We study a variant of the reachability problem with constraints of asymptotic character on the choice of controls. More exactly, we consider a control problem in the class of impulses of given intensity and vanishingly small length. The situation is complicated by the presence of discontinuous dependences, which produce effects of the type of multiplying a discontinuous function by a generalized function. The constructed extensions in the special class of finitely additive measures make it possible to present the required solution, defined as an asymptotic analog of a reachable set, in terms of a continuous image of a compact, which is described with the use of the Stone space corresponding to the natural algebra of sets of the control interval.
One of the authors had the honor of communicating with Nikolai Nikolaevich Krasovskii for many years and discussed with him problems that led to the statement considered in the paper. Krasovskii’s support of this research direction provided possibilities for its fruitful development. His disciples and colleagues will always cherish the memory of Nikolai Nikolaevich in their hearts.
A new strongly time-consistent (dynamically stable) optimality principle is proposed in a cooperative differential game. This is done by constructing a special subset of the core of the game. It is proposed to consider this subset as a new optimality principle. The construction is based on the introduction of a function V ^ V^ that dominates the values of the classical characteristic function in coalitions. Suppose that V(S,x ¯ (τ),T−τ) V(S,x¯(τ),T−τ) is the value of the classical characteristic function computed in the subgame with initial conditions x ¯ (τ) x¯(τ) , T−τ T−τ on the cooperative trajectory. Define
V ^ (S;x 0 ,T−t 0 )=max t 0 ≤τ≤T V(S;x ∗ (τ),T−τ)V(N;x ∗ (τ),T−τ) V(N;x 0 ,T−t 0 ). V^(S;x0,T−t0)=maxt0≤τ≤TV(S;x∗(τ),T−τ)V(N;x∗(τ),T−τ)V(N;x0,T−t0).
Using this function, we construct an analog of the classical core. It is proved that the constructed core is a subset of the classical core; thus, we can consider it as a new optimality principle. It is proved also that the newly constructed optimality principle is strongly time-consistent.