More precisely, given a solving the convexified p -th order differential inclusion
a.e., consider the general problem consisting in finding bang-bang solutions (i.e.
a.e.) under the same boundary-data,
&
(k=0,1,…,p−1); but restricted, moreover, by a pointwise state constraint of the type
(e.g. ω=(1,0,…,0) yielding
).
Previous results in the scalar d=1 case were the pioneering Amar & Cellina paper (dealing with a056ea60420">), followed by Cerf & Mariconda results, who solved the general case of linear differential operators Lp of order p≥2 with
-coefficients.
This paper is dedicated to: focus on the missing case p=1, i.e. using ; generalize the dimension of x(⋅), from the scalar case d=1 to the vectorial d∈N case; weaken the coefficients, from continuous to integrable, so that A(⋅) now becomes a d×d-integrable matrix; and allow the directional vector ω to become a moving AC function ω(⋅).
Previous vectorial results had constant ω , no matrix (i.e. A(⋅)≡0) and considered: constant control-vertices (Amar & Mariconda) and, more recently, integrable control-vertices (ourselves).