Time-optimal torus theorem and control of spin systems

Previous article Next article The Closed-Loop Time Optimal Control. II: StabilityPavol BrunovskýPavol Brunovskýhttps://doi.org/10.1137/0314013PDFBibTexSections ToolsAdd to favoritesExport CitationTrack CitationsEmail SectionsAboutAbstractThe problem, to which extent does the closed-loop time-optimal control of a linear system fulfill its task if the system is subject to small perturbations, is studied.[1] Pavol Brunovský, The closed-loop time-optimal control. I. Optimality, SIAM J. Control, 12 (1974), 624–634 MR0355719 (50:8193) 0301.49004 LinkISIGoogle Scholar[2] Pavol Brunovský, On the best stabilizing control under a given class of perturbations, Czechoslovak Math. J., 15 (90) (1965), 329–369 MR0181518 (31:5747) 0154.10005 Google Scholar[3] H. Hermes, J. Hale and , J. LaSalle, Discontinuous vector fields and feedback controlDifferential Equations and Dynamical Systems (Proc. Internat. Sympos., Mayaguez, P. R., 1965), Academic Press, New York, 1967, 155–165 MR0222424 (36:5476) 0183.15905 Google Scholar[4] T. Ważewski, On an optimal control problem, Differential Equations and Their Applications (Proc. Conf., Prague, 1962), Publ. House Czechoslovak Acad. Sci., Prague, 1965, 229–242 MR0201758 (34:1640) 0133.06005 Google Scholar[5] B. N. Pšeničnyj, Line jny je differencial'ny je igry, Avtomat. i Telemeh., (1968), 65–78 Google Scholar[6] A. F. Filippov, On certain questions in the theory of optimal control, Vestnik Moskov. Univ. Ser. I Mat. Meh., (1959), 25–32, English transl., this Journal, 1 (1962), pp. 76–84 Google Scholar Previous article Next article FiguresRelatedReferencesCited ByDetails Discrete and Continuous Dynamical Systems, Vol. 35, No. 9 | 2015 Cross Ref Measurement stability of time-optimal feedback control of two-input strictly normal linear systemsJournal of Differential Equations, Vol. 86, No. 1 | 1 Jul 1990 Cross Ref Local Time-Optimal Feedback Control of Strictly Normal Two-Input Linear SystemsSIAM Journal on Control and Optimization, Vol. 27, No. 1 | 14 July 2006AbstractPDF (3497 KB)Sensitivity Analysis of Time Optimal Control SystemsIFAC Proceedings Volumes, Vol. 17, No. 2 | 1 Jul 1984 Cross Ref Multivalued mappingsJournal of Soviet Mathematics, Vol. 24, No. 6 | 1 Mar 1984 Cross Ref Stability with regime switchingJournal of Economic Theory, Vol. 29, No. 1 | 1 Feb 1983 Cross Ref Measurement stability of third-order time-optimal control systemsJournal of Differential Equations, Vol. 36, No. 1 | 1 Apr 1980 Cross Ref Discontinuous differential equations, IIJournal of Differential Equations, Vol. 32, No. 2 | 1 May 1979 Cross Ref Volume 14, Issue 1| 1976SIAM Journal on Control and Optimization1-188 History Submitted:20 July 1973Published online:03 August 2006 InformationCopyright © 1976 © Society for Industrial and Applied MathematicsPDF Download Article & Publication DataArticle DOI:10.1137/0314013Article page range:pp. 156-162ISSN (print):0363-0129ISSN (online):1095-7138Publisher:Society for Industrial and Applied Mathematics

In the paper, the method of straight lines approximately solves one class of optimal control problems for systems, the behavior of which is described by a nonlinear equation of parabolic type and a set of ordinary differential equations. Control is carried out using distributed and lumped parameters. Distributed control is included in the partial differential equation, and lumped controls are contained both in the boundary conditions and in the right-hand side of the ordinary differential equation. The convergence of the solutions of the approximating boundary value problem to the solution of the original one is proved when the step of the grid of straight lines tends to zero, and on the basis of this fact, the convergence of the approximate solution of the approximating optimal problem with respect to the functional is established. A constructive scheme for constructing an optimal control by a minimizing sequence of controls is proposed. The control of the process in the approximate solution of a class of optimization problems is carried out on the basis of the Pontryagin maximum principle using the method of straight lines. For the numerical solution of the problem, a gradient projection scheme with a special choice of step is used, this gives a converging sequence in the control space. The numerical solution of one variational problem of the mentioned type related to a one-dimensional heat conduction equation with boundary conditions of the second kind is presented. An inequality-type constraint is imposed on the control function entering the right-hand side of the ordinary differential equation. The numerical results obtained on the basis of the compiled computer program are presented in the form of tables and figures. The described numerical method gives a sufficiently accurate solution in a short time and does not show a tendency to «dispersion». With an increase in the number of iterations, the value of the functional monotonically tends to zero

A fourth-order model is developed for a crane-load system. From this model a time optimal control scheme with input constraints is investigated for transferring the crane load with swing angle regulation. Rather than following a conventional numerical method approach for solving the two point boundary value problem, a rule-based approach inspired by the manual crane operation is proposed. It is shown that the time optimal control problem becomes the determination of the switching times. Based on the heuristics obtained, a set of expert rules is generated for the adjustment of the switching times. A simple iteration method for obtaining the optimal switching times is presented which provides an alternative solution when there is no analytical expression for a solution. From collected data a lookup table is generated. By incorporating an interpolation algorithm it is possible to transfer the crane load an arbitary distance without load pendulation at the destination in minimum time.

An optimal control problem for systems described by Fourier problem for nonlinear parabolic equations is studied. Control functions occur in the coefficients of the state equations. The existence of the optimal control in the case of final observation is proved.

In this paper, the bang–bang property and the existence of the solution for the time optimal control of a specific kind of microwave heating problem are studied. The mathematical model for the time optimal control problem (P) of microwave heating is formulated, in which the governing equations of the controlled system are the weak coupling system of Maxwell equations and the heat equation. Based on a new observability estimate of the heat controlled system, the controllability of the system is derived by proving the null controllability of an equivalent system. The existence of the time optimal control of microwave heating is acquired under some stated assumptions, enabling the bang–bang property of the time optimal control problem (P) to be achieved.

This paper is devoted to study time optimal control problem for deployment of a gravity gradient stabilized space structure. Schematic model of the space structure is a rigid main body equipped with mass-less booms tipped with point masses. Numerical results for three initial pitch angles show that time-optimal deployment control to eliminate pitch motion is the bang-bang control that consists of five phases;(a) simultaneous extension towards yaw and roll axis, (b) extension toward yaw axis, (c) suspension of extension, (d) extension towards roll axis, and finally (e) simultaneous extension towards yaw and roll axis, again.

Hierarchical time-optimal control of a continuous copolymerization reactor during start-up or grade change operation using genetic algorithms

We prove the following theorem: Let K be a connected symmetric subgroup of the group of isometries of a compact connected globally Riemannian symmetric space M.Then, K acts in a variationally complete manner on M. (We refer to the work of R. Bott and H. Samelson [l;2] for a definition of this concept and applications to topology.)Let G be a compact, connected Lie group, L a connected symmetric subgroup of G. Bott and Samelson have proved the following: (1) L acts in a variationally complete manner on G/L, the right coset space, (2) L acts via the linear isotropy representation in a variationally complete way on the tangent space at a point of G/L.Our theorem then generalizes (1), but does not imply (2).The method is essentially the same as Bott's [l], but uses a Lie algebra approach in a stronger way.The proof then provides an exposition of Bott's very important result from a Lie-algebra point of view.Notice also that it suggests the general program of studying the Jacobi-fields on a symmetric space as a problem of Lie algebra theory, even in the nonvariational complete case.1.

On Symmetric Cauchy–Riemann Manifolds

A Time and Energy Optimal Control of Industrial Robots

A time-optimal single-step velocity response control scheme for field-oriented induction machines is implemented, taking magnetic saturation of the rotor flux into account. The scheme is based on the analytical solutions of the time-optimal control problem. High velocity response is achieved, despite field-weakening, for low acoustic noise or high efficiency drives. Optimal control is realized by using a lookup table that is included in the indirect field-oriented control system. The problems of rotor speed and constraint on rate of change of acceleration or deceleration are also discussed. The merit of the time-optimal control scheme as compared with other control strategies is described. >

This paper is mainly concerned with a class of optimal control problems of systems governed by the nonlinear dynamic systems on time scales. Introducing the reasonable weak solution of nonlinear dynamic systems, the existence of the weak solution for the nonlinear dynamic systems on time scales and its properties are presented. Discussing L1-strong-weak lower semicontinuity of integral functional, we give sufficient conditions for the existence of optimal controls. Using integration by parts formula and Hamiltonian function on time scales, the necessary conditions of optimality are derived respectively. Some examples on continuous optimal control problems, discrete optimal control problems, mathematical programming and variational problems are also presented for demonstration.

In this paper, the finite difference method is applied to the optimal control problem of system governed by non-linear Schrödinger equation. For this purpose, the finite difference scheme is constituted for considered optimal control problem. Obtained an estimation for the solution of this difference scheme, the error of the difference scheme is evaluated. Finally, the convergence according to the functional of the finite difference approximations is proved.

Optimal control problems in hybrid systems with active singularities

Time-varying bang–bang property of time optimal controls for heat equation and its application