Exact controllability of a string to rest with a moving boundary

• Autorzy: Gugat Martin
• Języki publikacji: EN

Abstrakty

EN

We consider the problem of steering a finite string to the zero state in finite time from a given initial state by controlling the state at one boundary point while the other boundary point moves. As a possible application we have in mind the optimal control of a mining elevator, where the length of the string changes during the transportation process. During the transportation process, oscillations of the elevator-cable can occur that can be damped in this way. We present an exact controllability result for Dirichlet boundary control at the fixed end of the string that states that there exist exact controls for which the oscillations vanish after finite time. For the result we assume that the movements are Lipschitz continuous with a Lipschitz constant, whose absolute value is smaller than the wave speed. In the result, we present the minimal time, for which exact controllability holds, this time depending on the movement of the boundary point. Our results are based upon travelling wave solutions. We present a representation of the set of successful controls that steer the system to rest after finite time as the solution set of two point-wise equalities. This allows for a transformation of the optimal control problem to a form where no partial differential equation appears. This representation enables interesting insights into the structure of the successful controls. For example, exact bang-bang controls can only exist if the initial state is a simple function and the initial velocity is zero.

Słowa kluczowe

EN

pde constrained optimization; optimal control of pdes; optimal boundary control; wave equation; analytic solution; exact controllability; moving boundaries; mining elevator

• Wydawca: Systems Research Institute, Polish Academy of Sciences
• Czasopismo: Control and Cybernetics
• Rocznik: 2019
• Tom: Vol. 48, No. 1
• Strony: 69--87
• Opis fizyczny: Bibliogr. 27 poz., rys.

Twórcy

autor

Gugat Martin

• Friedrich-Alexander Universit¨at Erlangen-Nurnberg, Department Mathematik, Cauerstr. 11, 91058 Erlangen, Germany

Bibliografia

• [1] Avdonin, S. A. and Ivanov, S. S. (1995) Families of Exponentials. Cambridge University Press.
• [2] Bardos, C. and Chen, G. (1981) Control and stabilization for the wave equation. III: Domain with moving boundary. SIAM J. Control Optimization 19: 123-138.
• [3] Cui, L. and Song, L. (2014) Controllability for a Wave Equation with Moving Boundary. Journal of Applied Mathematics, Article ID 827698.
• [4] Delfour, M.C. and Zolesio, J.P. (2001) Shapes and Geometries: Analysis, Differential Calculus and Optimization. SIAM, Philadelphia.
• [5] Gugat, M. (2002) Analytic solutions of L∞–optimal control problems for the wave equation. Journal of Optimization Theory and Applications, 114: 397–421.
• [6] Gugat, M. (2003) Boundary Controllability between Sub– and Supercritical Flow. SIAM J. Control and Optimization, 42: 1056–1070.
• [7] Gugat, M. (2005) L1-optimal boundary control of a string to rest in finite time Recent Advances in Optimization. Proceedings of the 12th French-German-Spanish Conference on Optimization. Lecture Notes in Economics and Mathematical Systems. Springer-Verlag.
• [8] Gugat, M. and Leugering, G. (2002) Solutions of Lp–norm–minimal control problems for the wave equation. Computational and Applied Mathematics, 21: 227–244.
• [9] Gugat, M., Leugering, G. and Sklyar, G. (2005) Lp-optimal boundary control for the wave equation. SIAM J. Control and Optimization 44, 49-74.
• [10] Gugat, M. (2007) Optimal Energy Control in Finite Time by varying the Length of the String. SIAM J. Control and Optimization 46, 1705–1725.
• [11] Gugat, M. (2008) Optimal boundary feedback stabilization of a string with moving boundary. IMA Journal of Mathematical Control and Information 25, 111–121.
• [12] Gugat, M. and Sigalotti, M. (2010) Stars of vibrating strings: Switching boundary feedback stabilization. Networks and Heterogeneous Media 5, 299–314.
• [13] Gugat, M. and Sokolowski, J. (2013) A note on the approximation of Dirichlet boundary control problems for the wave equation on curved domains. Applicable Analysis 92, 2200-2214.
• [14] Gugat, M. and Tucsnak, M. (2011) An example for the switching delay feedback stabilization of an infinite dimensional system: The boundary stabilization of a string Systems and Control Letters 60, 226-233.
• [15] He, W., Ge, S.S. and Huang, D. (2015) Modeling and Vibration Control for a Nonlinear Moving String With Output Constraint. IEEE/ASME Transactions on Mechatronics 20, 1886-1897.
• [16] Krabs, W. (1982) Optimal control of processes governed by partial differential equations part ii: Vibrations. Zeitschrift fuer Operations Research, 26:63– 86.
• [17] Krabs, W. (1992) On Moment Theory and Controllability of One– Dimensional Vibrating Systems and Heating Processes. Lecture Notes in Control and Information Science 173, Springer–Verlag, Heidelberg.
• [18] LeVeque, R. J. (1999) Numerical Methods for Conservation Laws. Birkhaeuser, Basel.
• [19] Lions, J. L. (1988) Exact controllability, stabilization and perturbations of distributed systems. SIAM Review, 30: 1–68.
• [20] Malanowski, K. (1969) On time–optimal control of a vibrating string (Polish). Arch. Automat. Telemech., 14: 33–44.
• [21] Naumann, J. (2005) Transformation of Lebesgue Measure and Integral by Lipschitz mappings, Preprint Nr 2005 - 8. Humboldt-Universitat zu Berlin, Mathematisch-Naturwissenschaftliche Fakultat II, Institut fur Mathematik.
• [22] Russell, D. L. (1967) Nonharmonic Fourier Series in the Control Theory of Distributed Parameter Systems. Journal of Mathematical Analysis and Applications, 18, 542–560.
• [23] Truchi, C. and Zolesio, J. P. (1988) Wave equation in time periodical domain. Stabilization of flexible structures, Proc. ComCon Workshop, Montpellier, France, 1987, 282-294.
• [24] Valein, J. and Zuazua, E. (2009) Stabilization of the Wave Equation on 1-d Networks. SIAM J. Control and Optimization 48, 2771-2797.
• [25] Wang, J., Pi, Y. and Krstic, M. (2018) Balancing and suppression of oscillations of tension and cage in dual-cable mining elevators. Automatica 98, 223–238.
• [26] Wang, J., Tang, S.-X., Pi, Y. and Krstic, M. (2018) Exponential regulation of the anti-collocatedly disturbed cage in a wave PDE-modeled ascending cable elevator. Automatica 95, 122–136.
• [27] Zuazua, E. (2004) Optimal and Approximate Control of Finite-Difference Approximation Schemes for the 1D Wave equation. Rend. Mat. Appl. 24, 201–237.

Uwagi

PL

Opracowanie rekordu ze środków MNiSW, umowa Nr 461252 w ramach programu "Społeczna odpowiedzialność nauki" - moduł: Popularyzacja nauki i promocja sportu (2020).