The Reduction Method for Optimal Impulsive Control Problems with Nonstandard State Constraints

Elena Goncharova and Maxim Staritsyn*

We consider an optimal impulsive control problem with trajectories of bounded variation and vector-valued measures as controls under state and mixed constraints. The main feature of the problem consists in the presence of nonstandard state constraints imposed on a state trajectory at the atoms of the corresponding control measure and over the intervals, on which its continuous component is supported. This kind of state constraints has been first proposed in our last works. Note that such constraints seem very useful in modeling of real processes, where instant control impacts of high intensity imply jumps of a state trajectory. Furthermore, the states before, after and during a jump are subject to certain conditions. This is typical for many hybrid models arising in medical applications, robotics etc. In this paper, we propose an adequate modification of the discontinuous time reparameterization technique, which allows us to incorporate the nonstandard state constraints to an auxiliary reduced optimal control problem with measurable, essentially bounded controls. The latter problem is equivalent to the original one in the sense that the existence of a solution in a one of the problems guarantees the existence of a solution in another one with the same minimal cost. We discuss some mechanical applications of the obtained results. The work is supported by RFBR, grants no. 12-01-31252, 13-08-00441.

Mathematics Subject Classification: 49N25 49K99

Keywords: optimal impulsive control; discontinuous time reparameterization

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