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Control of PDEs with Boundaries Governed by ODEs

By Miroslav Krstic

The following article is based on the author’s W.T. and Idalia Reid Prize Lecture at the 2019 SIAM Conference on Control and its Applications, which took place earlier this year in Chengdu, China.

Control Systems

For dynamical systems modeled by ordinary or partial differential equations (PDEs) with significantly fewer input variables than state variables—like a scalar input variable for a PDE with a spatially-distributed or infinite-dimensional state—control theory formulates the input as a function(al) of the state. This achieves stability for the dynamical system, where “stability” in a technically rigorous sense refers to a set of properties and forces the state to converge to zero as time approaches infinity.

Constructing such input functions—also called “feedback laws” because input depends on the measurable state—is part of the design of most technological systems. A simple example is the Segway, whose driver would nosedive or fall backward in the absence of a feedback system that feeds the pitch angle measurements into the wheel angle inputs to keep the apparatus and rider upright. Less obvious feedback systems include those that developed through evolution to both keep organisms alive and prevent them from making drastic changes to themselves, regardless of how much they desire said modifications. For instance, feedback systems that regulate metabolism prevent people from achieving significant weight loss by starving themselves over several days. We generated these feedback systems to maintain our energy reserves in periods of famine and strenuous travel.

PDE Control on Moving Domains

Classical control theory devised for ordinary differential equations (ODEs) requires remarkable sophistication in the design of feedback laws for nonlinear systems. Feedback synthesis for PDEs poses even greater challenges, namely in transitioning from the finite to infinite system dimension. The greatest achievements in nonlinear ODE control occurred in the 1980s and 90s, whereas PDE control has blossomed during the last two decades.

Not all physical systems are modeled by ODEs of a fixed order or PDEs on fixed domains. Some important applications—including traffic, opinion dynamics, and climate science—involve processes whose dimensions or domains depend on the size of the process state. For instance, the state vector dimension can increase with the size of the state. Or a higher temperature in its PDE spatial domain may cause the domain to grow, as with melting ocean ice.

Classical control techniques are unequipped to deal with such dimension-varying dynamics. In fact, these possibilities have rarely even occurred to the control research community, which has been preoccupied in recent years with already-difficult nonlinear, infinite-dimensional, stochastic, and hybrid phenomena in fixed dimension.

Among the simplest and most elegant problems with the state’s dimension that varies with the state’s size are those involving a connected ODE and PDE, so that the PDE’s state acts as an input to the ODE, whose state thus represents the PDE’s boundary location. Such PDE-ODE systems may involve either parabolic or hyperbolic PDEs.

Control of the Stefan System (Parabolic)

An example of a parabolic PDE-ODE system in which the ODE state represents the PDE’s boundary location is the so-called Stefan system. Developed and analytically solved by Slovenian-Austrian physicist Josef Stefan (of Stefan-Boltzmann fame) in the late 1800s, the system models melting and freezing.

Figure 1. Temperature profiles and phase interface in a partial differential equation-ordinary differential equation (PDE-ODE) system involving a liquid, a solid, and rightward melting with the aid of heat flux applied by a laser on the left boundary. Figure courtesy of Shumon Koga.
Researchers have recently used the Stefan system to model numerous other physical phenomena, including additive manufacturing with both polymers and metals; growth of neuronal axons; tumor proliferation; cancer treatment via cryosurgeries; spread of invasive species in ecology; lithium-ion batteries; domain walls in ferroelectric thin films; and information propagation in social networks.

Stefan’s model gives rise to several control and state estimation problems; here we focus on one that is both simple and difficult. The goal is to regulate the liquid-solid interface position \(s(t)\) to a setpoint \(s_r>0\). This is shown in Figure 1, where \(T_s(x,t)\) and \(T_l(x,t)\) respectively represent the spatiotemporal temperatures in the solid and liquid. Heat PDEs govern the temperatures and a scalar ODE—whose inputs are the heat fluxes at the PDEs’ boundaries—governs the interface position.

Using the backstepping approach for PDE-ODE systems [3], we design and implement a feedback law \(q_c(s, T_l, T_s)\) by employing a laser to apply a heat flux to the liquid. This backstepping feedback is proportional to the error between the measured thermal energy and the thermal energy at the melting/freezing point, plus the interface tracking error \(s - s_r\). The backstepping approach entails construction of a Volterra transformation of the temperature state and a Lyapunov functional based on the transformed temperature state.

This control law achieves global stabilization for all initial conditions where the liquid temperature is above melting and the solid temperature is below freezing; both temperatures remain in these states for all time. In physical terms, this means that no solid islands form within the liquid and no pools of liquid form within the solid. The maximum principle for the heat equation establishes this result [1-2].

Control of Moving Shock in Congested Traffic

Figure 2. Free traffic (upstream/left) and congested traffic (downstream/right) are separated by shock, depicted as a sharp increase in density. Modulating the durations of the red and green lights on the on-ramps regulate the shock location to a desired position. Figure courtesy of Huan Yu.
The analog to the Stefan system’s parabolic PDE phenomenon is the hyperbolic PDE phenomenon that arises in traffic. This originates with a moving shock that delineates the free traffic (upstream of shock) from the congested traffic (downstream from shock), as seen in Figure 2. The hyperbolic nonlinear Lighthill-Whitham-Richards PDE, which acts as a simple delay for small deviations, models the traffic flow. A scalar ODE governs the shock motion, and the traffic densities of the congested and free traffic at the shock location form the ODE’s inputs. This ODE represents the Rankine-Hugoniot jump condition that is common in compressible gas models.

Researchers again use the PDE backstepping design to devise a feedback law that regulates the moving shock’s position to a setpoint. Without this type of control law to arrest the shock at a desired location on the freeway, congested traffic would continue to propagate upstream until it consumes the entire road. The feedback law is implemented via “ramp metering,” which involves modulation of the red and green lights on the freeway on-ramps around steady durations that correspond to the desired location of the shock.

Analyzing the PDE-ODE system with the feedback law once again employs a backstepping/Volterra transformation of the traffic density PDE’s state, along with a resulting Lyapunov functional. Like with the Stefan system, stability occurs in the \(H_1\) Sobolev norm. However, while stability for the Stefan system holds for all physically meaningful initial conditions, it only remains true locally—for small deviations of the density field around its equilibrium profile—for the traffic problem. 

[1] Koga, S., Diagne, M., & Krstic, M. (2019). Control and state estimation of the one-phase Stefan problem via backstepping design. IEEE Transac. Autom. Con., 64, 510-525.
[2] Koga, S., Karafyllis, I., & Krstic, M. (2018). Input-to-state stability for the control of Stefan problem with respect to heat loss at the interface. In 2018 American Control Conference. Milwaukee, WI.
[3] Krstic, M. (2009). Delay Compensation for Nonlinear, Adaptive, and PDE Systems. Boston, MA: Birkhauser.

Miroslav Krstic is a distinguished professor of mechanical and aerospace engineering at the University of California, San Diego. He is a coauthor of two books with SIAM (one of which received the International Federation of Automatic Control (IFAC) Chestnut Textbook Award) and has written 11 other books on control theory. Krstic is a Fellow of SIAM, the Institute of Electrical and Electronics Engineers, the American Society of Mechanical Engineers (ASME), IFAC, and three other societies. He is the recipient of ASME’s Rufus Oldenburger Medal, IFAC’s Nonlinear Control Systems Award, the American Automatic Control Council’s John R. Ragazzini Education Award, and several other accolades.

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