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''in control theory'' ...ttainable from it at time $T$ has non-empty interior. For example, for the control system defined in the $(x,y)$-plane by the two fields of admissible velocit

...atic control theory|Automatic control theory]]) if there is a delay in the control mechanism.

...mples of complex systems are: the nervous system, the brain, a computer, a control system in human society, etc. ...ir names, such as systems analysis, system-technology, and general systems theory, among others.

''for a linear stationary control system (an automatic control system)'' of the input of the system (the action, the control) $ u( t) $

is described by several systems of differential equations ...etails see [[#References|[3]]]). The participation of several differential systems $ ( S _ {i} ) $

...tive interpretation of them. Here are some (informal) examples of physical control system. ...attention to the more detailed internal properties. Therefore, two control systems, having in some sense identical behaviour and identical purposes, are not r

...ion of the initial elements are such that the operation of complex control systems can be described with the aid of Boolean functions. Boolean functions are a ...cube (cf. [[Boolean functions, metric theory of|Boolean functions, metric theory of]]), as well as of the various algebras of Boolean functions (cf. [[Many-

''control'' ...www.encyclopediaofmath.org/legacyimages/c/c025/c025990/c0259907.png" />. A control is called admissible if for each <img align="absmiddle" border="0" src="htt

$#C+1 = 62 : ~/encyclopedia/old_files/data/T092/T.0902870 Time\AAhoptimal control problem ...f. [[Optimal control, mathematical theory of|Optimal control, mathematical theory of]]), consisting in the determination of the minimum time

...class of systems. However, a precise and universal definition of adaptive control is still elusive. ...hen, numerous authors have tackled and solved these problems, and adaptive control has matured.

...it and network design; architectural design; control (cf. [[Control system|Control system]]); signal processing; selection of most reliable populations (in [[ ...arametric approach for selecting the most reliable population" ''Queueing Systems'' , '''24''' (1996) pp. 169–176</td></tr></table>

...the point of view of mathematical logic, as models of logics, that is, as systems of propositions with logical operations on them; and from the point of view ...ems of many-valued logic. Among the most important problems for functional systems are those of completeness, of the complexity of expressing some functions i

$#C+1 = 1 : ~/encyclopedia/old_files/data/A014/A.0104080 Automata, theory of ...[[Algorithms, theory of|Algorithms, theory of]]), particularly so with the theory of abstract machines, since automata can be regarded as a special case of s

...is important to note that if the equations of some of the elements of the systems are unknown, the Nyquist diagram can be constructed experimentally, by feed ...ce been developed for multivariable, infinite-dimensional and sampled-data systems, e.g. [[#References|[5]]], , , .

A parametric polynomial defined by a degree $n$, and a sequence of $n + 1$ control points $P_{0}, \ldots, P_{n}$. ...that Bézier curves are a parametric vector-valued representation based on control points, whereas Bernstein polynomials are usually thought of as a sequence

...at states conditions for the solvability of the Lur'e equations in control theory: ...te stability [[#References|[2]]], [[#References|[4]]]–[[#References|[6]]], control and adaptation (see, for example, [[#References|[7]]]–[[#References|[9]]]

...e phase coordinates and control functions. The original problem of optimal control by discretization (with respect to time $t$ and phase vector $x$) can be re ...representation of the trajectory $x(t)$ of the original problem of optimal control. The length $I(x_0,\dots,x_N)$ of this line is the sum of the (generalized)

...f. [[Optimal control, mathematical theory of|Optimal control, mathematical theory of]]), several approaches to the establishment of sufficient conditions for Let a problem of optimal control be posed in which the minimum of the functional

...there is also an analogue of Mikhailov's criterion for non-linear control systems (see [[#References|[6]]]). ...i, L.E. El'sgol'ts, "Mathematical foundations of the theory of control systems" , Moscow (1969) (In Russian)</td></tr><tr><td valign="top">[6]</td> <td

...e [[Optimal control, mathematical theory of|mathematical theory of optimal control]]. It was first formulated in 1956 by L.S. Pontryagin <ref name="Pontryagin ...ntryagin maximum principle corresponds to the following problem of optimal control. Given a system of [[Differential equation, ordinary|ordinary differential

$#C+1 = 69 : ~/encyclopedia/old_files/data/O068/O.0608460 Optimal programming control ...matical theory of|Optimal control, mathematical theory of]]), in which the control $ u= u( t) $

...tion introduced by W. Hamilton (1834) to describe the motion of mechanical systems. It is used, beginning with the work of C.G.J. Jacobi (1837), in the classi ...ions for the action function (cf. [[Hamilton–Jacobi theory|Hamilton–Jacobi theory]]) can be written in terms of a Hamilton function.

The Pontryagin maximum principle for discrete-time control processes. For such a process the maximum principle need not be satisfied, For example, consider the optimal control problem

...pursuer (hunter) $P$ and evader (prey) $E$, whose motions are described by systems of differential equations: ...ng a pursuit game means finding an equilibrium (cf. [[Saddle point in game theory]]).

...equations may be employed to find approximate solutions of the non-linear systems. which is typical in the theory of quasi-linear oscillations, the harmonic balance method consists in repla

...nning and design. Stochastic programming methods can also be used to adapt systems and algorithms to random changes in the state of the medium in which they o In order to construct realistic stochastic models of control processes, it is necessary to have at one's disposal not only the statistic

...of the classical [[Lyapunov function|Lyapunov function]] in the theory of systems of differential equations. ...p"> V.V. Kalashnikov, "Qualitative analysis of the behaviour of complex systems by the method of test functions" , Moscow (1978) (In Russian)</TD></TR></

...concerning the qualitative behaviour of linear equations and second-order systems (see [[#References|[3]]], [[#References|[4]]]). ...], and [[#References|[6]]], [[#References|[7]]]). For example, the optimum control $ u _ {0} $

There are systems of other types, where each request must necessarily be served, such as, for ...processes. The simplest queues can be described by two-dimensional random control sequences

...be an optimal method of controlling a system when a minimizing sequence of control functions does not have a limit in the class of Lebesgue-measurable functio ...ard to the constraining condition (2), the boundary conditions (3) and the control constraint (4), is satisfied by the trajectory

...[3]]], [[#References|[2]]]. The equation which is most often considered in control problems is where $x=x(t)$ is the vector function sought, while $u=u(t)$ is the control, i.e. a vector function which may be arbitrarily chosen out of all permissi

...6]]]. L. Neustadt has shown the significance of these operators in control theory [[#References|[a6]]]. ...on abstract Volterra operators and associated equations. However, a solid theory of this kind of operator does not yet exist.

...solutions of the corresponding initial-boundary value problems for a fixed control, etc. For example, the problem of optimal control consisting of the minimization of the functional

...ce of a dynamical system (cf. [[Automatic control theory|Automatic control theory]]) is appraised in terms of the $ H ^ \infty $- The theory of $ H ^ \infty $

...f $(A,B)$ as (the essential data of) a discrete-time time-invariant linear control system: ...$R = \mathbf{R}$ or $\mathbf{C}$, a continuous-time time-invariant linear control system:

''optimal singular control'' An [[Optimal control|optimal control]] for which, at a certain period of time, the conditions

...theory of non-linear oscillations, to physics, to the theory of automatic control, to astrodynamics, and to others. Averaging methods have often provided app 1) Standard systems in the sense of N.N. Bogolyubov

...l]] of small dimensions, but with constraints on the phase coordinates and control functions. After discretization of the functional and the system of differe ...ethod of]]) is difficult because of the labour involved in determining the control function.

In the method of local variations the original problem of optimal control, given as a [[Lagrange problem|Lagrange problem]], is made discrete with re

...ical with discrete analysis. Such structures may include certain algebraic systems, infinite graphs, certain types of calculating media (e.g. homogeneous stru ...mathematics|computational mathematics]]; [[Probability theory|probability theory]], and certain other disciplines the subject of which is discrete.

...~/encyclopedia/old_files/data/O068/O.0608430 Optimal control, mathematical theory of ...|mathematical programming]] and game theory (cf. [[Games, theory of|Games, theory of]]).

...the development of an analytical method for modeling and designing hybrid systems is to employ a max-plus algebra model of the manufacturing work-cell and ex ...oupling feedback. An excellent exposition of the state of the art in robot control is given in [[#References|[a5]]] and [[#References|[a4]]].

[[Pole assignment problem|pole assignment problem]] for linear control systems

...cal systems'' , '''I: Ordinary differential equations and smooth dynamical systems''' , Springer (1988) pp. 159ff (Translated from Russian)</TD></TR></tabl

...rticularly useful in the study of the "structural" properties of control systems. .../TD> <TD valign="top"> O.E. Lupanov, "On the synthesis of some class of control problems" ''Probl. Kibernet.'' , '''10''' (1963) pp. 63–97 (In Russia

...g the theory and the methods of solution of multi-step problems of optimal control. ...ructure of which consists of several steps, or else a process in which the control is subdivided into a sequence of successive stages (steps), which usually c

...n the areas of flexible manufacturing, communication networks and logistic systems. ...dresses the synthesis of controllers (i.e. supervisors) for discrete event systems to satisfy a set of qualitative specifications on the admissible orderings

...opedia/old_files/data/R081/R.0801170 Reliability and inspection of control systems, ''problems in the reliability of control systems''

...ontrol theory|automatic control theory]]. A systematic construction of the theory of ordinary differential equations with deviated arguments was begun in 194 ...h a deviation of the argument, permitting the construction of a meaningful theory, has been introduced. Several properties of such equations are directly ana

...y–Hamilton theorem can be also extended to singular two-dimensional linear systems described by Roesser-type or Fomasini–Marchesini-type models [[#Reference ...heorems may be also extended to two-dimensional continuous-discrete linear systems [[#References|[a5]]].

...a powerful tool for understanding the outside world and for prediction and control. The analysis of a mathematical model allows the essence of a phenomenon to ...) could not be measured, which resulted in quantitative conclusions of the theory not in proper correspondence with observations, so Copernicus was forced to

''parametrized measure, relaxed control, stochastic kernel'' The main area where Young measures have recently been used is optimization theory. Optimization problems where a local, integral cost functional is to be min

The design, development, tuning, and operation of computer and communication systems heavily rely on mathematical techniques which are usually indicated as perf ...from such areas as combinatorial optimization (scheduling) and stochastic control.

This problem has been greatly generalized in the theory of stationary stochastic processes (cf. [[Stationary stochastic process|Sta For systems of stochastic differential equations the problem of interpolation of some c

$#C+1 = 33 : ~/encyclopedia/old_files/data/H046/H.0406230 Hamilton\ANDJacobi theory ...o-called Hamilton–Jacobi equation. The fundamentals of the Hamilton–Jacobi theory were developed by W. Hamilton in the 1820s for problems in wave optics and

...assical (elementary) projective and analytic geometry one speaks of linear systems of curves, surfaces, quadrics, etc. These are families of curves, surfaces, ...heory: for linear input/output systems, linear dynamical systems or linear control system.

...ems theory. For systems in the sense of logics, see [[Formal_system|formal systems]]. ...cussed at the interplay between them (e.g. cybernetic models of biological systems). A more fundamental definition of a system was required encompassing the

...heory one considers boundary value problems with a functional parameter: a control. For example, consider the equation ...align="top">[5]</TD> <TD valign="top"> V.I. Zubov, "Lectures in control theory" , Moscow (1975) (In Russian)</TD></TR><TR><TD valign="top">[6]</TD> <TD

...a certain class. The term "absolute stability" assumes given a class of systems and an indication of the sense in which stability and uniformity are to be Numerous studies of particular non-linear systems have led to the understanding that in the first place one should take in co

...= 12 : ~/encyclopedia/old_files/data/K110/K.1100100 Kharitonov polynomial theory ...motivation for this theory derives from the issue of robust stability for systems of linear time-invariant differential equations. For a system of linear dif

...f a priori bounds on the computational complexity). This requires that the control, information and frequency relations arising in these processes between sta ...other zones, or a hammock (a subgraph connected with the other part of the control flow graph at precisely two vertices — the input and output; the input ve

...lifting theorem can also be used to solve problems in [[H^infinity-control-theory| $ H ^ \infty $ control theory]] and inverse scattering [[#References|[a1]]], [[#References|[a2]]].

...rgy minimization problems is presented. For the optimal control problem of systems governed by variational inequalities see [[#References|[a13]]]. ...TR><TR><TD valign="top">[a13]</TD> <TD valign="top"> V. Barbu, "Optimal control of variational inequalities" , ''Res. Notes Math.'' , '''100''' , Pitman (

...ary to consider statistical states of systems; for classical (non-quantum) systems they are described by probability distributions on the phase space. The rul ...ferentiable [[Dynamical system|dynamical system]] (which includes the main systems considered in analytical mechanics and classical statistical physics), the

...f natural parallelism occurring in computer-modelling and computer-control systems and processes; and b) a parallelization of information processing in multi- ...and subprograms; and independent problems and programs in multi-processor systems.

$#C+1 = 128 : ~/encyclopedia/old_files/data/O068/O.0608500 Optimal synthesis control ...n of a positional strategy makes possible an a posteriori realization of a control $ u $,

...op"> G.E.P. Box, G.M. Jenkins, "Time series analysis. Forecasting and control" , '''1–2''' , Holden-Day (1976)</TD></TR><TR><TD valign="top">[2]</TD> ...TD valign="top">[a5]</TD> <TD valign="top"> L. Ljung, T. Söderström, "Theory and practice of recursive identification" , M.I.T. (1983)</TD></TR></table

...ies of the system itself (see also [[Oscillations, theory of|Oscillations, theory of]]). The term "auto-oscillation" was introduced by A.A. Andronov (see [[# ...mous auto-oscillating systems with one degree of freedom may be defined as systems whose equations of motion have one or more limit cycles in the phase plane

...orks), which is oriented to the qualitative analysis and synthesis of such systems (discovering deadlocks or conflict situations and bottlenecks, computer-aid ...ine is the use of Petri nets as the basis of models for discrete dynamical systems in information technology, economics, digital engineering, etc.

...ments, a rule for combining them and a means of using the structure of the control system to define the function which it realizes are given. ...s of functions. A synthesis problem originally consisted of constructing a control system realizing a given function from this class.

...of dynamical processes with memory requires the further development of the theory of functional equations with abstract Volterra operators. Numerous applicat ...equations involving classical integral operators of Volterra type when the theory is framed in abstract spaces, see [[#References|[a2]]], [[#References|[a7]]

...yclopedia/old_files/data/P071/P.0701560 Parametric resonance, mathematical theory of The branch of the theory of ordinary differential equations that studies the phenomenon of parametri

...at follows it will be assumed that such an extension has been made for all control sequences. For systems in which $ \{ \tau _ {j} ^ {s} - \tau _ {j} ^ {e} \} \in G _ {I} $

...duction, $L$-systems constituted a significant part of [[formal language]] theory, allowing one to compare parallel rewriting to a more classical sequential ...d with these notions, in particular with D$0$L-sequences, which made D$0$L-systems mathematically very fruitful. The most famous problem is the D$0$L-sequence

...rmational processes in [[#References|[a11]]], and in networks of cognitive systems in [[#References|[a15]]] and [[#References|[a16]]]. ...o be used in more complex social processes, involving many self-organizing systems.

...of motion (understood in the same wide form). Thus, stability theory is a theory in the widest sense of this word. Among the different concepts of the stabi ...]]) and having a large intersection with the theory of linear systems, the theory of Lyapunov characteristic exponents (cf. [[Lyapunov characteristic exponen

...cience concerned with the construction of mathematical models of dynamical systems from measured input/output data. The constructed models are mostly of finit ...tworks form an example of common non-linear black-box models for dynamical systems.

...ried applications as system identification, adaptive control, transmission systems, adaptive filtering for signal processing, and several aspects of pattern r The study of adaptive algorithms is generally made through the theory of [[Stochastic approximation|stochastic approximation]].

...nsformational grammars; some of them are intended for processing component systems, others for processing hierarchy trees. As an example, one can quote the so grammars, which are finite systems of elementary transformations of the form $ t _ {1} \Rightarrow t _ {2} \

...g range of computer applications. The term is often understood to mean the theory of numerical methods and algorithms for the solution of typical mathematica ...als with the simplification of the man-computer interaction, including the theory and practice of [[Programming|programming]] computer problems and of [[Auto

The first significant results in the theory of equations (4) were obtained in [[#References|[11]]], where an effective ...dity of the majority of results listed above has also been established for systems of equations of type (4); however, in contrast to the case of a single equa

...f [[Orbit stability|orbit stability]] of closed trajectories of autonomous systems of differential equations one has to apply necessary and sufficient conditi ...">[10]</TD> <TD valign="top"> E. Jury, "Inners and stability of dynamic systems" , Wiley (1974)</TD></TR></table>

...ed space]]), in problems concerning the extension of functions, and in the theory of differential equations. In optimization over a finite or infinite horizon, e.g. in the theory of optimal economic growth, one aims to maximize a future return function

A code intended for the control of the functioning of an adder. When adding numbers represented in the bina ...rror-correcting code|Error-correcting code]]). In the study of problems of control of an adder one usually considers an encoding $ f: \{ 0 \dots M - 1 \} \

...ems, in particular, in circuit theory [[#References|[a4]]] and for control theory problems involving an $ H ^ \infty $- ...[[#References|[a1]]]. Generalizations of the Nevanlinna–Pick interpolation theory have also been developed for functions of several variables (see [[#Referen

are obtained in a less general form than for single-server systems, and are connected with a condition on the existence of so-called renewal e must be satisfied. If for such systems, in triangular arrays, the distributions of $ \tau _ {j} ^ {(n)} e $

...ces|[a21]]], [[#References|[a22]]] for general surveys of linear filtering theory. ...metrization, and this has led to extensive research on characterizing such systems and exploring the connection, uncovered by R.W. Brockett and J.M.C. Clark [

...s generalizations and modifications which reflect various features of real systems. In the case of a finite automaton $ (A, S, B, \phi , \psi ) $ ..., algebraic theory of|Automata, algebraic theory of]]); problems in coding theory generate the concepts of self-regulating automata, reversible automata, etc

...rcuit is one of the first classes of control systems (cf. [[Control system|Control system]]) considered from the mathematical point of view, as well as one of ...valign="top"> V.I. Shestakov, "On a logical calculus applicable to the theory of relay-contact circuits" ''Uchen. Zap. Moskov. Gosudarstv. Univ. Mat.''

...te evolution of the process is observed by the controller. A corresponding theory has also been developed in the case of partial observations (incomplete dat ...w.encyclopediaofmath.org/legacyimages/c/c026/c026010/c02601030.png" /> and control action <img align="absmiddle" border="0" src="https://www.encyclopediaofmat

...of finite-dimensional vector spaces, fields, algebraic systems, functional systems, etc.) which is independent of the particular computer used and which may b ...arious programming methods. A computational algorithm contains a number of control parameters, which remain indefinite in the first part, form a fixed part of

...bjects, such as complex spaces, topological manifolds, algebras, etc. Such systems occur in may parts of mathematics both as tools and as objects to be invest parameter systems of smooth curves $ \{ \sigma ^ {1} \dots \sigma ^ {n} \} $

...(from their own specific standpoints). By revealing the general aspects in systems of such different nature, cybernetics provides a general and moreover new m In many specific cybernetic systems, time is regarded as a parameter taking only a discrete set of values (usua

...cal systems with discrete states and changes of states called events; such systems arise e.g. in digital network communication protocols. ...l calculus" D. Franke (ed.) F. Kraus (ed.) , ''Design Methods of Control Systems'' , '''2''' , Pergamon (1991) pp. 723–728</TD></TR></table>

In control theory one studies equations $ \dot{x} = f( x, u) $ with a control parameter $ u $;

...[[Krein space|Krein space]]). The results are used in [[H^infinity-control-theory| $ H ^ \infty $ control theory]] (see [[#References|[a8]]]), and when the data are Fourier coefficients of

...uters and of programming them relies, to a large extent, on ideas from the theory of algorithms and mathematical cybernetics. In turn, practical work with co Common to all abstract machines is the presence of a finite control device that may be in one of the states $ q _ {1} \dots q _ {k} $.

The modern theory of linear inequalities has been constructed on the basis of this generaliza ...onomic technology and economic planning reduce to the solution of specific systems of linear inequalities; this has significantly determined the modern trend

...ts. Actually, Wold introduced (a1) as a joint representation for AR and MA systems (cf. also [[Mixed autoregressive moving-average process|Mixed autoregressiv ...esponding to stable and miniphase ARMA or (finite-dimensional) state space systems is by far the most important one. In this case there is a wide class of ide

...rators which is one of the most important classes of operators in function theory; it has many applications in different fields of mathematics and applied ma ...Hankel operators. The following realization is very important in function theory. Given a function $\phi \in L ^ { \infty }$, one defines the Hankel operato

..."non-classical" constraints of inequality type, in the theory of optimal control, a necessary condition for a strong extremum of the functional (*) is given ...of canonical transformations and [[Hamilton–Jacobi theory|Hamilton–Jacobi theory]]. Transition to the Hamiltonian form is also useful in the quantization of

The theory of $ L $- ...the work of A. Lindenmayer, [[#References|[a1]]]. The original aim of this theory was to provide mathematical models for the development of simple filamentou

$#C+1 = 288 : ~/encyclopedia/old_files/data/A014/A.0104090 Automatic control theory ...al apparatus of differential equations and especially an apparatus for the theory of steady motion [[#References|[2]]]; engineers had discovered the idea of

...the theory of power series, linear operators, random processes, and in the theory of extremal and approximation problems. ...owever, their applications in harmonic analysis are connected not with the theory of Fourier series, but with Fourier transforms.

...n of a [[Control system|control system]] and of special classes of control systems (contact schemes, diagrams or functional elements), of transition diagrams Synthesis of electrical networks, or realizability theory, refers to the problem of finding an electrical network with (partly) given

...ion of an [[Inverse matrix|inverse matrix]]. As for the solution of linear systems, methods for numerical inversion can be subdivided into direct and iterativ ...se matrix. The accuracy of the computed solution, as in the case of linear systems, depends on the rate of growth of the matrix entries in the intermediate st

...f. [[Optimal control, mathematical theory of|Optimal control, mathematical theory of]]). Consider the following problem in optimal control: Find a trajectory $ x( t) $

Oscillations in physical systems described by non-linear systems of ordinary differential equations ...also with the use of methods of optimal control of non-linear oscillating systems. The basic general problems of non-linear oscillations are: the search for

is only implicit in it — in terms of the control parameter $ \alpha ( t) = \mathop{\rm arg} f _ {t} ( z _ {t} ) $, ...n="top">[6]</TD> <TD valign="top"> V.I. Popov, "Quantization of control systems" ''Soviet Math. Dokl.'' , '''13''' : 6 (1972) pp. 1668–1672 ''Dokl.

...e one of the possible ways of action, e.g. in solving numerous problems of control and planning of production processes as well as in problems of design and l At the basis of the theory of convex, and, in particular, linear and quadratic, programming lie the [[

...ta processing, [[Dynamic programming|dynamic programming]], discrete-event systems, computer science, discrete mathematics, [[Mathematical logic|mathematical ...e leads to a unified approach to various optimization problems and optimal control problems with discrete or continuous state space as well to the correspondi

...ons occur simultaneously, and the elaboration and fulfillment of plans and control solutions at various levels, including the creation of conditions for the a ...vity of economic systems at various levels; factor models of productivity; systems of demand functions for groups of consumers and specific functions of consu

...mes (cf. [[Games, theory of|Games, theory of]]). The first studies in this theory appeared in the mid-1950s. ==Formulations of problems in the theory of differential games.==

...n of the input-output behaviour of non-linear systems, especially bilinear systems); cf. the collection [[#References|[a1]]] for a first idea.

...estimating the step error is to obtain formulas of Runge–Kutta type with a control term which approximates the principal term of the step error of the method ...p. These methods are suited for the solution of boundary value problems of systems of differential equations. Finite difference analogues of these methods are

...etc.). The following have had a decisive influence on the creation of the theory of orthogonal series: ...em of moments (mid-19th century), which led to his creation of the general theory of [[Orthogonal polynomials|orthogonal polynomials]];

...ions of general Padé approximation problems. In [[#References|[a19]]] this theory is first generalized and then applied to the inversion of (block) Hankel ma ...he theory of Hankel matrices to engineering problems of system and control theory, see [[#References|[a14]]] and [[#References|[a7]]].

...ounced by R. Thom [[#References|[1]]] in terms of their gradient dynamical systems and the analogous list of critical points of codimension $ \leq 4 $( ...t in an individual object, but in a collection of them, depending on some "control" parameters. Degenerate singularities which are removable for each fixed va

...a unique definition for each definite difference scheme. For an effective control of the viscosity to be possible, it is advisable to examine classes of diff ..."top">[7]</TD> <TD valign="top"> B.L. Rozhdestvenskii, N.N. Yanenko, "Systems of quasilinear equations and their applications to gas dynamics" , Amer. Ma

...tural numbers. Based on an elaborate system of verbal calculation, written systems of calculation arose and slowly the methods of doing the four arithmetic op ...mathematical theory. From arithmetic gradually grew [[Number theory|number theory]]. The systematic study of magnitudes and measurements was created. The pro

...e one of the most important types of control systems (cf. [[Control system|Control system]]). Substantially, a finite automaton may be described as a system w ...Depending on the particular point of view, such system may include formal systems (cf. [[Formal system|Formal system]]), real automata, living organisms, etc

A test for a given table, with control target $ \mathfrak N $ ...plexity. The construction of minimal tests is a central problem of testing theory. The reasons for this can be demonstrated by the example of an absolute tes

$#C+1 = 56 : ~/encyclopedia/old_files/data/Q076/Q.0706850 Queueing theory The branch of probability theory in which one studies mathematical models of various kinds of real queues (c

...stigation of these models was an important stage in the development of the theory of many-valued logics. ...m the point of view of algebra, models of many-valued logics are algebraic systems (cf. [[Algebraic system|Algebraic system]]) having both applied and theoret

...ve. This is the Vitali–Hahn–Saks theorem. Another striking result from the theory of vector measures is the so-called Nikodým boundedness theorem: For a col ...4]]]); see [[#References|[a5]]] for the role of vector measures in control theory.

...1]</TD> <TD valign="top"> D.R. Brillinger, "Time series. Data analysis and theory" , Holt, Rinehart &amp; Winston (1975) {{MR|0443257}} {{ZBL|0321.62004}} </ ...valign="top">[a4]</TD> <TD valign="top"> L. Ljung, "System identification theory for the user" , Prentice-Hall (1987) {{MR|1157156}} {{ZBL|0615.93004}} </TD

...ns in mathematical analysis are all equations of the type (1), as are also systems of algebraic equations, both finite and infinite, finite-difference equatio ...tems of compatible functional equations. Moreover, functional equations or systems of functional equations can contain a greater number of essential, essentia

==Oscillatory properties of solutions of linear Hamiltonian systems.== ...–(4) are involved in a number of problems in variational calculus, optimum control, studies on the properties of the spectrum of the corresponding differentia

...f. [[Optimal control, mathematical theory of|Optimal control, mathematical theory of]]) in fact arose in this manner. ...t this equation is solvable, a special apparatus had to be developed — the theory of the [[Riemann integral|Riemann integral]].

...( \theta ) = H ^ { 2 } \ominus \theta H ^ { 2 }$. There also has evolved a theory of so-called $C _ { 0 }$ operators, for which the operator $T ( \theta )$ ( ...nnections with engineering applications, such as signal processing, robust control and system identification, have led to new questions, different points of v

...Tucker conditions are applicable to a wide range of problems, from optimal control problems in engineering [[#References|[a5]]], [[#References|[a6]]], [[#Refe ..."> M.S. Bazaraa, H.D. Sherali, C.M. Shetty, "Nonlinear programming: Theory and algorithms" , Wiley (1993) (Edition: Second)</TD></TR>

...of mathematics it is stimulating research in pure mathematics (e.g. coding theory, $*$-algebras, quantum topology) as well as requiring and providing many op ...ach which led to the basic definitions of computer science and information theory as formulated by A. Church, A.M. Turing, C. Shannon and others in the first

...<TD valign="top"> Ya.Z. Tsypkin, "Foundations of the theory of learning systems" , Acad. Press (1973) (Translated from Russian)</TD></TR></table> ...ung, "Analysis of recursive stochastic algorithms" ''IEEE Trans. Autom. Control'' , '''AC-22''' (1977) pp. 551–575</TD></TR></table>

...near functions on sets of an $n$-dimensional [[vector space]] specified by systems of linear inequalities and equations; linear programming is one of the bran The basis for studying properties of linear programming problems is the theory of duality. The dual problem to the problem \eqref{eq:1}–\eqref{eq:3} is

...ally, the Lyapunov equation has many applications in stability and control theory; efficient numerical algorithms for solving it are available. ...<td valign="top"> A. Halanay, D. Wexler, "Qualitative theory of pulse systems" , Nauka (1971) (In Russian)</td></tr><tr><td valign="top">[a5]</td> <td

...after [[#References|[a3]]] have witnessed an explosive development of the theory of functional-differential equations and their applications (see, e.g. [[#R ...he boundary value problem, the existence of periodic solutions, stability, control, etc. For example, the Cauchy problem (also called the initial problem or b

This problem has been greatly generalized in the theory of stationary stochastic processes (cf. [[Stationary stochastic process|Sta For systems of stochastic differential equations, the problem of predicting some compon

[[Intersection theory|intersection theory]] provides a satisfactory modern framework. Enumerative geometry deals with {{Cite|Fu2}} for a complete reference on intersection theory; for historical surveys and a discussion of enumerative geometry, see

...st interested in algorithms of this kind. Fairly effective are interactive systems combining the advantages of the human eye assessments with the computer's a ...R><TD valign="top">[6]</TD> <TD valign="top"> I.V. Sobolev, "Production control in the timber industry" , Petrozavodsk (1976) (In Russian)</TD></TR><TR><

...-matrix. Together these transformations make up the feedback group (of the control system <img align="absmiddle" border="0" src="https://www.encyclopediaofmat ...top">[a10]</TD> <TD valign="top"> A. Mead, E. Ruch, A. Schönhofer, "Theory of chirality functions, generalized for molecules with chiral ligands" ''T

''theory of programming'' ...nvestigating models of programs. The formal combinatorial methods form the theory of program schemes, which studies properties of a program that are invarian

Solution of linear systems of equations. Given a matrix $A \in \mathbf{R} ^ { n \times n }$ and a vect ...sparse matrices or compute eigenvalues of sparse matrices or both. Linear systems are either solved by direct sparse methods or by iterative methods. A surve

...as well as from theoretical interest. See also [[Queueing theory|Queueing theory]]. ....g., a user data field) is served in pre-emptive resume manner because the control information has already been acquired, and the last portion (e.g., a messag

''in the theory of differential equations'' ...tements have been obtained on the basis of the comparison principle in the theory of the stability of motion (see [[#References|[4]]]–[[#References|[6]]]).

is the length of the boundary layer. All systems of type (1) for which the conditions a) and b) are satisfied simultaneously ...differential systems, lead to an increase in stiffness. Stiff differential systems require special methods of solution.

...esis problems for antennas and other physical systems); and various other control problems described by differential equations (in particular, differential ...sign of antennas and other systems or constructions, problems of optimal control and many others) are also called synthesis problems.

$#C+1 = 79 : ~/encyclopedia/old_files/data/R081/R.0801180 Reliability theory ...ost, etc.) are random. Other widely used methods are those of optimization theory, mathematical logic, etc.

A representing object for a generalized cohomology theory. The notion was introduced in [[#References|[1]]] (cf. also [[Generalized c ...construct the homotopy category of spectra [[#References|[2]]]. Postnikov systems (cf. [[Postnikov system|Postnikov system]]) of spectra of spaces have also

...atisfy certain compatibility conditions. Generally, almost all known Euler systems satisfy the condition ES) described below. Let $K$ be a number field. Fix a ...tion connected with the corresponding arithmetic object. In this way Euler systems establish (the sought for) relations between arithmetic objects and corresp

$#C+1 = 174 : ~/encyclopedia/old_files/data/Q076/Q.0706240 Qualitative theory of differential equations ...the modern theory of stability of motion (see [[Stability theory|Stability theory]]).

...]]). Context-free languages are also sometimes defined by so-called normal systems of equations in languages, which are another form of notation of context-fr ...valign="top">[1]</TD> <TD valign="top"> S. Ginsburg, "The mathematical theory of context-free languages" , McGraw-Hill (1966)</TD></TR></table>

$#C+1 = 144 : ~/encyclopedia/old_files/data/B017/B.0107310 Boundary\AAhlayer theory In the theory of systems of ordinary differential equations

...te noise. The precise description of such a filtering problem requires the theory of stochastic differential equations (e.g., [[#References|[a4]]]; see also ...n="top"> R.E. Mortensen, "Optimal control of continuous-time stochastic systems" ''PhD Diss. Univ. California, Berkeley'' (1966)</td></tr><tr><td valign=

...which the true classification is a priori known. In developing the general theory of recognition algorithms the most complete results have been obtained with ...top">[3]</TD> <TD valign="top"> V.N. Vapnik, A.Ya. Chervonenkis, "The theory of pattern recognition" , Moscow (1974) (In Russian)</TD></TR><TR><TD val

...earch are considered in the theory of games (cf. [[Games, theory of|Games, theory of]]). If an information state contains several physical states, and if the ...ion problems, as well as problems with preference relations, relate to the theory of games; they are classified according to their game-theoretic properties.

$#C+1 = 55 : ~/encyclopedia/old_files/data/S083/S.0803330 Scheduling theory ...scheduling theory include management, production, transportation, computer systems, construction, etc.

plans and contributions to Statistical Systems of other countries as control for changes in weather on the surface of the earth was

...ical applications, such as domain decomposition methods for solving linear systems of equations, and certain multi-grid schemes used for solving elliptic part ...inert, "Error bounds for the method of alternating projections" ''Math. Control Signals Syst.'' , '''1''' (1988) pp. 43–59</td></tr><tr><td valign="top

...tion of]]; [[Boolean functions, metric theory of|Boolean functions, metric theory of]]) $ | M | > 2 ^ {2 ^ {n} } $. ...cheduling theory]]), extremal problems on graphs (cf. [[Graph theory|Graph theory]]), and problems on the minimal representation of Boolean functions and fun

...fundamental work of C. Shannon [[#References|[1]]] published in 1948. The theory of alphabetical coding is a branch of applied mathematics that consists in ...exity of a circuit realization by automata goes beyond the scope of coding theory.) The so-called prefix codes (that is, codes for which no word of $ V $

...pparatus. A delicate question is how to express the type of socio-economic systems which are modelled, taking account of the social structure. It often happen ...y if the cost of the outputs is equal to the cost of the inputs. The given theory of efficient production methods and their characterization using $ \pi $

[[Category:Linear and multilinear algebra; matrix theory]] led to the matrix calculus — the subject matter of the theory of

...ingle species models, age structure is considered in many multi-population systems. In the domain of demography, two-sex models are of special interest and in ...ons; however, in recent years, a functional-analytic approach based on the theory of semi-groups of operators (cf. also [[Semi-group of operators|Semi-group

...um hypothesis is independent of the Zermelo–Frankel axioms. See also [[Set theory]]. ...alism]], [[Formal system]], [[Axiomatic method]], [[Consistency]], [[Proof theory]], and [[Unsolvability]]. For the history of mathematics associated with th

...e traditional logical problems by algebraic methods. With the birth of the theory of sets (in the 1870s), propositions and logical operations on them became ...logic, which is a part of the problem of synthesis of control (switching) systems. The minimization of functions of the algebra of logic consists in construc

...earning involves sequences of actions and the task consists of acquiring a control strategy for choosing appropriate actions to achieve a goal, this mode is s ...uncertainty as to the correct answer (cf. [[Probability theory|Probability theory]]). They calculate the probability distribution for the unknown variables,

$#C+1 = 20 : ~/encyclopedia/old_files/data/G045/G.0405010 Graph theory ...Methods were developed for solving a number of extremal problems in graph theory; one such problem is the construction of the maximum flow across a network

Unlike formal languages, natural languages are dynamic, by nature. Rule systems and vocabularies of natural languages continuously change over time, the le ...ned unification is $\cal N P$-complete (cf. [[Complexity theory|Complexity theory]]). On the other hand, carefully crafted "mildly context-sensitive" gramm

...e number of computations, so efficient approximation techniques attempt to control the number of calculations for both computational and accuracy effectivenes See also [[Errors, theory of|Errors, theory of]]; [[Error|Error]].

see [[#References|[8]]]). With practically no changes, the theory of Smith normal forms goes over to the case when $ K $ ...icular, this holds for the theory of finitely-generated Abelian groups and theory of the Jordan normal form (see below).

...of assignment of the value of an expression to a variable, the operator of control transfer (unconditional or conditional jump, or goto), the operator of proc ...tion of an intermediate form, are employed in optimizing translators or in systems with mixed generation for various types of machines; in certain very comple

...extremum is guaranteed only for convex and related unimodal functions. The theory of finding a global extremum is still (1989) in the initial phase of develo ...><TD valign="top">[15]</TD> <TD valign="top"> , ''The current state of the theory of Operations Research'' , Moscow (1979) (In Russian)</TD></TR><TR><TD va

The classification adopted in the theory of linear differential operators refers mainly to linear differential opera ...rential operator of|Principal type, partial differential operator of]]). A theory of local solvability and smoothness of solutions has been developed for suc

but is also suitable for systems subject to uncertainties of the form $ \Delta A = \epsilon | A | $ th eigenvalue alone. From first-order perturbation theory one finds, [[#References|[a1]]],

...ng processes, Krein spaces, the geometry of complex Banach spaces, control theory, and optimization. These semi-groups can be considered natural non-linear a ...|[a18]]]. He used this approach to study several important problems in the theory of linear operators on indefinite metric spaces. Note that, generally speak

...tationary stochastic process is widely used in applications of probability theory in various areas of natural science and technology, since these processes a In the mathematical theory of stationary stochastic processes, an important role is played by the mome

$#C+1 = 33 : ~/encyclopedia/old_files/data/A011/A.0101910 Algorithms, theory of ...ers, cf. [[Turing machine|Turing machine]]). Subsequent development of the theory of algorithms is due to the studies of Kleene, Post [[#References|[6]]], [[

...4th ACM STOC Symp.'' (1982) pp. 147–152 (Revised version: Inform. &amp; Control 68 (1986), 86-104)</td></tr><tr><td valign="top">[a25]</td> <td valign="top

...r, this is not achievable anymore in a perfect way. The effects of quantum theory may begin to have an influence on the outcome of the calculation. Quantum theory would not be probabilistical without measurements. The probabilistic nature

..."arithmetization of analysis" refers to 19th century efforts to create a "theory of real numbers ... using set-theoretic constructions, starting from the na # the creation of the theory of real numbers.