Konstantin V. Semenikhin

Methods to design optimal control of Markov process with finite state set in the presence of constraints

The problem of optimal control of a nonuniform Markov process with a finite state set over a fixed interval in the presence of inequality-like constraints was considered. The design of control relies on the principle of dynamic programming in combination with the methods of convex programming and the duality theory. Two types of conditions under which it is possible to select a Markov optimal control were proposed.

Filtering of the Markov jump process given the observations of multivariate point process

The problem of optimal filtering of the Markov process with finite number of states through the discrete observations arriving at random time instants was formulated and resolved. It was established that the desired estimate obeys a finite-dimensional differential-difference system which admits an explicit solution. The theoretical results obtained are applicable to the problem of monitoring the telecommunication link.

The Optimality of Linear Estimation Algorithms in Minimax Identification

The optimality of linear estimates in minimax estimation of a stochastically uncertain vector in a linear observation model by mean-square criterion is studied. In the Gaussian case, a uniformly optimal linear estimate is shown to exist in the class of all unbiased estimates. Moreover, it is minimax in the class of all nonlinear estimates if the nonrandom parameters of the observation model are unbounded. If the a priori information on random parameters are given as constraints on the covariance matrix, linear estimates are shown to be minimax.

Optimal Channel Choice for Lossy Data Flow Transmission

We consider the optimal control problem for the load of several communication channels defined by independent Markov jump processes. Implicit information on the state of a channel is available in the form of a flow of losses whose intensity is proportional to the controllable load of this channel. The optimized functionals take into account the total throughput of channels and energy costs for data transmission over a fixed interval of time. We obtain optimal filtering equations for joint estimation of channel states. We construct a locally optimal strategy that explicitly depends on the set of state estimates.

Methods for minimax estimation under elementwise covariance uncertainty

We consider the minimax estimation problem in the linear regression model under elementwise constraints imposed on the covariance matrix of the random parameters vector. Minimax estimates are designed using several approaches to the numerical solution of the dual problem, namely, the semidefinite programming method, the conditional gradient method and its modification with the Lagrange multipliers and regularization. The efficiency of the suggested methods is illustrated by the example of path restoration for a maneuvering target with a statistically uncertain acceleration.

Optimal control problem regularization for the Markov process with finite number of states and constraints

The optimal control problem is considered for a system given by the Markov chain with integral constraints. It is shown that the solution to the optimal control problem on the set of all predictable controls satisfies Markov property. This optimal Markov control can be obtained as a solution of the corresponding dual problem (in case if the regularity condition holds) or (in other case) by means of proposed regularization method. The problems arising due to the system nonregularity along with the way to cope with those problems are illustrated by an example of optimal control problem for a single channel queueing system.

Minimax linear filtering of random sequences with uncertain covariance function

Consideration was given to the development of a numerical method for determination of the minimax filter in the linear stochastic difference system studied over a finite horizon in the presence of an uncertain covariance function in the model of useful signal. Selection of the considered uncertainty sets relied on the form of the corresponding confidence regions. The developed iterative procedure was applied to filtering of the position of a maneuvering target with inexactly given acceleration covariance function.

Minimax estimation methods under ellipsoidal constraints

We consider the minimax estimation problem in a linear observation model under ellipsoidal constraints on the vector of unknown parameters. To solve the problem, we use dual optimization and semidefinite programming methods. The developed algorithms are applied to constructing motion parameter estimates for a maneuvering flying vehicle under constraints on the acceleration vector.

The Conditionally Minimax Nonlinear Filtering Method and Modern Approaches to State Estimation in Nonlinear Stochastic Systems

We consider, in chronological order, the main results that have defined the concept of conditionally minimax nonlinear filtering. This would let us to follow all the evolution stages of this universal method, from a particular application, through basic mathematical concepts, to an advanced theory able to solve a wide class of robust estimation problems in linear and nonlinear stochastic systems.