint. On one hand, there is now an established theory for smooth solutions and multidimensional systems of conservation and balance laws; such solutions are known to exist given data with sufficiently small \(H^s\) norm or sufficiently small control horizon, respectively. Their construction is often conducted using the theory of characteristics or energy estimates in adapted weighted norms. The particularities of such solutions are also reflected by the tools used for controllability and stabilization. The second approach, which uses Lyapunov–type arguments, solves for the stabilization of linear and nonlinear hyperbolic balance laws in one or more spatial dimensions and even on stratified domains (i.e., networks). Since most physically relevant equations are nonlinear, the classical tools of linear control theory rarely apply, leading to an emphasis on the development of new methods; strong results are available on the controllability and stabilization, particularly in the spatially one-dimensional (1D) setting [3, 10]. Recent developments investigate turnpike properties where the optimal control equals the optimal control of the associated static optimal problem over an extended time horizon — a phenomenon originally introduced by Nobel Prize winner Paul Anthony Samuelson in the context of financial markets and more recently used in the context of nonlinear balance laws. Numerical schemes that mirror analytical approaches for stabilization (and turnpike controls) have also been developed and their theoretical decay rates recovered. This is relevant for numerical schemes that are grid based—as opposed Lagrangian or characteristic methods—due to the presence of numerical dissipation.

Conversely, even for smooth initial data in the scalar nonlinear case, smooth solutions cease to exist beyond finite time. Thus, the formation of discontinuities requires a different set of analytical and numerical tools. Theoretically, progress has been made in the case of spatially 1D, nonlinear systems using the notion of weak entropic solutions, but due to the weak notion of solutions, results on controllability and stabilization in this setting are sparse compared to the smooth case [8]. The presence of discontinuities also required the development of a novel notion of analytical sensitivities for the solution with respect to control variates. Extensive work on alternative notions of differentiability began thirty years ago [4], since the evolution operator generated by the conservation law is generically non-differentiable in \(L^1\), even in the scalar case. New concepts like shift-differentiability or tangent vectors have proven to be a suitable theoretical tool for the description of differentials. 

Today, in the spatially 1D case, results on directional differentiability, and in the scalar case, on Fréchet differentiability, are available; these concepts also led to a theory for optimal control problems in the presence of (finitely many) discontinuities. Within this, calculus adjoint equations have posed additional challenges, being a PDE in a non-conservative form with possibly discontinuous coefficient. Even with the theoretical framework established, the development of suitable numerical schemes in multiple spatial dimensions and up to high-order for the optimal control problem is still under development. One reason for this limitation is that defining the differential requires the solution to have a particular structure of piecewise \(C^1\) solutions with located, finitely many (typically non-intersecting) discontinuities. This structure is in general not obtained using classical discontinuous—Discontinuous Galerkin or Finite-Volume schemes—due to the presence of numerical dissipation and the Gibbs phenomena close to discontinuities. The latter requires additional care, as numerical error typically influences the results on the optimal control. Promising results for low-order, finite-volume schemes to control the dissipation in the spatially 1D setting exist, however, even in the case of 1D systems (and the multi-dimensional case), they are still an area of active research [9].

The aforementioned dichotomy requires a new set of adapted nonlinear tools making the control of hyperbolic PDEs an interesting and vibrant field of research. Furthermore, the field of optimization and control of nonlinear hyperbolic PDEs is rapidly expanding in many different directions. Significant theoretical and numerical contributions remain in optimization calculus and control, including problems involving optimization calculus in the presence of stratified geometries such as networks or manifolds; optimal control for multi-dimensional hyperbolic PDEs and control based on measure-valued solutions; and the controllability for general systems in the presence of shocks. The possibilities extend far further than the given list, and with the increase in understanding of the underlying concepts, the applications for control problems have increased significantly (see Figure 1). The interaction with applications has proven to raise new and interesting control questions for the ubiquitous transport phenomena. The many unexplored areas—theoretical, numerical, or applied—most certainly merit the interest and attention of the mathematics community. In the future, we expect that the interdisciplinary fields of control and hyperbolic PDEs will lead to exciting and new results with a wide range of applications.

References 
[1] Ancona, F., & Nguyen, K.T. (2021). On the global controllability of scalar conservation laws with boundary and source controls. SIAM J. Control Optim., 59(6), 4314-4338.
[2] Bastin, G., & Coron, J.M. (2016). Stability and boundary stabilization of 1-D hyperbolic systems. In Nonlinear differential equations and their applications (Vol. 88). Cham, Switzerland: Springer.
[3] Bressan, A., & Marson, A., (1995). A variational calculus for discontinuous solutions of systems of conservation laws. Commun. Part. Differ. Equ., 20(9-10), 1491-1552.
[4] D’Apice, C., Göttlich, S., Herty, M., & Piccoli, B. (2010). Modeling, simulation, and optimization of supply chains. Philadelphia, PA: Society for Industrial and Applied Mathematics.
[5] Euler, L. (1757). Principes généraux du mouvement des fluides. Mémoires de l’académie des sciences de Berlin, 11, 274-315.
[6] Glass, O. (2007). On the controllability of the 1-D isentropic Euler equation. J. Eur. Math. Soc., 9(3), 427-486.
[7] Herty, M., & Thein, F. (2024). Stabilization of a multi-dimensional system of hyperbolic balance laws. Preprint, arxiv:2207.12006.
[8] Herty, M., & Ulbrich, S. (2023). Numerics and control of conservation laws. In E. Trélat & E. Zuazua, (Eds.), Handbook of numerical analysis (Vol. 24, pp. 473-509). Elsevier.
[9] Li, T. (2010). Controllability and observability for quasilinear hyperbolic systems. In AIMS on applied mathematics (Vol. 3). Springfield, MO: American Institute of Mathematical Sciences.
[10] Motsch, S., & Tadmor, E. (2014). Heterophilious dynamics enhances consensus. SIAM Rev., 56(4), 577-621.
[11] Stern, R.E., Cui, S., Delle Monache, M.L., Bhadani, R., Bunting, M., Churchill, M., … Work, D.B. (2018). Dissipation of stop-and-go waves via control of autonomous vehicles: Field experiments. Transp. Res. Part C Emerg. Technol., 89, 205-221.