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Jingrui Sun - One of the best experts on this subject based on the ideXlab platform.

  • open loop and closed loop solvabilities for stochastic linear quadratic optimal control problems
    2016
    Co-Authors: Jingrui Sun, Jiongmin Yong
    Abstract:

    This paper is concerned with a stochastic linear quadratic (LQ) optimal control problem. The notions of open-loop and closed-loop solvabilities are introduced. A simple example shows that these two solvabilities are different. Closed-loop solvability is established by means of solvability of the corresponding Riccati equation, which is implied by the uniform convexity of the quadratic cost functional. Conditions ensuring the convexity of the cost functional are discussed, including the issue of how negative the control weighting matrix-valued function $R(\cdot)$ can be. Finiteness of the LQ problem is characterized by the convergence of the solutions to a family of Riccati equations. Then, a Minimizing Sequence, whose convergence is equivalent to the open-loop solvability of the problem, is constructed. Finally, some illustrative examples are presented.

  • mean field stochastic linear quadratic optimal control problems open loop solvabilities
    2015
    Co-Authors: Jingrui Sun
    Abstract:

    This paper is concerned with a mean-field linear quadratic (LQ, for short) optimal control problem with deterministic coefficients. It is shown that convexity of the cost functional is necessary for the finiteness of the mean-field LQ problem, whereas uniform convexity of the cost functional is sufficient for the open-loop solvability of the problem. By considering a family of uniformly convex cost functionals, a characterization of the finiteness of the problem is derived and a Minimizing Sequence, whose convergence is equivalent to the open-loop solvability of the problem, is constructed. Then, it is proved that the uniform convexity of the cost functional is equivalent to the solvability of two coupled differential Riccati equations and the unique open-loop optimal control admits a state feedback representation in the case that the cost functional is uniformly convex. Finally, some examples are presented to illustrate the theory developed.

  • open loop and closed loop solvabilities for stochastic linear quadratic optimal control problems
    2015
    Co-Authors: Jingrui Sun, Jiongmin Yong
    Abstract:

    This paper is concerned with a stochastic linear quadratic (LQ, for short) optimal control problem. The notions of open-loop and closed-loop solvabilities are introduced. A simple example shows that these two solvabilities are different. Closed-loop solvability is established by means of solvability of the corresponding Riccati equation, which is implied by the uniform convexity of the quadratic cost functional. Conditions ensuring the convexity of the cost functional are discussed, including the issue that how negative the control weighting matrix-valued function R(s) can be. Finiteness of the LQ problem is characterized by the convergence of the solutions to a family of Riccati equations. Then, a Minimizing Sequence, whose convergence is equivalent to the open-loop solvability of the problem, is constructed. Finally, an illustrative example is presented.

Alexander J. Zaslavski - One of the best experts on this subject based on the ideXlab platform.

  • existence of a Minimizing Sequence of trajectory control pairs with bounded controls for linear control problems
    2009
    Co-Authors: Alexander J. Zaslavski
    Abstract:

    In this paper we study two large classes of finite-dimensional linear control systems which are identified with the corresponding complete metric spaces of integrands satisfying a growth condition. For most elements of the first space of integrands (in the sense of Baire category) we establish the existence of a Minimizing Sequence of trajectory-control pairs with bounded controls. We also establish that for most elements of the second space (in the sense of Baire category) the infimum on the full admissible class of trajectory-control pairs is equal to the infimum on a subclass of trajectory-control pairs whose controls are bounded by a certain constant.

  • NONOCCURRENCE OF THE LAVRENTIEV PHENOMENON FOR MANY INFINITE DIMENSIONAL LINEAR CONTROL PROBLEMS WITH NONCONVEX INTEGRANDS
    2008
    Co-Authors: Alexander J. Zaslavski
    Abstract:

    In this paper we establish nonoccurrence of gap for two large classes of innite- dimensional linear control systems in a Hilbert space with nonconvex integrands. These classes are identied with the corresponding complete metric spaces of integrands which satisfy a growth condition common in the literature. For most elements of the rst space of integrands (in the sense of Baire category) we establish the existence of a Minimizing Sequence of trajectory-control pairs with bounded controls. We also establish that for most elements of the second space (in the sense of Baire category) the inm um on the full admissible class of trajectory-control pairs is equal to the inm um on a subclass of trajectory-control pairs whose controls are bounded by a certain constant. AMS (MOS) Subject Classication. 49J27. Hilbert spaces with nonconvex integrands. These classes are identied with the corresponding complete metric spaces of integrands which satisfy a growth condition common in the literature. For most elements of the rst space of integrands (in the sense of Baire category) we establish the existence of a Minimizing Sequence of trajectory-control pairs with bounded controls. We also establish that for most elements of the second space (in the sense of Baire category) the inm um on the full admissible class of trajectory-control pairs is equal to the inm um on a subclass of trajectory-control pairs whose controls are bounded by a certain constant. The results of the paper show that for these classes of integrands the Lavrentiev

  • Nonoccurrence of gap for infinite-dimensional control problems with nonconvex integrands
    2006
    Co-Authors: Alexander J. Zaslavski
    Abstract:

    In this article we establish nonoccurrence of gap for two classes of infinite-dimensional control systems with nonconvex integrands. For the first class of integrands we show the existence of a Minimizing Sequence of trajectory-control pairs with bounded controls while for the second class we establish that an infimum on the full admissible class is equal to the infimum on a set of trajectory-control pairs with controls which are bounded by the same constant.

Jiongmin Yong - One of the best experts on this subject based on the ideXlab platform.

  • open loop and closed loop solvabilities for stochastic linear quadratic optimal control problems
    2016
    Co-Authors: Jingrui Sun, Jiongmin Yong
    Abstract:

    This paper is concerned with a stochastic linear quadratic (LQ) optimal control problem. The notions of open-loop and closed-loop solvabilities are introduced. A simple example shows that these two solvabilities are different. Closed-loop solvability is established by means of solvability of the corresponding Riccati equation, which is implied by the uniform convexity of the quadratic cost functional. Conditions ensuring the convexity of the cost functional are discussed, including the issue of how negative the control weighting matrix-valued function $R(\cdot)$ can be. Finiteness of the LQ problem is characterized by the convergence of the solutions to a family of Riccati equations. Then, a Minimizing Sequence, whose convergence is equivalent to the open-loop solvability of the problem, is constructed. Finally, some illustrative examples are presented.

  • open loop and closed loop solvabilities for stochastic linear quadratic optimal control problems
    2015
    Co-Authors: Jingrui Sun, Jiongmin Yong
    Abstract:

    This paper is concerned with a stochastic linear quadratic (LQ, for short) optimal control problem. The notions of open-loop and closed-loop solvabilities are introduced. A simple example shows that these two solvabilities are different. Closed-loop solvability is established by means of solvability of the corresponding Riccati equation, which is implied by the uniform convexity of the quadratic cost functional. Conditions ensuring the convexity of the cost functional are discussed, including the issue that how negative the control weighting matrix-valued function R(s) can be. Finiteness of the LQ problem is characterized by the convergence of the solutions to a family of Riccati equations. Then, a Minimizing Sequence, whose convergence is equivalent to the open-loop solvability of the problem, is constructed. Finally, an illustrative example is presented.

Komendarczyk R. - One of the best experts on this subject based on the ideXlab platform.

  • On Woltjer's force free minimizers and Moffatt's magnetic relaxation
    2021
    Co-Authors: Komendarczyk R.
    Abstract:

    In this note, we exhibit a situation where a stationary state of Moffatt's ideal magnetic relaxation problem is different than the corresponding force-free $L^2$ energy minimizer of Woltjer's variational principle. Such examples have been envisioned in Moffatt's seminal work on the subject and involve divergence free vector fields supported on collections of essentially linked magnetic tubes. Justification of Moffatt's examples requires the strong convergence of a Minimizing Sequence. What is proven in the current note is that there is a gap between the global minimum ({\em Woltjer's minimizer}) and the minimum over the weak $L^2$ closure of the class of vector fields obtained from a topologically non-trivial field by energy-decreasing diffeomorphisms. In the context of Taylor's conjecture, our result shows that the Woltjer's minimizer cannot be reached during the ideal MHD relaxation phase if the initial field has nontrivial topology. The result also applies beyond Moffatt's relaxation to any other relaxation process which evolves a divergence free field by means of energy-decreasing diffeomorphisms, such processes were proposed by Vallis et.al and more recently by Nishiyama.Comment: 7 pages, 1 figure, a terrible typo in the title corrected, plus small updates, submitted versio

  • On Wojtier's force free minimizers and Moffatt's magnetic relaxation
    2020
    Co-Authors: Komendarczyk R.
    Abstract:

    In this note, we exhibit a situation where a stationary state of Moffatt's ideal magnetic relaxation problem is different than the corresponding force-free $L^2$ energy minimizer of Wojtier's variational principle. Such examples have been envisioned in Moffatt's seminal work on the subject and involve divergence-free vector fields supported on collections of essentially linked magnetic tubes. Justification of Moffatt's examples requires a strong convergence of a Minimizing Sequence. What is proven in the current note is that there is a gap between the global minimum ({\em Wojtier's minimizer}) and the minimum over the weak $L^2$ closure of the class of vector fields obtained from a topologically non-trivial field by energy-decreasing diffeomorphisms. Consequently, our result applies beyond the Moffatt's relaxation to any other relaxation process which evolves a divergence-free field by means of energy-decreasing diffeomorphisms, such processes were proposed by Vallis et.al and more recently by Nishiyama.Comment: 7 pages, 1 figure, a few editorial changes, submitted versio

Luo Guimei - One of the best experts on this subject based on the ideXlab platform.