A pointwise constrained version of the Liapunov convexity theorem for vectorial linear first-order control systems

We generalize the Liapunov convexity theorem's version for vectorial control systems driven by linear ODEs of first-order p=1p=1, in any dimension d∈Nd∈N, by including a pointwise state-constraint.More precisely, given a x‾(⋅)∈Wp,1([a,b],Rd) solving the convexified p  -th order differential inclusion Lpx‾(t)∈co{u0(t),u1(t),…,um(t)} a.e., consider the general problem consisting in finding bang-bang solutions (i.e. Lpxˆ(t)∈{u0(t),u1(t),…,um(t)} a.e.) under the same boundary-data, xˆ(k)(a)=x‾(k)(a) & xˆ(k)(b)=x‾(k)(b) (k=0,1,…,p−1k=0,1,…,p−1); but restricted, moreover, by a pointwise state constraint of the type 〈xˆ(t),ω〉≤〈x‾(t),ω〉∀t∈[a,b] (e.g. ω=(1,0,…,0)ω=(1,0,…,0) yielding xˆ1(t)≤x‾1(t)).Previous results in the scalar d=1d=1 case were the pioneering Amar & Cellina paper (dealing with Lpx(⋅)=x′(⋅)), followed by Cerf & Mariconda results, who solved the general case of linear differential operators LpLp of order p≥2p≥2 with C0([a,b])-coefficients.This paper is dedicated to: focus on the missing case p=1p=1, i.e. using Lpx(⋅)=x′(⋅)+A(⋅)x(⋅); generalize the dimension of x(⋅)x(⋅), from the scalar case d=1d=1 to the vectorial d∈Nd∈N case; weaken the coefficients, from continuous to integrable, so that A(⋅)A(⋅) now becomes a d×dd×d-integrable matrix; and allow the directional vector ω   to become a moving AC function ω(⋅)ω(⋅).Previous vectorial results had constant ω  , no matrix (i.e. A(⋅)≡0A(⋅)≡0) and considered: constant control-vertices (Amar & Mariconda) and, more recently, integrable control-vertices (ourselves).