Sparsity and <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" display="inline" id="d1e715" altimg="si6.svg"> <mml:msup> <mml:mrow> <mml:mi>L</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>1</mml:mn> </mml:mrow> </mml:msup> </mml:math> -optimal contr

For linear time-invariant systems, we take an interest in sparse controls, i.e., those whose support (or L 0 -norm) is of minimal Lebesgue measure. We propose to extend this research to Radon measures and, in particular, to controls presenting Dirac impulses. Then, we can find a minimizer which is purely impulsive (and so with a null support) and prove that there is no gap when we extend the problem from L 1 -controls to Radon measures. To refine our research, this reasoning leads us to study the minimization problems for L 1 and measure norms. We prove that impulsive controls are still pertinent for these problems: once again, we can find a minimizer and prove that there is no gap. In addition to the existence of a solution, we show that, by extending the research to Radon measures, we can find a solution of L 0 and L 1 -minimization problems in the form of a finite linear combination of Dirac impulses, and we give an expression of the maximal number of these impulses. We also propose an algorithm in order to compute an impulsive solution of the measure norm minimization problem.