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Section: Scientific Foundations

Optimal control of partial differential equations

The field has been strongly influenced by the work of J.L. Lions, who started its systematic study of optimal control problems for PDEs in [139] , in relation with singular perturbation problems [140] , and ill-posed problems [141] . A possible direction of research in this field consists in extending results from the finite-dimensional case such as Pontryagin's principle, second-order conditions, structure of bang-bang controls, singular arcs and so on. On the other hand PDEs have specific features such as finiteness of propagation for hyperbolic systems, or the smoothing effect of parabolic sytems, so that they may present qualitative properties that are deeply different from the ones in the finite-dimensional case.

Optimal control of variational inequalities

Unilateral systems in mechanics, plasticity theory, multiphases heat (Stefan) equations, etc. are described by inequalities; see Duvaut and Lions [108] , Kinderlehrer and Stampacchia [130] . For an overview in a finite dimensional setting, see Harker and Pang [119] . Optimizing such sytems often needs dedicated schemes with specific regularization tools, see Barbu [44] , Bermúdez and Saguez [54] . Nonconvex variational inequalities can be dealt as well in Controllability of such systems is discussed in Brogliato et al. [90] .

Sensitivity and second-order analysis

As for finite-dimensional problems, but with additional difficulties, there is a need for a better understanding of stability and sensitivity issues, in relation with the convergence of numerical algorithms. The second-order analysis for optimal control problems of PDE's in dealt with in e.g. [68] , [160] . No much is known in the case of state constraints. At the same time the convergence of numerical algorithms is strongly related to this second-order analysis.

Coupling with finite-dimensional models

Many models in control problems couple standard finite dimensional control dynamics with partial differential equations (PDE's). For instance, a well known but difficult problem is to optimize trajectories for planes land-off, so as to minimize, among others, noise pollution. Noise propagation is modeled using wave like equations, i.e., hyperbolic equations in which the signal propagates at a finite speed. By contrast when controlling furnaces one has to deal with the heat equation, of parabolic type, which has a smooting effect. Optimal control laws have to reflect such strong differences in the model.


Let us mention some applications where optimal control of PDEs occurs. One can study the atmospheric reentry problem with a model for heat diffusion in the vehicle. Another problem is the one of traffic flow, modeled by hyperbolic equations, with control on e.g. speed limitations. Of course control of beams, thin structures, furnaces, are important.

Our past contributions in control and optimal control of PDEs

An overview of sensitivity analysis of optimization problems in a Banach space setting, with some applications to the control of PDEs of elliptic type, is given in the book [83] . See also [68] .

We studied various regularization schemes for solving optimal control problems of variational inequalities: see Bonnans and D. Tiba [84] , Bonnans and E. Casas [69] , Bergounioux and Zidani [53] . The well-posed of a “nonconvex” variational inequality modelling some mechanical equilibrium is considered in Bonnans, Bessi and Smaoui [71] .

In Coron and Trélat [93] , [94] , we prove that it is possible, for both heat like and wave like equations, to move from any steady-state to any other by means of a boundary control, provided that they are in the same connected component of the set of steady-states. Our method is based on an effective feedback procedure which is easily and efficiently implementable. The first work was awarded SIAM Outstanding Paper Prize (2006).


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