L1limit solutions for control systems

Abstract

Abstract. For a control Cauchy probleṁgα(x)uα, x(a) =x, on an interval [a, b], we propose a notion of limit solution x, verifying the following properties: i) x is defined for L 1 (impulsive) inputs u and for standard, bounded measurable, controls v; ii) in the commutative case (i.e. when [gα, g β ] ≡ 0, for all α, β = 1, . . . , m), x coincides with the solution one can obtain via the change of coordinates that makes the gα 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.

A Higher-Order Maximum Principle for Impulsive Optimal Control Problems

“…Definition 2.2. We call a feasible strict-sense process (T ,ū,ā,x,v) a local strict-sense minimizer of (P) if there exists δ > 0 such that for every feasible strict-sense process (T, u, a, x, v) verifying d (T, x, v), (T ,x,v) < δ, where d is the distance defined in (5). If relation (7) is satisfied for all feasible strict-sense processes, we say that (T ,ū,ā,x,v) is a global strict-sense minimizer.…”

“…Let us call equivalent any two space-time processes (S,w 0 ,w,α,ỹ 0 ,ỹ,β), (S, w 0 , w, α, y 0 , y, β) as above. The following result is quite straightforward: 5 Since every L 1 -equivalence class contains Borel measurable representatives, we tacitly assume that all L 1 -maps we are considering are Borel measurable when necessary.…”

“…Let us point out that one can equivalently give a t-based description of this extension using bounded variation trajectories as in[5,31,7].…”

Normality and gap phenomena in optimal unbounded control

“…and (ȳ 0 ,ȳ,ν) = (ȳ 0 ,ȳ 1 ,ȳ 2 ,ȳ 3 ,ν) = (s, 1, 0, 0, 0)χ [0,1] + (1, 2 − s, 0, 0, s − 1)χ [1,2] ,…”

“…(s) = 0, dp 2 ds (s) = −p 2 (s)χ [0,1] (s) + p 3 (s)χ [1,2] (s) , a.e. s ∈ [0, 2], dp 3 ds (s) = 0 .…”