L1L1 limit solutions for control systems

For a control Cauchy problemx˙=f(t,x,u,v)+∑α=1mgα(x)u˙α,x(a)=x¯, on an interval [a,b][a,b], we propose a notion of limit solution x, verifying the following properties: i) x   is defined for L1L1 (impulsive) inputs u and for standard, bounded measurable, controls v  ; ii) in the commutative case (i.e. when [gα,gβ]≡0[gα,gβ]≡0, for all α,β=1,…,mα,β=1,…,m), x   coincides with the solution one can obtain via a change of coordinates that makes the gαgα simultaneously constant; 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 issues call for a unified notion of trajectory. For instance, this is the case of optimal control problems, possibly with state and endpoint constraints, for which no extra assumptions (like e.g. coercivity, bounded variation, commutativity) are made in advance.