Control of Partial Differential Equations

The purpose of these Notes is to present some recent advances on stabilization for wave-like equations together with some well-known methods of stabilization. This course will give several references on the subject but do not pretend to exhaustivity. The spirit of these Notes is more that of a research monograph. We aim to give a simplified overview of some aspects of stabilization, on the point of view of energy methods, and insist on some of the methodological approaches developed recently. We will focus on nonlinear stabilization, memory-damping and indirect stabilization of coupled PDE’s and present recent methods and results. Energy methods have the advantage to handle and deal with physical quantities and properties of the models under consideration. For nonlinear stabilization, our purpose is to present the optimal-weight convexity method introduced in (Alabau-Boussouira, Appl. Math. Optim. 51(1):61–105, 2005; Alabau-Boussouira, J. Differ. Equat. 248:1473–1517, 2010) which provides a whole methodology to establish easy computable energy decay rates which are optimal or quasi-optimal, and works for finite as well as infinite dimensions and allow to treat, in a unified way different PDE’s, as well as different types of dampings: localized, boundary. Another important feature is that the upper estimates can be completed by lower energy estimates for several examples, and these lower estimates can be compared to the upper ones. Optimality is proved in finite dimensions and in particular for one-dimensional semi-discretized wave-like PDE’s. These results are obtained through energy comparison principles (Alabau-Boussouira, J. Differ. Equat. 248: 1473–1517, 2010), which are, to our knowledge, new. This methodology can be extended to the infinite dimensional setting thanks to still energy comparison principles supplemented by interpolation techniques. The optimal-weight convexity method is presented with two approaches: a direct and an indirect one. The first approach is based on the multiplier method and requires the assumptions of the multiplier method on the zone of localization of the feedback. The second one is based on an indirect argument, namely that the solutions of the corresponding undamped systems satisfy an observability inequality, the observation zone corresponding to the damped zone for the damped system. The advantage is that, this observability inequality holds under the sharper optimal Geometric Control Condition of Bardos et al. (SIAM J. Contr. Optim. 30:1024–1065, 1992). The optimal-weight convexity method also extends to the case memory-damping, for which the damping effects are nonlocal, and leads to nonautonomous evolution equations. We will only state the results in this latter case. Indirect stabilization of coupled systems have received a lot of attention recently. This subject concerns stabilization questions for coupled PDE’s with a reduced number of feedbacks. In practice, it is often not possible to control all the components of the vector state, either because of technological limitations or cost reasons. From the mathematical point of view, this means that some equations of the coupled system are not directly stabilized. This generates mathematical difficulties, which requires to introduce new tools to study such questions. In particular, it is important to understand how stabilization may be transferred from the damped equations to the undamped ones. We present several recent results of polynomial decay for smooth initial data. These results are based on energy methods, a nondifferential integral inequality introduced in (Alabau, Compt. Rendus Acad. Sci. Paris I 328:1015–1020, 1999) [see also (Alabau-Boussouira, SIAM J. Contr. Optim. 41(2):511–541, 2002; Alabau et al. J. Evol. Equat. 2:127–150, 2002)] and coercivity properties due to the coupling operators.

In these notes we motive the study of Liouville equations having control terms using examples from problem areas as diverse as atomic physics (NMR), biological motion control and minimum attention control. On one hand, the Liouville model is interpreted as applying to multiple trials involving a single system and on the other, as applying to the control of many identical copies of a single system; e.g., control of a flock. We illustrate the important role the Liouville formulation has in distinguishing between open loop and feedback control. Mathematical results involving controllability and optimization are discussed along with a theorem establishing the controllability of multiple moments associated with linear models. The methods used succeed by relating the behavior of the solutions of the Liouville equation to the behavior of the underlying ordinary differential equation, the related stochastic differential equation, and the consideration of the related moment equations.

We prove Carleman estimates for elliptic and parabolic operators, using several methods: a microlocal approach where the main tool is the Gårding inequality and a more computational direct approach. Carleman estimates are proven locally and we describe how they can be patched together to form a global estimate. We expose how they can be used to provide unique continuation properties, as well as approximate and null controllability results.

In these Notes we make a self-contained presentation of the theory that has been developed recently for the numerical analysis of the controllability properties of wave propagation phenomena and, in particular, for the constant coefficient wave equation. We develop the so-called discrete approach. In other words, we analyze to which extent the semidiscrete or fully discrete dynamics arising when discretizing the wave equation by means of the most classical scheme of numerical analysis, shear the property of being controllable, uniformly with respect to the mesh-size parameters and if the corresponding controls converge to the continuous ones as the mesh-size tends to zero. We focus mainly on finite-difference approximation schemes for the one-dimensional constant coefficient wave equation. Using the well known equivalence of the control problem with the observation one, we analyze carefully the second one, which consists in determining the total energy of solutions out of partial measurements. We show how spectral analysis and the theory of non-harmonic Fourier series allows, first, to show that high frequency wave packets may behave in a pathological manner and, second, to design efficient filtering mechanisms. We also develop the multiplier approach that allows to provide energy identities relating the total energy of solutions and the energy concentrated on the boundary. These observability properties obtained after filtering, by duality, allow to build controls that, normally, do not control the full dynamics of the system but rather guarantee a relaxed controllability property. Despite of this they converge to the continuous ones. We also present a minor variant of the classical Hilbert Uniqueness Method allowing to build smooth controls for smooth data. This result plays a key role in the proof of the convergence rates of the discrete controls towards the continuous ones. These results are illustrated by means of several numerical experiments.