Optimal Energy Control in Finite Time by varying the Length of the String

We consider a finite string where, at both end points, a homogeneous Dirichlet boundary condition holds. One boundary point is fixed, and the other is moving; hence the length of the string is changing in time. The string is controlled through the movement of this boundary point. We consider movements of the boundary that are Lipschitz continuous. Only movements for which at the given finite terminal time the string has the same length as at the beginning are admissible. Moreover, we impose an upper bound for the Lipschitz constant of the movement that is smaller than the speed of wave propagation. We consider the optimal control problem to find an admissible movement for which at the given terminal time the energy of the string is minimal. We give a sufficient condition for the existence and uniqueness of an optimal movement and construct an optimal control movement.

[1]  Werner Krabs,et al.  Optimal control of processes governed by partial differential equations part II: Vibrations , 1982, Z. Oper. Research.

[2]  Sergei Avdonin,et al.  Families of Exponentials: The Method of Moments in Controllability Problems for Distributed Parameter Systems , 1995 .

[3]  Carlos Castro,et al.  Unique Continuation and Control for the Heat Equation from an Oscillating Lower Dimensional Manifold , 2004, SIAM J. Control. Optim..

[4]  Jean-Paul Zolésio,et al.  Shape stabilization of wave equation , 1988 .

[5]  P. Duclos,et al.  STABILITY AND INSTABILITY OF THE WAVE EQUATION SOLUTIONS IN A PULSATING DOMAIN , 1998 .

[6]  J. Zolésio.  Boundary Control and Boundary Variations , 1987 .

[7]  M. Thoma,et al.  On Moment Theory and Controllability of One-Dimensional Vibrating Systems and Heating Processes , 1992 .

[8]  Dittrich,et al.  Instability in a classical periodically driven string. , 1994, Physical review. E, Statistical physics, plasmas, fluids, and related interdisciplinary topics.

[9]  David L. Russell,et al.  Nonharmonic Fourier series in the control theory of distributed parameter systems , 1967 .

[10]  A. Khapalov,et al.  Controllability of the wave equation with moving point control , 1995 .

[11]  Gignoux,et al.  Exponential growth of the energy of a wave in a 1D vibrating cavity: Application to the quantum vacuum. , 1996, Physical review letters.

[12]  Martin Gugat,et al.  Optimal boundary control of a string to rest in finite time with continuous state , 2006 .

[13]  M. Delfour,et al.  Shapes and Geometries: Analysis, Differential Calculus, and Optimization , 1987 .

[14]  Observability and stabilization of the vibrating string equipped with bouncing point sensors and actuators , 2001 .

[15]  Claude Bardos,et al.  Control and Stabilization for the Wave Equation, Part III: Domain with Moving Boundary , 1981 .

[16]  J. Lions.  Exact controllability, stabilization and perturbations for distributed systems , 1988 .

[17]  On the solution of the wave equation with moving boundaries , 1961 .