Optimal Control of Non-convex Measure-driven Differential Inclusions

Abstract

Necessary conditions for optimality in control problems with differentialinclusion dynamics have recently been developed in the non-convex case by Clarke, Vinter, and others. Using appropriate reparametrizations of the time variable, we extend these results to systems whose dynamics involve a differential inclusion where a vector-valued measure appears. An auxiliary result central to our proof is an extension of existing free end-time necessary conditions to Clarke's stratified framework.

On some extension of optimal control theory

“…The pair ðx; cÞ is called control process if (8) holds true. A control process is said to be admissible if the endpoint constraints (9) and state constraints (10) are satisfied. Let us denote the set of all admissible processes by P. An admissible process ðx n ; c n Þ is said to be optimal or a solution to (P) if the least possible finite value of the integral in (7) over all elements from P is reached by ðx n ; c n Þ.…”

“…Overall, the line of investigation undertaken here is, of course, a road well traveled, and based on the so-called "graph completion" or "discontinuous time variable change" technique combined with the Gamkrelidze compactification, or convexification, technique. Besides the above-mentioned sources, our main line of research also goes along the works in [9,10,12,[15][16][17]24,25,29,32]. This list, though, is far from being complete.…”

Measure Differential Inclusions: Existence Results and Minimum Problems

“…As an application of the compactness result, we deduce (Theorem 6) a condition for minimizing a functional This is an extension of a classical optimal control problem (which cannot be solved in the class of absolutely continuous functions for solutions, respectively of L 1 controls) to the space of functions of bounded variation for solutions, respectively of measures for controls (see [29,31,32] treating the single-valued case or [23,28]). For optimality conditions in the set-valued setting, but concerning different optimal control problems, we can refer to [39] (with convex velocity sets) or [14].…”

On the properness of an impulsive control extension of dynamic optimization problems

“…Impulsive control systems, in its various guises including existence of solution, well-posedness of solution, sensitivity analysis, etc., have been investigated by various researchers. It is relevant to mention the works by Aronna and Rampazzo [1], Arutyunov, Dykhta, Jacimovic and Pereira [4,5], Bressan and Rampazzo [11,12], Code and Loewen [13], Code and Silva [14], Dykhta and Samsonyuk [15], Forcadel, Rao and Zidani [16], Kurzhanskii and Daryin [17], Miller and Rubinovich [19], Pereira and Silva [23], Pereira, Silva and de Oliveira [24], Rishel [30], Silva and Vinter [26,27], Vinter and Pereira [29], Warga [31], Wolenski and Zabic [32,33], among others. The concept of impulsive control considered in this work is being developed along the lines of [7][8][9].…”