# Controllability and Local Asymptotic Stabilizability of Control Systems

By Paul Davis

Does SIAM award a prize for balancing a walking cane better than Jimmy Stewart, stabilizing a baby carriage (*sans* baby), and leveling surges in the Sambre and Meuse rivers? Indeed it does, for those are but three of many examples illustrating the profound work for which Jean-Michel Coron received the W.T. and Idalia Reid Prize in Mathematics at the 2017 SIAM Annual Meeting, held in Pittsburgh, Pa., last summer.

**Figure 1.** Coron’s illustration of small-time local controllability. The nonlinear control system has an equilibrium at (*y*_{e},u_{e}); the initial state *y*^{0} and the target state *y*^{1} are very close to *y*_{e}; the state remains close to *y*_{e}; the control remains close to *u*_{e}; and the time is small. Figure credit: Jean-Michel Coron.

Coron is currently a full professor at Sorbonne Université (Paris 6) and a member of the French Academy of Sciences. The Reid Prize citation specifically recognized the importance of “the Coron return method for feedback stabilization of nonlinear systems using time-varying controls,” and his prize lecture offered an artful and entirely modest tour of this approach and other substantial contributions.

Coron’s work centers on two essential properties of control systems: controllability, or the existence of a control strategy that guides a system from one specified state to another, and local asymptotic stabilizability, or the existence of a feedback law—a control depending on the state—that imparts asymptotic stability to the corresponding closed-loop system. He offered the following example to illustrate the importance of stability. A satellite’s orientation can be controlled if it has two or more rocket motors. But the satellite loses local asymptotic stabilizability when only two motors are functioning, and soon drifts out of the desired orientation. Coron noted dryly that the mathematical outcome—the absence of a stabilizing feedback law—appears to warrant national news coverage if the satellite is sufficiently expensive or important.

Coron illustrated the breadth of his work with a variety of applications, both whimsical and practical. He opened his talk with a video displaying feedback control of a moving cart stabilizing an inverted double pendulum into perfect vertical rigidity. This was followed by a saucy clip from Alfred Hitchcock’s *Vertigo* in which Jimmy Stewart struggles to balance a vertical walking cane—a mere single pendulum—on his palm.

**Figure 2.** The idea underlying Coron’s return method to stabilize the empty baby carriage—and more generally, driftless control systems—is to construct a preliminary *T*-periodic feedback *u*(*t,y*), leading to *T*-periodic trajectories with controllable linearizations. Using these linearizations, one can then assemble a *T*-periodic perturbation *u*(*t,y*)+*v*(*t,y*) of *u*(*t,y*), such that the trajectories are now converging to 0. Figure credit: Jean-Michel Coron

Although Stewart’s balancing efforts did not inspire confidence, the rigidly-vertical inverted double pendulum left no doubt as to its stabilizability. Previewing both the foundation of his own work and what he calls “Louis Nirenberg’s advice to depressed mathematicians: Have you tried to linearize?”, Coron demonstrated the controllability of the inverted pendulum system linearized around its vertical equilibrium. If a linear system is controllable, a pole-shifting argument reveals the presence of a linear feedback control that renders its zero state globally asymptotically stable. From this point, one can argue that a nonlinear system with a controllable linearization is both small-time locally controllable (Figure 1 illustrates this notion of controllability) and locally asymptotically stabilizable. Stewart’s problem with balancing a cane vertically is entirely his own.

**Figure 3.** The idea underlying Coron’s return method for assessing controllability of nonlinear systems whose linearization about the equilibrium of interest is not controllable: linearize instead about a nearby trajectory (red) that leaves from and returns to the troublesome equilibrium. Figure credit: Jean-Michel Coron.

Unfortunately, a continuous feedback law cannot locally stabilize all small-time locally controllable systems. Coron’s methodical dissection by example cleanly separated the two properties—controllable and stabilizable—and demonstrated the advantage of

*time-dependent* feedback for stabilization.

A baby carriage is an example of a controllable system that cannot be stabilized by a continuous feedback law. This system is small-time locally controllable at the origin, but cannot be asymptotically locally stabilized because it does not satisfy Roger Brockett’s necessary condition for local asymptotic stabilizability. However, a time-dependent feedback law (see Figure 2) can restore local asymptotic stability of the carriage. Coron cautioned that his analysis is limited to the carriage *without* a baby; with a baby, he could not guarantee small perturbations!

Similar phenomena manifest themselves in both a satellite with less than three functioning thrusters and a quadcopter confined to a plane (also known as a slider). Both eventually crash because they are small-time locally controllable but not locally asymptotically stabilizable by means of continuous feedback laws. Again, time-dependent feedback laws save the day.

**Figure 4.** Water flows below the sluice gates (see Figure 6 for schematic). The back of the figure shows the gates moving up and down according to feedback laws created by Jean-Michel Coron and his colleagues. In the front of the figure, the gates are motionless. A much faster convergence to the desired height of the water, represented by the red line, is achieved by feedback laws. Image credit: Jonathan de Halleux.

Motivated perhaps by his well-placed admiration for Nirenberg, many of Coron’s theoretical contributions can be seen as remedies for the gaps and limitations of linearization. His eponymous return method, for instance, avoids a potentially embarrassing failure: what if one could say nothing about the controllability of a nonlinear system near an equilibrium point when the linearized version is *not* controllable? Lie brackets offer an alternative tool in finite dimensions, but can fail for many important partial differential equations (PDEs).

Coron’s return method examines the nonlinear system when it is linearized around nontrivial trajectories that begin and end at the problematic equilibrium and for which the linearized system is controllable. One can retrieve local controllability of the nonlinear system via an inverse mapping theorem argument (see Figure 3). With such trajectories in hand, the tools of linear control suffice to show that systems like the baby carriage are indeed controllable.

**Figure 5.** On the Meuse River, gates move according to feedback laws constructed by Jean-Michel Coron and his colleagues. Photo credit: Jean-Michel Coron.

Three quick visuals, two of them quite unassuming, summarized Coron’s substantial achievements in understanding the control of PDEs: an animation showing rapid attenuation of one-dimensional shallow water waves in a pool by controlled motion of hydraulic gates at the two ends of the pool (see Figure 4); a photo of gates on the Meuse River moving according to feedback laws constructed by Coron and his colleagues (see Figure 5); and a cross-sectional view of the control devices themselves — an adjustable sluice gate and its partner, an adjustable spillway (see Figure 6).

These examples help describe Coron’s success in controlling the Saint-Venant shallow water equations. He modestly drew the audience’s attention to Adhémar Jean Claude Barré de Saint-Venant’s derivation of those PDEs at the age of 74, rather than to his own subsequent accomplishments with them. And Coron barely mentioned his other successes with such fundamentals of fluid mechanics as the Euler equations and the Navier-Stokes equations.

**Figure 6.** A sluice gate and spillway, control devices that Coron and his colleagues use to rapidly stabilize the water level on the Sambre and Meuse rivers. Figure credit: Georges Bastin.

Mirroring the breadth of his analytic contributions, the practical reach of Coron’s research goes far beyond movie clips and animations of suppressed waves. He and his collaborators have implemented a stabilizing control strategy for the Sambre and Meuse rivers in Belgium. Coron began working on this subject in 2003, and his feedback laws were implemented a few years later on the Sambre and only recently on the Meuse.

*Coron’s Reid Prize presentation is available from SIAM as slides with synchronized audio or as a PDF of slides only.*

*The Reid Prize began with awards by SIAM in 1994, 1996, and 1998, all funded by John Narcisco, nephew of Idalia Reid. Since 2000, SIAM has awarded the prize annually with support from a bequest from Idalia Reid in memory of her late husband, William T. Reid. W.T. Reid worked in differential equations, the calculus of variations, and optimal control, sharing naming rights for the workhorse Gronwall-Reid-Bellman inequality. He held faculty appointments at the University of Chicago, Northwestern University, the University of Iowa, the University of Oklahoma, and the University of Texas. He was an important figure in the optimal control community and a beloved mentor to his students.*^{1}

^{1} John Burns, a student of Reid’s and the 2010 recipient of the Reid Prize, provided a personal account entitled “William T. and Idalia Reid: His Mathematics and Her Mathematical Family” as his Reid Lecture. A

PDF of his lecture and slides with synchronized audio are available from SIAM.

Paul Davis is professor emeritus of mathematical sciences at Worcester Polytechnic Institute.