\`x^2+y_1+z_12^34\`
Advanced Search
Article Contents
Article Contents

Optimal control of dynamical systems with polynomial impulses

Abstract / Introduction Related Papers Cited by
  • The paper is devoted to the $BV$-relaxation of a dynamical system, whose right-hand side is a $p$th degree polynomial with rational powers of control under a uniform bound on its $L_p$-norm, and coefficients containing usual measurable bounded control.
        Under natural convexity assumptions, we give an explicit representation of generalized solutions to the control system by a measure differential equation. The main results concern an optimal impulsive control problem for the relaxed system: We establish the existence of a minimizer, and give necessary optimality conditions in the form of a Maximum Principle.
    Mathematics Subject Classification: Primary: 49N25, 49K99; Secondary: 49J99.

    Citation:

    \begin{equation} \\ \end{equation}
  • [1]

    A. Arutyunov, D. Karamzin and F. Pereira, On constrained impulsive control problems, J. Math. Sci., 165 (2010), 654-688.doi: 10.1007/s10958-010-9834-z.

    [2]

    A. Arutyunov, D. Karamzin and F. Pereira, A nondegenerate Maximum Principle for the impulse control problem with state constraints, SIAM J. Control Optim., 43 (2005), 1812-1843.doi: 10.1137/S0363012903430068.

    [3]

    J.-P. Aubin and A. Cellina, Differential Inclusions, Springer-Verlag, Berlin, 1984.doi: 10.1007/978-3-642-69512-4.

    [4]

    A. Bressan, Impulsive control of Lagrangian systems and locomotion in fluids, Discr. Cont. Dynam. Syst., 20 (2008), 1-35.doi: 10.3934/dcds.2008.20.1.

    [5]

    A. Bressan and M. Motta, A class of mechanical systems with some coordinates as controls. A reduction of certain optimization problems for them. Solution methods, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Mem., 2 (1993), 30pp.

    [6]

    A. Bressan and F. Rampazzo, Moving constraints as stabilizing controls in classical mechanics, Arch. Ration. Mech. Anal., 196 (2010), 97-141.doi: 10.1007/s00205-009-0237-6.

    [7]

    A. Bressan and F. Rampazzo, On systems with quadratic impulses and their application to Lagrangean mechanics, SIAM J. Control Optim., 31 (1993), 1205-1220.doi: 10.1137/0331057.

    [8]

    A. Bressan and F. Rampazzo, Impulsive control systems without commutativity assumptions, J. Optim. Theory Appl., 81 (1994), 435-457.doi: 10.1007/BF02193094.

    [9]

    V. Dykhta, Impulse-trajectory extension of degenerate optimal control problems, IMACS Ann. Comput. Appl. Math., 8 (1990), 103-109.

    [10]

    V. Dykhta and O. Samsonyuk, A maximum principle for smooth optimal impulsive control problems with multipoint state constraints, Comput. Math. Math. Phys., 49 (2009), 942-957.doi: 10.1134/S0965542509060050.

    [11]

    V. Gurman, On optimal processes with unbounded derivatives, Autom. Remote Control, 17 (1972), 14-21.

    [12]

    A. Ioffe and V. Tikhomirov, Theory of Extremal Problems, North-Holland, Amsterdam, 1979.

    [13]

    D. Karamzin, Necessary conditions of the minimum in an impulse optimal control problem, J. Math. Sci., 139 (2006), 7087-7150.doi: 10.1007/s10958-006-0408-z.

    [14]

    B. Miller, The generalized solutions of nonlinear optimization problems with impulse control, SIAM J. Control Optim., 34 (1996), 1420-1440.doi: 10.1137/S0363012994263214.

    [15]

    B. Miller and E. Rubinovich, Impulsive Control in Continuous and Discrete- Continuous Systems, Kluwer Academic / Plenum Publishers, New York, 2001.

    [16]

    M. Motta and F. Rampazzo, Space-time trajectories of nonlinear systems driven by ordinary and impulsive controls, Differential Integral Equations, 8 (1995), 269-288.

    [17]

    M. Motta and C. Sartori, Asymptotic problems in optimal control with a vanishing Lagrangian and unbounded data, (2014) [published online as arXiv:1406.7655v1].

    [18]

    P. Pedregal and J. Tiago, Existence results for optimal control problems with some special nonlinear dependence on state and control, SIAM J. Control Optim., 48 (2009), 415-437.doi: 10.1137/08071805X.

    [19]

    F. Rampazzo and C. Sartori, Hamilton-Jacobi-Bbellman equations with fast gradient-dependence, Indiana Univ. Math. J., 49 (2000), 1043-1077.doi: 10.1512/iumj.2000.49.1736.

    [20]

    R. Rishel, An extended Pontryagin principle for control systems whose control laws contain measures, J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1965), 191-205.doi: 10.1137/0303016.

    [21]

    R. Vinter and F. Pereira, A maximum principle for optimal processes with discontinuous trajectories, SIAM J. Control Optim., 26 (1988), 205-229.doi: 10.1137/0326013.

    [22]

    J. Warga, Relaxed variational problems, J. Math. Anal. Appl., 4 (1962), 111-128.doi: 10.1016/0022-247X(62)90033-1.

    [23]

    J. Warga, Optimal Control of Differential and Functional Equations, Academic Press, New York, 1972.

    [24]

    J. Warga, Variational problems with unbounded controls, J. SIAM Control Ser. A, 3 (1965), 424-438.doi: 10.1137/0303028.

    [25]

    S. Zavalischin and A. Sesekin, Dynamic Impulse Systems: Theory and Applications, Kluwer Academic Publishers, Dorderecht, 1997.doi: 10.1007/978-94-015-8893-5.

  • 加载中
SHARE

Article Metrics

HTML views() PDF downloads(109) Cited by(0)

Access History

Other Articles By Authors

Catalog

    /

    DownLoad:  Full-Size Img  PowerPoint
    Return
    Return