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On measure differential inclusions in optimal control theory. (English) Zbl 1028.49002

In this paper the authors consider the following measure differential inclusion problem \[ \frac{d\mu _a}{d\lambda}(t)\in Q(t),\quad\lambda\text{-a.e. }t\in I, \]
\[ \frac{d\mu _s}{d|\mu _s|}(t)\in [Q(t)]_\infty,\quad \mu_s\text{a.e. }t\in I, \] where \(\mu=\mu_a+\mu_s\) is a Borel measure on \(I\subset \mathbb{R}\), \(\mu_a\) is the absolutely continuous part of \(\mu\) with respect to the Lebesgue measure \(\lambda\), \(\mu_s\) the singular part, \(t\to Q(t)\) a multifunction with \(Q(t)\) a non-empty closed and convex subset of \(\mathbb{R}^n\) and \([Q(t)]_\infty\) the asymptotic cone of \(Q(t)\). The main result of this paper is a closure result under weak convergence of measure, and it is based on the proof of the equivalence of the strong and weak formulation of the problem. In addition an application of optimal control problems in the BV setting is given.

MSC:

49J24 Optimal control problems with differential inclusions (existence) (MSC2000)
49J45 Methods involving semicontinuity and convergence; relaxation
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