ADAPTIVE DYNAMIC EQUILIBRIUMS IN PROBLEMS of STOCHASTIC
CONTROL UNDER UNCERTAINTY

 

V.V. Baranov

                        The problem of stochastic control and decisions making under uncertainty concerning to state and transient function, definite on outcomes, and ambiguously of given function of utility  is considered. The base  apriory of data is determined by set of objects

            CBG = {X, a0 (X),Y, [YxÌY, xÎX], Z,[XzÌ X,zÎZ],G,qg(Z|Z´X´Y),wg(Z´X´Y), gÎG},

where X is set of states, inaccessible to supervision; a0(X) -the apriory distribution on X; Y - set of allowable controls; YxÌY - restriction on admission of controls in dependences on state xÎX; Z - set of random outcomes;  XzÌ X - restriction on admission of state as alternatives of their diagnostic in dependence on outcomes zÎZ;  qg (Z |Z´X´Y) - transient function, determining probability of transitions from Z´X´Y to Z, which, however, is not given; G - set of  hypotheseses about transient function; w g (Z´X´Y) - function of utility, presenting preferences on controls y ÎYx pursuant to condition:  y¢  y Û wg (z, x, y¢) > w g (z, x, y).

            The solution of problem is based on principle of decomposition: at uncertainty the problem is solved by redution to set interdependent of tasks, inducing by corresponding to nature of uncertainty, and in joint their solutions.

            In particularly, in conditions of base CBG the problem is reduced to three tasks: diagnostics of state, identification of transient function and selection of rule of control.

            The mutual dependence of such tasks induces the game content of problem with unopposite interests.

            The problem is considered in some assumptions and designs.

            From received here results the following final statements follow:

            1. Problem of stochastic control at uncertainty in conditions of  base CBG is solvable in principle and constructionally.

            2. Principle of decomposition and structural transformations determine the effective
method of formalization the problem, inducing the game statement of problem with unopposite  interests, solution of which is described by equilibrium strategy.

            3. In considered conditions the equilibrium strategy asymptotical stationary, adaptive and is computed consecutively on trajectory of outcomes. Adaptability provides the convergence of process to stationary equilibriums, achievable at known true transient function.

            The developed approach and methods permit to solution enough the wide class of pra-ctical problems of control in complex systems at various variants of uncertainty, in particular, task of directed development, control of socio - economic processes and  many other ones.