On a guaranteed guidance problem under incomplete information

We discuss the problem of guaranteed guidance of a linear control system by a fixed time under the assumption that the system is subject to an unknown disturbance. We consider the case when a part of state coordinates are measured and the set of unknown initial states is finite. We specify a solution algorithm based on the combination of the package approach, the theory of dynamic inversion, and the extremal shift method.     

Numerical encoding of sampled controls and an approximation metric criterion for the solvability of a guidance game problem

A game problem of guaranteed guidance in the class of positional strategies is considered for a conflict-controlled system with affine scalar controls in the equation of the system. Simplified sampled analogs of quasi-strategies, i.e., of nonanticipating program responses of the first player to the controls of the second player, are introduced. The nonanticipating property is characterized in metric terms with the use of numerical images (codes) of argument controls and response controls. A class of nonanticipating transformations is introduced that is approximately equivalent by the criterion of the solvability of the game problem to the class of positional strategies. The elements of this class as transformations of the numerical codes of controls are characterized by the 1-Lipschitzness property. A numerical algorithm for checking the solvability of the problem in this class is described. The complexity order of the algorithm is close to that of the approximation variant of the classical program construction.     

Method for solving a boundary problem for a nonlinear control system with incomplete information

An easy-to-implement numerical method is proposed for synthesizing a feedback control function that transfers a wide class of nonlinear stationary systems from an initial state to a given terminal state with allowance for measured data. A constructive criterion is obtained for choosing terminal states for which this transfer is guaranteed. The problem of an interorbital flight is considered and numerically simulated.     

A control problem under incomplete information for a linear stochastic differential equation

A problem of guaranteed closed-loop control under incomplete information is considered for a linear stochastic differential equation (SDE) from the viewpoint of the method of open-loop control packages worked out earlier for the guidance of a linear control system of ordinary differential equations (ODEs) to a convex target set. The problem consists in designing a deterministic open-loop control providing (irrespective of a realized initial state from a given finite set) prescribed properties of the solution (being a random process) at a terminal point in time. It is assumed that a linear signal on some number of realizations is observed. By the equations of the method of moments, the problem for the SDE is reduced to an equivalent problem for systems of ODEs describing the mathematical expectation and covariance matrix of the original process. Solvability conditions for the problems in question are written.     

An optimal stopping problem with two levels of incomplete information

The following variant of a classical selection problem is considered. Offers of random size are successively available to a decision-maker at discrete time instants, and he has to select one without having the possibility of recall. At any time he can accept or reject the current offer based on some initial information (just received) on its still unknown size, or he can demand additional information and take his decision thereafter. The costc ₀ > 0 for continuing to the next offer and the costd ₀ > 0 for obtaining the additional information are assumed to be constant. Under reasonable conditions on the underlying stochastic structure we find and characterize strategies maximizing the expected net reward.This research was supported in part by a grant from the MINERVA Foundation.     

Linear quadratic control problem with a terminal convex constraint for discrete-time distributed systems

The present work deals with the linear quadratic control problemfor a discrete distributed system with terminal convex constraint.Using techniques of perturbation by feedback, it is shown thatthe resolution of the considered problem is equivalent to thatof a controllability, one so-called Extended Exact Controllabilitywith time-varying operators. The Hilbert uniqueness method approachis then extended to this case to provide an explicit form forthe optimal control. In the same framework, the inequality constraintcase is examined for which a practical numerical resolutionis given. Finally, the results obtained are used to treat aminimum-time reachability problem.     

Linear optimal control problem for discrete 2-D systems with constraints

Optimal control problem for linear two-dimensional (2-D) discrete systems with mixed constraints is investigated. The problem under consideration is reduced to a linear programming problem in appropriate Hubert space. The main duality relations for this problem is derived such that the optimality conditions for the control problem are specified by using methods of the linear operator theory. Optimality conditions are expressed in terms of solutions for conjugate system.     

On solvability of a two-sided singular control problem

We study a two-sided singular control problem in a general linear diffusion setting and provide a set of conditions under which an optimal control exists uniquely and is of singular control type. Moreover, under these conditions the associated value function can be written in a quasi-explicit form. Furthermore, we investigate comparative static properties of the solution with respect to the volatility and control parameters. Lastly we illustrate the results with two explicit examples.     

A criterion of solvability of the elliptic Cauchy problem in a multi-dimensional cylindrical domain

In this paper, we consider the Cauchy problem for multidimensional elliptic equations in a cylindrical domain. The method of spectral expansion in eigenfunctions of the Cauchy problem for equations with deviating argument establishes a criterion of the strong solvability of the considered elliptic Cauchy problem. It is shown that the ill-posedness of the elliptic Cauchy problem is equivalent to the existence of an isolated point of the continuous spectrum for a self-adjoint operator with deviating argument.     

A criterion for the solvability of the multiple interpolation problem by simple partial fractions

Using reduction to polynomial interpolation, we study the multiple interpolation problem by simple partial fractions. Algebraic conditions are obtained for the solvability and the unique solvability of the problem. We introduce the notion of generalized multiple interpolation by simple partial fractions of order ≤ n. The incomplete interpolation problems (i.e., the interpolation problems with the total multiplicity of nodes strictly less than n) are considered; the unimprovable value of the total multiplicity of nodes is found for which the incomplete problem is surely solvable. We obtain an order n differential equation whose solution set coincides with the set of all simple partial fractions of order ≤ n.     

Nonlinear Goursat-Darboux control problem: Conditions for the preservation of global solvability

We consider the nonlinear vector Goursat-Darboux problem of general form with a control on the right-hand side in the differential equation. We consider the case in which its solution necessarily belongs to the class of functions with p-integrable (for some p > 1) mixed and first partial derivatives. In this connection, we use the fact that the first derivatives of such a solution are elements of spaces with a mixed norm; these spaces are much narrower than L _p . Sufficient conditions for the preservation of the global solvability of the problem under control perturbations are proved.     

On a class of control problems with incomplete information

For a class of nonlinear control systems with constrained control, we consider a terminal control problem of reaching a target point in which the initial point of the process and the vector of parameters of the system belong to known sets and there is no information on which point from the set of initial states is true and which parameter of the system from the set of parameters is true. We establish sufficient conditions for the existence of a solution to the problem in the class of guaranteeing program packages of Yu.S. Osipov and A.V. Kryazhimskiy. We also present the results of calculation of a model example.     

Local solvability of control systems with mixed constraints

We analyze the local solvability of a control system with mixed constraints and control constraints. We obtain a sufficient condition for the local solvability of the control system under the assumption of smoothness and 2-regularity of the mapping g specifying the mixed constraints. For the case in which the mapping g is not smooth, a sufficient condition for the local solvability is obtained under the assumption of coverability of g.     

On the solvability problem for equations with a single coefficient

It is proved that the general solvability problem for equations in a free group is polynomially reducible to the solvability problem for equations of the formw(x ₁, ...,x _n)=g, whereg is a coefficient, i.e., an element of the group, andw(x ₁, ...,x _n) is a group word in the alphabet of unknowns. We prove the NP-completeness of the solvability problem in a free semigroup for equations of the formw(x ₁,...,x _n)=g, wherew(x ₁,...,x _n) is a semigroup word in the alphabet of unknowns andg is an element of a free semigroup. Translated fromMatematicheskie Zametki, Vol. 59, No. 6, pp. 832–845, June, 1996. I wish to express my deep gratitude to S. I. Adyan and A. A. Razborov for the discussion of the present paper and for valuable remarks concerning the exposition. This research was partially supported by the International Science Foundation under grant MUV000.     

Feedback control of semilinear diffusion systems: inertial manifolds for closed-loop systems

An approach via inertial-manifold theory is presented as a wayto study the problem of stabilizing semilinear diffusion systemsusing finite-dimensional controllers. It is shown that a Sakawatype of controller plays an important role in the constructionof an inertial manifold for the closed-loop (controlled) semilineardiffusion system. This means that the use of a Sakawa type ofcontroller reduces the stabilization problem for the closed-loopsystem to the one on the inertial manifold. ^*This paper was partially presented at the 12th IFAC World Congress,Sydney, 18–23 July 1993.     

Linear stationary control systems over a Boolean semiring: Geometric properties and the isomorphism problem

In the present paper, we consider linear stationary dynamical systems over a Boolean semiring B. We analyze the complete observability, identifiability, reachability, and controllability of such systems. We define the notion of a “graph of modules” of completely controllable, completely reachable Boolean linear stationary systems by analogy with the spaces of modules in the case of systems over fields. We give a graph-theoretic interpretation of systems of this class. We solve the isomorphism problem in this class of systems.