Trainee Project
Sparse control
2023/03/15 - 2023/09/14
** General problem, Context

The search for sparse solutions has experienced a boom in the last decades, especially for image processing, cf. for example [12]. More recently, sparse approaches have been proposed for the controllability of multi-agent systems, cf. for example [1,2,13]. Let us also note that, in the presence of a unilateral constraint on the control, a minimal time control is naturally sparse, cf. for example [8]. More generally, one can refer to the article [11].

Consider the finite dimensional linear system,
(*) x'=Ax+Bu,
where u is the control (or input) and x is the state of the system. We assume that the system (*) is controllable, i.e. that for any time T>0, any initial state and any final state, there exists a control u such that the solution of (*) is equal to the final state at time T. Such a control can for instance be obtained by minimizing its norm, typically the L² norm. Such an approach leads to the HUM method (Hilbert Uniqueness Method), cf. [3].
On the other hand, the support of the obtained control is generically the whole interval [0,T].

The objective of a sparse control is to minimize the action time of the control.
Formally, instead of minimizing the L²-norm of the control, we will minimize its L⁰-norm, that is, the Lebesgue measure of its support.
A result from [6], ensures that the controllability of a linear system can be done using a finite number of Dirac masses. Thus, this minimization problem does not admit a minimizer in L², but can admit one in the measure space.

Another concern in this minimization problem is that the functional, the L⁰ norm, is not convex. An approximate convex problem, which in some cases is equivalent to the original problem, cf. [4,7,11], is to replace the pseudo-norm L⁰ by the norm L¹. As before, this minimization problem does not in general admit a minimizer in L¹. On the other hand, it admits a minimizer in the Radon measurement set.

** Objectives of the training course

The objective of this thesis is to better understand the sparse structure of such controls, to extend the results to the nonlinear setting, and to propose efficient numerical methods leading to sparse controls. It will therefore be appropriate to focus on:

-- the extension to the nonlinear setting of the results succinctly mentioned in the previous paragraph.
For this purpose, a typical model will be the Heisenberg system. In the nonlinear framework, it is also necessary to understand the impact of an impulsive control. For this we refer to the work done in [10];

-- the existence of a Lavrentiev jump.
More precisely, is the infimum for L¹ controls the infimum for measure controls? For this, one can be inspired by [9];

-- the location of the control medium.
More precisely, can the obtained control be expressed as an event triggered control?

-- the construction of numerical methods leading to sparse controls.
For this, one can be inspired by algorithms based on the Bregman distance, cf. [7,12], or on the reformulations proposed in [5].

This list of problems is of course not exhaustive.
The student may for example also be interested in the stabilization problem with sparse controls.

** Références
[1] M. Caponigro, M. Fornasier, B. Piccoli, and E. Trélat. Sparse stabilization and optimal control of the Cucker-Smale model. Math. Control Relat. Fields, 3(4) :447466, 2013.
[2] M. Caponigro, M. Fornasier, B. Piccoli, and E. Trélat. Sparse stabilization and control of alignment models. Math. Models Methods Appl. Sci., 25(3) :521564, 2015.
[3] J.-M. Coron. Control and nonlinearity., volume 136 of Math. Surv. Monogr. Providence, RI : American Mathematical Society (AMS), 2007.
[4] T. Ikeda and K. Kashima. On sparse optimal control for general linear systems. IEEE Transactions on Automatic Control, 64(5) :20772083, 2019.
[5] C. Kanzow, A. Schwarz, and F. Weiÿ. The sparse(st) optimization problem : Reformulations, optimality, stationarity, and numerical results. Working paper or preprint, 2022.
[6] E. B. Lee and L. Markus. Foundations of optimal control theory. The SIAM Series in Applied Mathematics. New York-London-Sydney : John Wiley and Sons, Inc. xii, 576 p. (1967)., 1967.
[7] Y. Li and S. Osher. Coordinate descent optimization for `1 minimization with application to compressed sensing ; a greedy algorithm. Inverse Probl. Imaging, 3(3) :487503, 2009.
[8] J. Lohéac, E. Trélat, and E. Zuazua. Nonnegative control of nite-dimensional linear systems. Ann. Inst. Henri Poincaré, Anal. Non Linéaire, 38(2) :301346, 2021.
[9] M. Motta, M. Palladino, and F. Rampazzo. Unbounded control, inmum gaps, and higher order normality. SIAM J. Control Optim., 60(3) :14361462, 2022.
[10] M. Motta and C. Sartori. On L1 limit solutions in impulsive control. Discrete Contin. Dyn. Syst., Ser. S, 11(6) :12011218, 2018.
[11] M. Nagahara. Sparse control for continuous-time systems. International Journal of Robust and Nonlinear Control, n/a(n/a).
[12] S. Osher, M. Burger, D. Goldfarb, J. Xu, and W. Yin. An iterative regularization method for total variation-based image restoration. Multiscale Model. Simul., 4(2) :460489, 2005.
[13] B. Piccoli, N. P. Duteil, and E. Trélat. Sparse control to prevent black swan clustering in collective dynamics. In 2018 Annual American Control Conference (ACC), pages 955960, 2018.

This internship subject is associated with a thesis subject of which a detailed description is available at :
Optimal control, impulsive control, sparse control
Control Identification Diagnosis