Generalized solutions for singular optimal control problems

This text presents a complete theory of existence/uniqueness and the structure of generalized solutions for singular linear-quadratic optimal control problems. Generalized optimal controls are distributions of order r and the corresponding generalized trajectories are distributions of order (r − 1). r is the “order of singularity” of the problem, an integer no greater than the dimension of the state space. Its value is obtained through a certain reduction procedure. In the final section, some perspectives and partial results concerning the extension of these results to nonlinear problems are briefly discussed. Translated from Sovremennaya Matematika. Fundamental’nye Napravleniya (Contemporary Mathematics. Fundamental Directions), Vol. 27, Optimal Control, 2007.     

An extension of the Caratheodory—Kalman variational method

We apply the Caratheodory method to variational problems posed in a Banach space. A treatment of the problem of Lagrange leads to the necessary conditions for constrained optimal control problems. Specialization of the space leads to recovery of the usual necessary conditions and extension of the necessary conditions to a class of problems for systems with distributed parameters. The spaces are not restricted to be reflexive, thus permitting consideration of control problems in queuing theory and transport theory. An application to certain queuing processes completes the paper.This work was performed partly under the auspices of the US Atomic Energy Commission and partly under National Science Foundation Grant No. GP-3900.     

Generalized solutions of initial-boundary value problems for second-order hyperbolic systems

The method of boundary integral equations is developed as applied to initial-boundary value problems for strictly hyperbolic systems of second-order equations characteristic of anisotropic media dynamics. Based on the theory of distributions (generalized functions), solutions are constructed in the space of generalized functions followed by passing to integral representations and classical solutions. Solutions are considered in the class of singular functions with discontinuous derivatives, which are typical of physical problems describing shock waves. The uniqueness of the solutions to the initial-boundary value problems is proved under certain smoothness conditions imposed on the boundary functions. The Green’s matrix of the system and new fundamental matrices based on it are used to derive integral analogues of the Gauss, Kirchhoff, and Green formulas for solutions and solving singular boundary integral equations.     

The unique solvability of certain multiplicative-convolution equations

In the class of distributions of slow (moderate) growth we consider a class of equations with operations of convolution and multiplication on the real axis. This class contains convolution equations, in particular, ordinary differential equations with constant coefficients, equations in finite differences, functional differential equations with constant coefficients and shifts, and pair differential equations. By virtue of the analytic representation theory for distributions of moderate growth (the Hilbert or Cauchy transform) the class of equations under consideration is equivalent to the class of boundary value problems of the Riemann type, where an equation corresponds to a boundary value condition in the sense of distributions of moderate growth. As a research technique we use the Fourier transform, the generalized Fourier transform (the Carleman-Fourier transform), and the theory of convolution equations in the space of distributions of moderate growth.     

A space of generalized distributions

In this paper we use a duality method to introduce a new space of generalized distributions. This method is exactly the same introduced by Schwartz for the distribution theory. Our space of generalized distributions contains all the Schwartz distributions and all the multipole series of physicists and is, in a certain sense, the smallest space containing all these series. To The Memory of Laurent Schwartz     

A new look at the Lagrange method for continuous-time stochastic optimization

We formulate a Lagrange method for continuous-time stochastic optimization in an appropriate normed space by using a proper stochastic process as the Lagrange multiplier.The obtained optimality conditions are applied to different types of problems.Some examples selected from control theory and economic theory are studied to test and illustrate the potential applications of the method.     

New versions of the Colombeau algebras

We construct some versions of the Colombeau theory. In particular, we construct the Colombeau algebra generated by harmonic (or polyharmonic) regularizations of distributions connected with a half‐space and by analytic regularizations of distributions connected with an octant. Unlike the standard Colombeau's scheme, our theory has new generalized functions that can be easily represented as weak asymptotics whose coefficients are distributions, i.e., in form of asymptotic distributions . The algebra of asymptotic distributions generated by the linear span of associated homogeneous distributions (in the one‐dimensional case) which we constructed earlier [9] can be embedded as a subalgebra into our version of Colombeau algebra. The representation of distributional products in the form of weak asymptotic series proved very useful in solving problems which arise in the theory of discontinuous solutions of hyperbolic systems of conservation laws [10]–[16], [49] and [50]. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

SEARCH THEORY IN NATURAL RESOURCE MODELING

The role of search theory in the exploitation of natural resources is discussed in this paper. After a brief history and taxonomy of search problems, the mathematics of search is discussed. This includes underlying probability distributions, the differential equations of search, Bayesian use of search information and optimization problems in search theory. The theory is illustrated by applications in fisheries, pest control, animal foraging, and oil and mineral exploration.     

A quantitative measure of compactness and similarity in a competitive space

Measures of similarity between objects in a metric space and a competitive space are described. The function of rival similarity is proposed as a similarity measure used in the problems of classification and pattern recognition. This function enables us to develop effective algorithms for solving all major problems of data mining, to obtain a quantitative estimate of pattern compactness and informativity of the feature space, and to construct simply interpreted decision rules. The method is applicable to the problems with arbitrary number of patterns and any character of their distributions, and it can be also used for solving ill-conditioned problems.     

Regularized extragradient method in multicriteria control problems with inaccurate data

The class of Hilbert space multicriteria optimization problems considered in the paper includes control problems for various dynamical systems with lumped as well as distributed parameters. An equilibrium point is sought under the assumption that the criteria and their derivatives are known approximately. We use a regularized extragradient method and prove its convergence. As a sample application of the general theory, we consider a control problem for a parabolic equation with two criteria.     

Stochastic optimization theory in Hilbert spaces—1

We study a class of stochastic optimization problems in which the state as well as the observation spaces are permitted to be (Hilbert spaces) of non-finite dimension. Although there have been previous attempts in the Hilbert space setting, our results, techniques, as well as applications, are totally different. We initiate the use of Gauss measure on a Hilbert space even though it is only finitely additive; and an associated theory of white noise, in contrast to the Wiener process theory, which is novel even in the finite dimensional case. We only treat time-invariant systems, but no strong ellipticity or coercivity conditions are used; we exploit the theory of semigroups of operators in contrast to the Lions-Magenes theory. A key result involves a far-reaching generalization of the Factorization theorem of Krein. We apply the results to the problem of boundary observation and control for partial differential equations. By the creation of a special state space, we can apply the theory to problems in which the state equations are finitedimensional but the noise does not have a rational spectrum. In a final section, we present a stochastic theory for inverse problems (System Identification) in the Hilbert space setting. The basic theoretical problem is the calculation of R-N derivatives for finitely additive measures. A fundamental result concerns Identifiability; in particular the identifiability of diffusion coefficients from boundary data is treated here for the first time.     

Robinsches und pragersches problem der potentialtheorie bei beliebiger dimension

The Robin problem of potential theory in three dimensions is concerned with the representation of circulation-free vector fields, given in a region B?R³, by distributions of sources on the region B and its boundary [d]B in the sense of the Newton-Coulomb law; the Parger problem is concerned with the representation of source-free vector fields by distributions of vortices in the sense of the Biot-Savart law. The Robin problem a corresponding result on the Prager problem. The aim of this paper is to generalize the Robin and Prager representation problems to skew-symmetric tensor fields of arbitrary rank in the Euclidean space of arbitrary dimension, and to obtain a better understanding of the duality of these problems. The main results are constructive proofs of the existence and uniqueness of the solutions of the general Robbin and Prager problems. This is done by reducing these representation problems to equivalent integral equation formulations and applying Fredholm's alternative. For this recent results of the author on the Dirichlet and Neumann problems is governed by the so-called Hodge duality operator: theorems on the Prager problem are obtained from corresponding theorems on the Robin problem by applying the Hodge duality operator.     

三维井眼轨道设计模型及应用

讨论了两种典型的三维井眼轨道设计问题 .建立了设计的一般数学模型 ,利用空间矢量分析理论得到了约束变量间的解析表达式和井眼轨道计算式 .这种新方法避免了求解多维非线性方程组 ,设计计算简单精确 .该设计模型成功地解决了多约束条件下的三维井眼轨道设计这一难题 ,具有普遍适用性 .可广泛用于定向井、水平井和多目标井的井眼轨道设计 ,为井眼轨道控制提供了更为准确的理论依据 .     

Quadratic control problems

The present paper is concerned with the study of quadratic control problems on linear spaces. In particular, we are concerned with the conditions under which a quadratic criterion function is positive on certain linear spaces. This involves the elementary theory of conjugate and focal points, the existence of a conjugate system with a nonvanishing determinant, and the existence of extremal fields. The results given are in part a translation into control language of known theory for the problem of Bolza. The method used is based on the Hilbert space techniques developed earlier by the author.     

Optimal Control and Applications to Aerospace: Some Results and Challenges

This article surveys the usual techniques of nonlinear optimal control such as the Pontryagin Maximum Principle and the conjugate point theory, and how they can be implemented numerically, with a special focus on applications to aerospace problems. In practice the knowledge resulting from the maximum principle is often insufficient for solving the problem, in particular because of the well-known problem of initializing adequately the shooting method. In this survey article it is explained how the usual tools of optimal control can be combined with other mathematical techniques to improve significantly their performances and widen their domain of application. The focus is put onto three important issues. The first is geometric optimal control, which is a theory that has emerged in the 1980s and is combining optimal control with various concepts of differential geometry, the ultimate objective being to derive optimal synthesis results for general classes of control systems. Its applicability and relevance is demonstrated on the problem of atmospheric reentry of a space shuttle. The second is the powerful continuation or homotopy method, consisting of deforming continuously a problem toward a simpler one and then of solving a series of parameterized problems to end up with the solution of the initial problem. After having recalled its mathematical foundations, it is shown how to combine successfully this method with the shooting method on several aerospace problems such as the orbit transfer problem. The third one consists of concepts of dynamical system theory, providing evidence of nice properties of the celestial dynamics that are of great interest for future mission design such as low-cost interplanetary space missions. The article ends with open problems and perspectives.     

Stochastic distributed systems with point observations and boundary control: an abstract theory

There already exists a fairly complete theory for the problems of estimation and stochastic optimal control for linear distributed parameter systems, with Gaussian or non Gaussian noise disturbance. In [8] and [12] generalizations of the familiar finite dimensional results of the Kalman-Bucy filter and the separation principle are obtained using an abstract input-output Hilbert space representation for the system. However, in [8] and [12] all the input operators are assumed to be bounded and so it does not cover the important practical cases of control and noise on submanifolds of the spatial domain or point observations. Here we introduce unbounded system operators in the abstract input-output Hilbert space representation and thus extend all the results of [8] and [12] to allow for point observations and noise and control on submanifolds including the boundary. The theory is illustrated by several examples.     

Vector-valued Modulation Spaces and Localization Operators with Operator-valued Symbols

We study the short-time Fourier transformation, modulation spaces, Gabor representations and time-frequency localization operators, for functions and tempered distributions that have as range space a Banach or a Hilbert space. In the Banach space case the theory of modulation spaces contains some modifications of the scalar-valued theory, depending on the Banach space. In the Hilbert space case the modulation spaces have properties similar to the scalar-valued case and the Gabor frame theory essentially works. For localization operators in this context symbols are operator-valued. We generalize two results from the scalar-valued theory on continuity on certain modulation spaces when the symbol belongs to an L^p,q space and M^∞, respectively. The first result is true for any Banach space as range space, and the second result is true for any Hilbert space as range space.     

Error Estimates for the Approximation of a Class of Optimal Control Systems Governed by Linear PDEs

This paper deals with a class of optimal control problems in which the system is governed by a linear partial differential equation and the control is distributed and with constraints. The problem is posed in the framework of the theory of optimal control of systems. A numerical method is proposed to approximate the optimal control. In this method, the state space as well as the convex set of admissible controls are discretized. An abstract error estimate for the optimal control problem is obtained that depends on both the approximation of the state equation and the space of controls. This theoretical result is illustrated by some numerical examples from the literature.     

On Numerical Approximations of Fractional and Nonlocal Mean Field Games

We construct numerical approximations for Mean Field Games with fractional or nonlocal diffusions. The schemes are based on semi-Lagrangian approximations of the underlying control problems/games along with dual approximations of the distributions of agents. The methods are monotone, stable, and consistent, and we prove convergence along subsequences for (i) degenerate equations in one space dimension and (ii) nondegenerate equations in arbitrary dimensions. We also give results on full convergence and convergence to classical solutions. Numerical tests are implemented for a range of different nonlocal diffusions and support our analytical findings.

     

Distributional solution theory for linear DAEs

A solution theory for switched linear differential–algebraic equations (DAEs) is developed. To allow for non–smooth coordinate transformation, the coefficients matrices may have distributional entries. Since also distributional solutions are considered it is necessary to define a suitable multiplication for distribution. This is achieved by restricting the space of distributions to the smaller space of piecewise–smooth distributions. Solution formulae for two special DAEs, distributional ordinary differential equations (ODEs) and pure distributional DAEs, are given. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)