Stationarity conditions in optimal control problems class related to dynamical objects group control

Abstract

We consider two classes of optimal control problems related dynamical oobjects group control. The first one is target orbiting of autonomous objects group with carrier object use (satellites, UAVs etc.). To describe the corrsponding situation, we use system of ODEs that consists of two subsystems: the "carrier" subsystem and a number of "payload" ones. They are determined on the "main" and "nested" time segments ordered w.r.t. start times. Start time of each "nested" segment is in fact time of the corresponding payload "decoupling". Our aim is to formulate first-order necessary conditions of extended weak minimum ("extended" means in fact that we consider variations of decoupling times also). To do this, we perfom the following scheme: first, introduce normalized time and replicate all constraints to reduce our OCP to classical form, then write necessary conditions, and finally, rewrite these conditions in original time. We also provide simple kinematic example of our conditions application (carrier moves along helix between two "orbits", payloads moves in passive way after decoupling and, in addition, provide problem statement related to group cooperation evasion). The second class is cooperative evasion of objects in gravity field. A number of controled "satellites" have to avoid collisions with both of uncontroled "space debri" elements and other satellites. We work with a number of "local" costs of maximin type, transform each of them to standard terminal cost by introducing undefined scalar variable and linear state constraint and then use the convolution with weights to work with "general" cost.