Linear stationary control systems over the Boolean semiring and their graphs of modules

We consider linear stationary dynamical systems over the Boolean semiring. We analyze the properties of complete observability, identifiability, attainability, and controllability of a system. We define the notion of the “graph of modules” of totally controllable totally attainable Boolean linear stationary systems by analogy with spaces of modules in the case of systems over fields. The above-mentioned graph is described in the simplest case of one-dimensional inputs and outputs. We prove the weak connectedness of this oriented graph.     

Linear stationary surface waves over a rough bottom

We consider the linear formulation of the 3-dimensional problem of a stationary flow with a free surface over a rough bottom of an irrotational capillary heavy ideal incompressible fluid. For fast near-critical flows we derive an approximate equation for the free surface elevation. We express a fundamental solution to the problem in terms of contour integrals and establish its asymptotic behavior at large distances from the origin.     

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 maps that strongly preserve regular matrices over the Boolean algebra

The set of all m × n Boolean matrices is denoted by $ \mathbb{M} $ \mathbb{M} _m,n . We call a matrix A ∈ $ \mathbb{M} $ \mathbb{M} _m,n regular if there is a matrix G ∈ $ \mathbb{M} $ \mathbb{M} _n,m such that AGA = A. In this paper, we study the problem of characterizing linear operators on $ \mathbb{M} $ \mathbb{M} _m,n that strongly preserve regular matrices. Consequently, we obtain that if min{m, n} ⩽ 2, then all operators on $ \mathbb{M} $ \mathbb{M} _m,n strongly preserve regular matrices, and if min{m, n} ⩾ 3, then an operator T on $ \mathbb{M} $ \mathbb{M} _m,n strongly preserves regular matrices if and only if there are invertible matrices U and V such that T(X) = UXV for all X ε $ \mathbb{M} $ \mathbb{M} _m,n , or m = n and T(X) = UX ^T V for all X ∈ $ \mathbb{M} $ \mathbb{M} _n .     

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.     

An Open-loop criterion for the solvability of a closed-loop guidance problem with incomplete information: Linear control systems

The method of open-loop control packages is a tool for stating the solvability of guaranteed closed-loop control problems under incomplete information on the observed states. In this paper, a solution method is specified for the problem of guaranteed closed-loop guidance of a linear control system to a convex target set at a prescribed point in time. It is assumed that the observed signal on the system’s states is linear and the set of its admissible initial states is finite. It is proved that the problem under consideration is equivalent to the problem of open-loop guidance of an extended linear control system to an extended convex target set. Using a separation theorem for convex sets, we derive a solvability criterion, which reduces to solving a finite-dimensional optimization problem. An illustrative example is considered.     

Matrices over a semiring of binary relations and V-variable fractals

In this paper we prove that V-variable fractal sets are limits of infinite products of matrices over the semiring of binary relations on a compact metric space.     

Star partitions and the graph isomorphism problem

A canonical basis of Rⁿ associated with a graph G on n vertices has been defined in [15] in connection with eigenspaces and star partitions of G. The canonical star basis together with eigenvalues of G determines G to an isomorphism. We study algorithms for finding the canonical basis and some of its variations. The emphasis is on the following three special cases; graphs with distinct eigenvalues, graphs with bounded eigenvalue multiplicities and strongly regular graphs. We show that the procedure is reduced in some parts to special cases of some well known combinatorial optimization problems, such as the maximal matching problem. the minimal cut problem, the maximal clique problem etc. This technique provides another proof of a result of L. Babai et al. [2] that isomorphism testing for graphs with bounded eigenvalue multiplicities can be performend in a polynomial time. We show that the canonical basis in strongly regular graphs is related to the graph decomposition into two strongly regular induced subgraphs. Examples of distinguishing between cospectral strongly regular graphs by means of the canonical basis are provided. The behaviour of star partitions of regular graphs under operations of complementation and switching is studied.     

Linear extensions of dynamic systems and the reductibility problem

The relation of linear extensions of smooth dynamic systems to cohomologies and to reducibility in the case of flow is investigated. A result is obtained concerning -cohomologies in the neighborhood of a constant cocycle for the case of an arbitrary closed subgroup of the group GL (k, C).Translated from Matematicheskie Zametki, Vol. 8, No. 4, pp. 451–462, October, 1970.     

Some remarks on Boolean control systems: Controllability domains and realization theory

We construct a theory of realizations and controllability domains for linear stationary systems in the category of finitely generated free semimodules over a Boolean semiring. We show that the classical realization theorems cannot be generalized to this case, and we prove some incomplete analogs of these theorems. We analyze the structure of controllability domains and the reachability and observability characteristics. In particular, we define a geometric object representing the reachability properties of a system, namely, the generalized reachability topology on the state space.     

Linear control systems: Controllability with constrained controls

We study the null controllability with bounded controls of an ordinary control process in ⁿ . To prove controllability conditions for the nonautonomous case, we need to generalize the definition of characteristic exponents of a linear system. Some theorems on characteristic exponents are stated.This work follows the program of the Gruppo Nazionale per l'Analisi Funzionale e le Sue Applicazioni, Consiglio Nazionale delle Ricerche, Rome, Italy.     

Some properties of the stationary states of two-level systems

It is shown that some function J(x) is a constant at any point on the x axis at arbitrary variation of the potential energy of a two-level system. The application range of the LCAO method for calculating the wave functions of a one-dimensional two-level system is estimated.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 108, No. 1. pp. 69–78, July, 1996.     

On an isomorphism problem on the closed-set lattice of a graph

Let (G) and V(G) be, respectively, the closed-set lattice and the vertex set of a graph G. Any lattice isomorphism : V(G)(G) induces a bijection : V(G)V(G) such that for each x in V(G), (x)=x' in V(G') iff ({x})={x'}. A graph G is strongly sensitive if for any graph G' and any lattice isomorphism : (G)(G), the bijection induced by is a graph isomorphism of G onto G'. In this paper we present some sufficient conditions for graphs to be strongly sensitive and prove in particular that all C ₄-free graphs and all covering graphs of finite lattices are strongly sensitive.     

Linear periodic control systems: Controllability with restrained controls

Consider the problem of null-controllability for a linear non-autonomous system of the form , whereX andU are Banach spaces,A(t) andB(t) are linear bounded operators for eacht 0, and is a subset containing 0 (but not necessarily as an interior point). Some necessary and sufficient conditions for local and global null-controllability are proved whenA(·) andB(·) are periodic continuous functions. The proofs of the main results are based on discretization and on consideration of the corresponding linear discrete-time systems.     

Optimal control problem for a stationary model of low concentrated aqueous polymer solutions

We study the existence of an optimal control for lower bounded upper semicontinuous functionals and weak solutions of a system of equations describing the stationary motions of low concentrated aqueous polymer solutions in a bounded domain with locally Lipschitz boundary.     

Algorithmic aspects of the substitution decomposition in optimization over relations,set systems and Boolean functions

In the last years, decomposition techniques have seen an increasing application to the solution of problems from operations research and combinatorial optimization, in particular in network theory and graph theory. This paper gives a broad treatment of a particular aspect of this approach, viz. the design of algorithms to compute the decomposition possibilities for a large class of discrete structures. The decomposition considered is thesubstitution decomposition (also known as modular decomposition, disjunctive decomposition, X-join or ordinal sum). Under rather general assumptions on the type of structure considered, these (possibly exponentially many) decomposition possibilities can be appropriately represented in acomposition tree of polynomial size. The task of determining this tree is shown to be polynomially equivalent to the seemingly weaker task of determining the closed hull of a given set w.r.t. a closure operation associated with the substitution decomposition. Based on this reduction, we show that for arbitrary relations the composition tree can be constructed in polynomial time. For clutters and monotonic Boolean functions, this task of constructing the closed hull is shown to be Turing-reducible to the problem of determining the circuits of the independence system associated with the clutter or the prime implicants of the Boolean function. This leads to polynomial algorithms for special clutters or monotonic Boolean functions. However, these results seem not to be extendable to the general case, as we derive exponential lower bounds for oracle decomposition algorithms for arbitrary set systems and Boolean functions.