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Perturbation theory of observable linear systems

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The present work is motivated by the asymptotic control theory for a system of linear oscillators: the problem is to design a common bounded scalar control for damping all oscillators in asymptotically minimal time. The motion of the system is described in terms of a canonical system similar to that of the Pontryagin maximum principle. We consider the evolution equation for adjoint variables as a perturbed observable linear system. Due to the perturbation, the unobservable part of the state trajectory cannot be recovered exactly. We estimate the recovering error via the L 1-norm of perturbation. This allows us to prove that the control makes the system approach the equilibrium state with a strictly positive speed.

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  1. A. K. Fedorov and A. I. Ovseevich, “Asymptotic control theory for a system of linear oscillators,” arXiv:1308.6090 (2013).

    Google Scholar 

  2. A. I. Ovseevich and A. K. Fedorov, “Asymptotically optimal feedback control for a systemof linear oscillators,” Doklady Mathematics 88 (2), 613–617 (2013).

    Article  MATH  MathSciNet  Google Scholar 

  3. L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, The Mathematical Theory of Optimal Processes (Interscience Publishers, New York–London, 1962).

    MATH  Google Scholar 

  4. E. V. Goncharova and A. I. Ovseevich, “Comparative analysis of the asymptotic dynamics of reachable sets to linear systems,” Journal of Computer and Systems Sciences International 46 (4), 505–513 (2007).

    Article  MATH  MathSciNet  Google Scholar 

  5. L. Hörmander, Notions of Convexity (Birkhäuser, Boston, 1994).

    MATH  Google Scholar 

  6. A. I. Ovseevich, “Singularities of attainable sets,” Russian J. Math. Phys. 5 (3), 389–398 (1997).

    MATH  MathSciNet  Google Scholar 

  7. R. E. Kalman, “On the general theory of control systems,” in Proceedings of the First IFAC World Congress, Moscow (Butterworths, London, 1960), Vol. 1, pp. 481–492.

    Google Scholar 

  8. R. E. Kalman, “Mathematical description of linear dynamical systems,” J. SIAM, Series A, Control 1 (2), 152–192 (1963).

    MATH  MathSciNet  Google Scholar 

  9. R. Kalman, P. Falb, and M. Arbib, Topics in Mathematical System Theory (McGraw-Hill, New York, 1969).

    MATH  Google Scholar 

  10. P. Brunovsky, “A classification of linear controllable systems,” Kybernetika 6 (3), 176–188 (1970).

    MathSciNet  Google Scholar 

  11. E. M. Stein, “Functions of exponential type,” Annals ofMathematics 53 (3), 582–592 (1957).

    Google Scholar 

  12. S. Agmon, A. Douglis, and L. Nirenberg, “Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I,” Commun. Pure Appl. Math. 12 (4), 623–727 (1959).

    Article  MATH  MathSciNet  Google Scholar 

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Correspondence to A. K. Fedorov.

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Fedorov, A.K., Ovseevich, A.I. Perturbation theory of observable linear systems. Math Notes 98, 216–221 (2015).

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