The problem of conflicting control with mixed constraints

The control problem for a linear dynamical system is considered at a minimax of the terminal quality index. Feasible controls are simultaneously restricted by geometrical constraints and by integrated momentum constraints, the latter being thought of as a store of control resources. The problem is formalized as a differential game 1–4] using concepts 5–8] developed at Ekaterinburg. Here, because of the geometrical constraints, the momentum formulation and its associated difficulties 2–4] do not appear. On the other hand the presence of the integral restrictions leads to the appearance of additional variables whose evolution describes the dynamics of the expenditure of the control resources. These variables are subject to phase restrictions, which is a peculiarity of the problem. A reasonably informative picture and a class of strategies for which the given game has a value and a saddle point are given. A constructive method for computing the value function of the game and constructing optimal strategies is presented. This method is conceptually related to the construction of a stochastic programming synthesis 5] and is based on the recursive construction of upper-convex envelopes for certain auxiliary functions. The possibility of exchanging the minimum and maximum operations over the resource parameters when calculating the value of the game using these procedure is established.