Konstantin V. Siemenikhin

Brief paper: Towards the optimal control of Markov chains with constraints

An optimal control problem with constraints is considered on a finite interval for a non-stationary Markov chain with a finite state space. The constraints are given as a set of inequalities. The optimal solution existence is proved under a natural assumption that the set of admissible controls is non-empty. The stochastic control problem is reduced to a deterministic one and it is shown that the optimal solution satisfies the maximum principle, moreover it can be chosen within a class of Markov controls. On the basis of this result an approach to the numerical solution is proposed and its implementation is illustrated by examples.

Optimal control of Markov chains with constraints

A problem of optimal control of Markov chain with finite state space is considered. We consider a non-stationary finite horizon problem with constraints, given as a set of inequalities. Basing on recent results on existence of optimal solution we suggest to use the dual approach to optimization and thereby an approach to effective numerical algorithms. The approach is illustrated by numerical examples.

On minimax identification: method of dual optimization

The problem of minimax affine identification of a linear uncertain stochastic multivariate model is considered. The minimax optimization problem together with the corresponding dual one are stated and examined. The necessary and sufficient conditions for the minimax affine estimate to exist and to be determined analytically via the dual problem solution are given. The algorithm of minimax stochastic estimation for the infinite-dimensional model given a finite number of observations is also considered. The numerical method for minimax estimation is described, and the results of computer modeling are presented.

Kalman Filtering by Minimax Criterion with Uncertain Noise Intensity Functions

The problem of minimax filtering is examined for linear continuous-time observation models with uncertain intensities of non-stationary white noises. For designing algorithms of minimax filtering, the method of dual optimization is used together with the techniques of the maximum principle. It is shown that the Kalman filter is a minimax one if its coefficients are defined by the least favorable noise intensity. The explicit form of the minimax filter is derived in the case of scalar state and observation processes with arbitrarily correlated disturbances. The results of numerical modeling are also presented.

Minimax estimation of random elements: theory and applications

The problem of minimax estimation for the infinite-dimensional stochastic model is considered. The prior information about the random elements involved is formulated in term of second-order moment characteristics. The minimax estimation procedure is described and the corresponding numerical algorithm is presented. It is proved that the least favorable distribution of the model random elements is Gaussian. The efficiency of the proposed estimation algorithms is illustrated by means of the examples related to the signal and field robust processing.

Linear stochastic programming with minimax quantile and probability criterions

The problems of linear stochastic model optimization are considered using quantile and probability criterions. The a priori information about the distribution law of the model random coefficients is defined by certain constraints on the first- and second-order moments. The concept of the minimax strategy is formulated and the last one is constructed using the convex programming duality theory. The analytic dependence of the minimax strategy on the solution of the dual problem is derived. A computational procedure for solving the dual problem is examined. The results of computer modeling are presented.

MINIMAX PARAMETER ESTIMATION FOR SINGULAR LINEAR MULTIVARIATE MODELS WITH MIXED UNCERTAINTY

Abstract The problem of minimax estimation is considered for the linear multivariate statistically indeterminate observation model with mixed uncertainty. It is shown that in the regular case the minimax estimate is defined explicitly via the solution of the dual optimization problem. For the singular models, the method of dual optimization is developed by means of using the technique of Tikhonov regularization. Several particular cases which are widely used in practice are also examined.