https://hal.inria.fr/hal-01068306Aronna, Maria SoledadMaria SoledadAronnaRampazzo, FrancoFrancoRampazzoDipartimento di Matematica Pura e Applicata [Padova] - Universita degli Studi di Padova$\mathcal{L}^1$ limit solutions for control systemsHAL CCSD2015[MATH.MATH-CA] Mathematics [math]/Classical Analysis and ODEs [math.CA][MATH.MATH-OC] Mathematics [math]/Optimization and Control [math.OC]Bouzat, EstelleSensitivity Analysis for Deterministic Controller Design - SADCO - - EC:FP7:PEOPLE2011-01-01 - 2014-12-31 - 264735 - VALID - 2014-09-25 13:54:202022-08-19 09:18:402014-09-25 13:54:20enJournal articles10.1016/j.jde.2014.10.0131For a control Cauchy problem $$ \dot x= {f}(t,x,u,v) +\sum_{\alpha=1}^m g_\alpha(x) \dot u_\alpha\quad x(a)=\bar x, $$ on an interval $[a,b]$, we propose the notion of {\em limit solution} $x$ that verifies the following properties: i) $x$ is defined for $\mathcal{L}^1$ (impulsive) inputs $u$ and for standard, bounded measurable, controls $v$; ii) in the commutative case (i.e. when $[g_{\alpha},g_{\beta}]\equiv 0,$ for all $\alpha,\beta=1,\dots,m$), $x$ coincides with the solution constructed via multiple fields' rectification; iii) $x$ subsumes former concepts of solution valid for the generic, noncommutative case. In particular, when $u$ has bounded variation, we investigate the relation between limit solutions and (single-valued) graph completion solutions. Furthermore, we prove consistency with the classical Carathéodory solution when $u$ and $x$ are absolutely continuous. Even though some specific problems are better addressed by means of special representations of the solutions, we believe that various theoretical and practical issues call for a unified notion of trajectory. For instance, this is the case of optimal control problems, possibly with state and end-point constraints, for which no extra assumptions (like e.g. coercivity, boundedness, commutativity) are made in advance.