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1.
In this paper a linearly perturbed version of the well-known matrix Riccati equations which arise in certain stochastic optimal control problems is studied. Via the
concepts of mean square stabilizability and mean square detectability we improve previous results on both the convergence properties of the linearly perturbed Riccati differential equation and the solutions of the linearly perturbed algebraic Riccati equation. Furthermore, our approach unifies, in some way, the study for this class of Riccati equations with the one for classical
theory, by eliminating a certain inconvenient assumption used in previous works (e.g., [10] and [26]). The results are derived
under relatively weaker assumptions and include, inter alia, the following: (a) An extension of Theorem 4.1 of [26] to handle systems not necessarily observable. (b) The existence of
a strong solution, subject only to the mean square stabilizability assumption. (c) Conditions for the existence and uniqueness of stabilizing
solutions for systems not necessarily detectable. (d) Conditions for the existence and uniqueness of mean square stabilizing
solutions instead of just stabilizing. (e) Relaxing the assumptions for convergence of the solution of the linearly perturbed
Riccati differential equation and deriving new convergence results for systems not necessarily observable.
Accepted 30 July 1996 相似文献
2.
G. M. Saha 《Annals of the Institute of Statistical Mathematics》1973,25(1):605-616
Summary Recently Saha and Das [10] constructed partially balanced incomplete block (PBIB) designs of two and more associate classes
by using confounded designs for 2
n
factorials. Several new designs of two associate classes were obtained through those methods. This paper generalizes one
of the earlier methods of construction to obtain several series ofT
m
-type (m≧2) PBIB designs, i.e., the designs havingm-dimensional triangular association schemes. Some more new designs of two associate classes (i.e.,T
2-type) are obtained through the generalized methods of construction. 相似文献
3.
An approach to minimize the control costs and ensuring a stable deviation control is the Riccati controller and we want to use it to control constrained dynamical systems (differential algebraic equations of Index 3). To describe their discrete dynamics, a constrained variational integrators [1] is used. Using a discrete version of the Lagrange-d’Alembert principle yields a forced constrained discrete Euler-Lagrange equation in a position-momentum form that depends on the current and future time steps [2]. The desired optimal trajectory (qopt, popt) and according control input uopt is determined solving the discrete mechanics and optimal control (DMOC) algorithm [3] based on the variational integrator. Then, during time stepping of the perturbed system, the discrete Riccati equation yields the optimal deviation control input uR. Adding uopt and uR to the discrete Euler-Lagrange equation causes a structure preserving trajectory as both DMOC and Riccati equations are based on the same variational integrator. Furthermore, coordinate transformations are implemented (minimal, redundant and nullspace) enabling the choice of different coordinates in the feedback loop and in the optimal control problem. (© 2016 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim) 相似文献
4.
Vejdi I. Hasanov 《Numerical Functional Analysis & Optimization》2013,34(12):1532-1559
The perturbation results for the solutions of two linearly perturbed algebraic Riccati equations are derived. We generalize the results of Sun [SIAM J. Matrix Anal. Appl., 19 (1998):39–65] for continuous (CARE) and discrete (DARE) algebraic Riccati equations, respectively. The results are illustrated by numerical examples. 相似文献
5.
Wolfgang Watzlawek 《Monatshefte für Mathematik》1985,100(1):67-76
The investigations ofRosenbloom-Widder [13] on expansions in terms of heat polynomials have been generalized for partial differential equations of second order with singular coefficients, equations of higher order etc. In the present paper we introduce a new method for the discussion of such problems. The technique is related to the treatment of the Cauchy problem on the base of the Ovcyannikov theorem. The method is used for the discussion of two classes of partial differential equations of arbitrary order. 相似文献
6.
J. Casti 《Journal of Optimization Theory and Applications》1975,17(1-2):169-175
In this article, a new equation is derived for the optimal feedback gain matrix characterizing the solution of the standard linear regulator problem. It will be seen that, in contrast to the usual algebraic Riccati equation which requires the solution ofn(n + 1)/2 quadratically nonlinear algebraic equations, the new equation requires the solution of onlynm such equations, wherem is the number of system input terminals andn is the dimension of the state vector of the system. Utilizing the new equation, results are presented for the inverse problem of linear control theory. 相似文献
7.
Summary This paper is concerned with the numerical solution of stiff initial value problems for systems of ordinary differential equations using Runge-Kutta methods. For these and other methods Frank, Schneid and Ueberhuber [7] introduced the important concept ofB-convergence, i.e. convergence with error bounds only depending on the stepsizes, the smoothness of the exact solution and the so-called one-sided Lipschitz constant . Spijker [19] proved for the case <0 thatB-convergence follows from algebraic stability, the well-known criterion for contractivity (cf. [1, 2]). We show that the order ofB-convergence in this case is generally equal to the stage-order, improving by one half the order obtained in [19]. Further it is proved that algebraic stability is not only sufficient but also necessary forB-convergence.This study was completed while this author was visiting the Oxford University Computing Laboratory with a stipend from the Netherlands Organization for Scientific Research (N.W.O.) 相似文献
8.
Klaus-Günther Strack 《Numerische Mathematik》1986,48(2):221-237
Summary In this paper we give bounds for the error constants of certain classes of stable implicit finite difference methods for first order hyperbolic equations in one space dimension. We consider classes of methods that user downwind ands upwind points in the explicit part andR downwind andS upwind points in the implicit part, respectively, and that are of optimal orderp=min (r+R+s+S, 2(r+R+1), 2(s+S)).In some cases the error constant of interpolatory methods [5] can be improved. The results are proved via the order star technique. They are further used to determine methods of optimal order that are stable. 相似文献
9.
There exist some useful methods for the calculation of Hilbert's function without using a free resolution of polynomial ideals (see for example [4], [10], [11] and the references in these papers). Using Bezout's theorem (in the sense ofW. Gröbner [3], 144.5) these methods are suited for a proof that special homogeneous polynomial ideals are imperfect, but not for the arithmetically Cohen-Macaulay property. It is the theorem of this paper that these gaps can be filled. This theorem therefore provides some proof that an arbitrary homogeneous polynomial ideal is perfect or imperfect. Our methods are demonstrated in three examples, taking the third example from the paper ofG. A. Reisner [7], p. 35 and, using our methods, we rather easily obtain the result of [7], that the Cohen-Macaulay property depends on the characteristic of the field. In the second example, we give some remarks on the usefulness of the definition for perfeet ideals ofF. S. Macaulay [5] (see also [6]). This also illustrates whyF. S. macaulay could only construct imperfect ideals-except such one obtainable by using ideals of the principal class.
Unserem Lehrer, Herrn Professor Dr. W. Gröbner, zum 80. Geburtstag in Verehrung gewidmet 相似文献
Unserem Lehrer, Herrn Professor Dr. W. Gröbner, zum 80. Geburtstag in Verehrung gewidmet 相似文献
10.
Prof. Dr. Jürgen Tölke 《manuscripta mathematica》1976,19(2):189-194
Since H.R. MüLLER [1] it is well known that spherical envelope pairs X,X to a given roll sliding number are solutions of two conjugate Riccati differential equations. For two special cases an integral is known [1,3]. By use of the fundamental differential equations of the polodes [2] we show the existence of two classes of integrable curves X,X.
Herrn Professor Dr. H.R. Müller zum 65. Geburtstag gewidmet 相似文献
Herrn Professor Dr. H.R. Müller zum 65. Geburtstag gewidmet 相似文献
11.
Laurent Padé-Chebyshev rational approximants,A
m
(z,z
−1)/B
n
(z, z
−1), whose Laurent series expansions match that of a given functionf(z,z
−1) up to as high a degree inz, z
−1 as possible, were introduced for first kind Chebyshev polynomials by Clenshaw and Lord [2] and, using Laurent series, by
Gragg and Johnson [4]. Further real and complex extensions, based mainly on trigonometric expansions, were discussed by Chisholm
and Common [1]. All of these methods require knowledge of Chebyshev coefficients off up to degreem+n. Earlier, Maehly [5] introduced Padé approximants of the same form, which matched expansions betweenf(z,z
−1)B
n
(z, z
−1)). The derivation was relatively simple but required knowledge of Chebyshev coefficients off up to degreem+2n. In the present paper, Padé-Chebyshev approximants are developed not only to first, but also to second, third and fourth
kind Chebyshev polynomial series, based throughout on Laurent series representations of the Maehly type. The procedures for
developing the Padé-Chebyshev coefficients are similar to that for a traditional Padé approximant based on power series [8]
but with essential modifications. By equating series coefficients and combining equations appropriately, a linear system of
equations is successfully developed into two sub-systems, one for determining the denominator coefficients only and one for
explicitly defining the numerator coefficients in terms of the denominator coefficients. In all cases, a type (m, n) Padé-Chebyshev approximant, of degreem in the numerator andn in the denominator, is matched to the Chebyshev series up to terms of degreem+n, based on knowledge of the Chebyshev coefficients up to degreem+2n. Numerical tests are carried out on all four Padé-Chebyshev approximants, and results are outstanding, with some formidable
improvements being achieved over partial sums of Laurent-Chebyshev series on a variety of functions. In part II of this paper
[7] Padé-Chebyshev approximants of Clenshaw-Lord type will be developed for the four kinds of Chebyshev series and compared
with those of the Maehly type. 相似文献
12.
Equations for the Missing Boundary Values in the Hamiltonian Formulation of Optimal Control Problems
Vicente Costanza Pablo S. Rivadeneira Ruben D. Spies 《Journal of Optimization Theory and Applications》2011,149(1):26-46
Partial differential equations for the unknown final state and initial costate arising in the Hamiltonian formulation of regular
optimal control problems with a quadratic final penalty are found. It is shown that the missing boundary conditions for Hamilton’s
canonical ordinary differential equations satisfy a system of first-order quasilinear vector partial differential equations
(PDEs), when the functional dependence of the H-optimal control in phase-space variables is explicitly known. Their solutions are computed in the context of nonlinear systems
with ℝ
n
-valued states. No special restrictions are imposed on the form of the Lagrangian cost term. Having calculated the initial
values of the costates, the optimal control can then be constructed from on-line integration of the corresponding 2n-dimensional Hamilton ordinary differential equations (ODEs). The off-line procedure requires finding two auxiliary n×n matrices that generalize those appearing in the solution of the differential Riccati equation (DRE) associated with the linear-quadratic
regulator (LQR) problem. In all equations, the independent variables are the finite time-horizon duration T and the final-penalty matrix coefficient S, so their solutions give information on a whole two-parameter family of control problems, which can be used for design purposes.
The mathematical treatment takes advantage from the symplectic structure of the Hamiltonian formalism, which allows one to
reformulate Bellman’s conjectures concerning the “invariant-embedding” methodology for two-point boundary-value problems.
Results for LQR problems are tested against solutions of the associated differential Riccati equation, and the attributes
of the two approaches are illustrated and discussed. Also, nonlinear problems are numerically solved and compared against
those obtained by using shooting techniques. 相似文献
13.
Summary. In this work we address the issue of integrating
symmetric Riccati and Lyapunov matrix differential equations. In
many cases -- typical in applications -- the solutions are positive
definite matrices. Our goal is to study when and how this property
is maintained for a numerically computed solution.
There are two classes of solution methods: direct and
indirect algorithms. The first class consists of the schemes
resulting from direct discretization of the equations. The second
class consists of algorithms which recover the solution by
exploiting some special formulae that these solutions are known to
satisfy.
We show first that using a direct algorithm -- a one-step scheme or
a strictly stable multistep scheme (explicit or implicit) -- limits
the order of the numerical method to one if we want to guarantee
that the computed solution stays positive definite. Then we show two
ways to obtain positive definite higher order approximations by
using indirect algorithms. The first is to apply a symplectic
integrator to an associated Hamiltonian system. The other uses
stepwise linearization.
Received April 21, 1993 相似文献
14.
This is the second and final part of a paper which appeared in a preceding issue of this journal. Herein the methods developed in the earlier sections of this paper are used first, in conjunction with some ideas of Krein, to develop models for simple, closed symmetric [resp. isometric]operators with finite and equal deficiency indices. A number of other related issues and applications are then discussed briefly. These include entropy inequalities, interpolation, parametrization ofJ inner matrices, the Schur algorithm and canonical equations. Finally, a list of misprints for Part I is incorporated at the end. 相似文献
15.
Anders Lindquist 《Applied Mathematics and Optimization》1976,3(1):1-13
In this paper we consider the problem of determining the error covariance matrix (and hence the gain) in Kalman-Bucy filtering, utilizing the smallest possible number of time-invariant, first-order differential equations. The traditional method requires the solution of a matrix Riccati equation containing 1/2n(n+1) such equations. Here we demonstrate that, under certain conditions, onlypn equations are needed, wherep is a number which often is much smaller thann. This is an improvement on the previous non-Riccati algorithms developed by Kailath and Lindquist. The reduction is achieved by exploiting certain time-invariant integrals of this system. This paper complements our previous papers [11, 12] in that the present method is more direct.This work was supported by the National Science Foundation under grant MPS 75-07028. 相似文献
16.
Optimal control of linear stochastic evolution equations in Hilbert spaces and uniform observability
Viorica Mariela Ungureanu 《Czechoslovak Mathematical Journal》2009,59(2):317-342
In this paper we study the existence of the optimal (minimizing) control for a tracking problem, as well as a quadratic cost
problem subject to linear stochastic evolution equations with unbounded coefficients in the drift. The backward differential
Riccati equation (BDRE) associated with these problems (see [2], for finite dimensional stochastic equations or [21], for
infinite dimensional equations with bounded coefficients) is in general different from the conventional BDRE (see [10], [18]).
Under stabilizability and uniform observability conditions and assuming that the control weight-costs are uniformly positive,
we establish that BDRE has a unique, uniformly positive, bounded on ℝ + and stabilizing solution. Using this result we find the optimal control and the optimal cost. It is known [18] that uniform
observability does not imply detectability and consequently our results are different from those obtained under detectability
conditions (see [10]).
相似文献
17.
C. M. Carballo C. A. Morales M. J. Pacifico 《Bulletin of the Brazilian Mathematical Society》2000,31(3):287-303
A transitive set of a vector fieldX ismaximal transitive if it contains every transitive set ofX intersecting it. We shall prove that ifX isC
1 generic then every singularity ofX with either only one positive or only one negative eigenvalue belongs to a maximal transitive set ofX. In particular, we characterize maximal transitive sets with singularities for genericC
1 vector fields on closed 3-manifolds in terms of homoclinic classes associated to a unique singularity. We apply our results to the examples introduced in [3] and [15].This work is partially supported by CNPq 001/2000, FAPERJ and PRONEX/Dynamical Systems, FINEP-CNPq. 相似文献
18.
Andreas W. M. Dress 《Aequationes Mathematicae》1985,29(1):222-243
The new regular polyhedra as defined by Branko Grünbaum in 1977 (cf. [5]) are completely enumerated. By means of a theorem of Bieberbach, concerning the existence of invariant affine subspaces for discrete affine isometry groups (cf. [3], [2] or [1]) the standard crystallographic restrictions are established for the isometry groups of the non finite (Grünbaum-)polyhedra. Then, using an appropriate classification scheme which—compared with the similar, geometrically motivated scheme, used originally by Grünbaum—is suggested rather by the group theoretical investigations in [4], it turns out that the list of examples given in [5] is essentially complete except for one additional polyhedron.So altogether—up to similarity—there are two classes of planar polyhedra, each consisting of 3 individuals and each class consisting of the Petrie duals of the other class, and there are ten classes of non planar polyhedra: two mutually Petrie dual classes of finite polyhedra, each consisting of 9 individuals, two mutually Petrie dual classes of infinite polyhedra which are contained between two parallel planes with each of those two classes consisting of three one-parameter families of polyhedra, two further mutually Petrie dual classes each of which consists of three one parameter families of polyhedra whose convex span is the whole 3-space, two further mutually Petrie dual classes consisting of three individuals each of which spanE
3 and two further classes which are closed with respect to Petrie duality, each containing 3 individuals, all spanningE
3, two of which are Petrie dual to each other, the remaining one being Petrie dual to itself.In addition, a new classification scheme for regular polygons inE
n
is worked out in §9. 相似文献
19.
Gloria Rinaldi 《Journal of Geometry》1995,54(1-2):148-154
A new transformation method for incidence structures was introduced in [8],an open problem is to characterize classical incidence structures obtained by transformation of others. In this work we give some, sufficient conditions to transform, with the procedure of [8],a unital embedded in a projective plane into another one. As application of this result we construct unitals in the Hall planes by transformation of the hermitian curves and we give necessary and sufficient conditions for the constructed unitals to be projectively equivalent. This allows to find different classes of not projectively equivalent Buekenhout's unitals, [2],and to find the class of unitals descovered by Grüning, [4],easily proving its embeddability in the dual of a Hall plane. Finally we prove that the affine unital associated to the unital of [4]is isomorphic to the affine hyperbolic hermitian curve.Work performed under the auspicies of G.N.S.A.G.A. and supported by 40% grants of M.U.R.S.T. 相似文献
20.
A. Gameiro Pais 《Aequationes Mathematicae》1986,30(1):223-238
The method presented in [4] for the solution of linear difference equations in a single variable is extended to some equations in two variables. Every linear combination of a given functionf and of its partial differences can be obtained by the discrete convolution product off by a suitable functionl (which depends on the considered linear combination), and we want to solve in a convolutional form difference equations in the whole plane. However, the convolution of two functions may not be possible if their supports contain half straight lines with opposite directions. To avoid this, we take four sets of functions corresponding to the quadrants such thatl belong to every set, every set endowed with the convolution and with the usual addition is a ring, and there is an inverse ofl in each of the four rings. This is attained by taking, for each ring, a set of functions whose supports belong to suitable cones. After choosing such rings, a very natural initial-value first-order Cauchy Problem (in partial differences) is reduced to a convolutional form. This is done either by a direct method or by introducing the forward difference functions
i
f(i=1,2) in a general way depending on the shape of the support off so that Laplace-like formulas with initial and final values) hold. Applications to difference equations in the whole plane and to partial differential problems are made. 相似文献