Optimal Internal Dissipation of a Damped Wave Equation Using a Topological Approach

Allaire, G., de Gournay, F., Jouve, F. and Toader, A. (2005). Structural optimization using topological and shape sensitivity analysis via a level-set method, Control and Cybernetics 34(1): 59-80.Search in Google Scholar

Allaire, G., Jouve, F. and Toader, A. (2004). Structural optimization using sensitivity analysis and a level-set method, Journal of Computational Physics 194(1): 363-393.10.1016/j.jcp.2003.09.032Search in Google Scholar

Banks, H., Ito, K. and Wang, B. (1991). Exponentially stable approximations of weakly damped wave equations, Estimation and Control of Distributed Parameter Systems (Vorau, 1990), International Series of Numerical Mathematics, Vol. 100, Birkhäuser, Basel, pp. 1-33.Search in Google Scholar

Bardos, C., Lebeau, G. and Rauch, J. (1992). Sharp sufficient conditions for the observation, control and stabilization from the boundary, SIAM Journal on Control and Optimization 30(5): 1024-1065.10.1137/0330055Search in Google Scholar

Burger, M. and Osher, S. (2005). A survey on level set methods for inverse problems and optimal design, European Journal of Applied Mathematics 16(2): 263-301.10.1017/S0956792505006182Search in Google Scholar

Cagnol, J. and Zolésio, J.-P. (1999). Shape derivative in the wave equation with Dirichlet boundary conditions, Journal of Differential Equations 158(2): 175-210.10.1006/jdeq.1999.3643Search in Google Scholar

Castro, C. and Cox, S.J. (2001). Achieving arbitrarily large decay in the damped wave equation, SIAM Journal on Control and Optimization 39(6): 1748-1755.10.1137/S0363012900370971Search in Google Scholar

Cohen, G.C. (2002). Higher-order Numerical Methods for Transient Wave Equations, Scientific Computation, Springer-Verlag, Berlin.10.1007/978-3-662-04823-8Search in Google Scholar

Degryse, E. and Mottelet, S. (2005). Shape optimization of piezoelectric sensors or actuators for the control of plates, ESAIM Control, Optimization and Calculus of Variations 11(4): 673-690.10.1051/cocv:2005025Search in Google Scholar

Delfour, M. and Zolesio, J. (2001). Shapes and Geometries— Analysis, Differential Calculus and Optimization, SIAM, Philadelphia, PA.Search in Google Scholar

Fahroo, F. and Ito, K. (1997). Variational formulation of optimal damping designs, Optimization Methods in Partial Differiential Equations (South Hadley, MA, 1996), Contemporary Mathematics, Vol. 209, American Mathematical Society, Providence, RI, pp. 95-114.Search in Google Scholar

Freitas, P. (1999). Optimizing the rate of decay of solutions of the wave equation using genetic algorithms: A counterexample to the constant damping conjecture, SIAM Journal on Control and Optimization 37(2): 376-387.10.1137/S0363012997329445Search in Google Scholar

Fulmanski, P., Laurain, A., Scheid, J.-F. and Sokołowski, J. (2008). Level set method with topological derivatives in shape optimization, International Journal of Computer Mathematics 85(10): 1491-1514.10.1080/00207160802033350Search in Google Scholar

Glowinski, R., Kinton, W. and Wheeler, M. (1989). A mixed finite element formulation for the boundary controllability of the wave equation, International Journal for Numerical Methods in Engineering 27(3): 623-635.10.1002/nme.1620270313Search in Google Scholar

Hébrard, P. and Henrot, A. (2003). Optimal shape and position of the actuators for the stabilization of a string. Optimization and control of distributed systems Systems and Control Letters 48(3-4): 199-209.Search in Google Scholar

Hébrard, P. and Henrot, A. (2005). A spillover phenomenon in the optimal location of actuators, SIAM Journal on Control and Optimization 44(1): 349-366.10.1137/S0363012903436247Search in Google Scholar

Henrot, A. and Pierre, M. (2005). Variation et optimisation de formes—Une analyse géométrique, Mathématiques et Applications, Vol. 48, Springer, Berlin.Search in Google Scholar

Lions, J. and Magenes, E. (1968). Problèmes aux Limites Non Homogènes et Applications, Dunod, Paris.Search in Google Scholar

López-Gómez, J. (1997). On the linear damped wave equation, Journal of Differential Equations 134(1): 26-45.10.1006/jdeq.1996.3209Search in Google Scholar

Maestre, F., Münch, A. and Pedregal, P. (2007). A spatio-temporal design problem for a damped wave equation, SIAM Journal on Applied Mathematics 68(1): 109-132.10.1137/07067965XSearch in Google Scholar

Münch, A. (2008). Optimal design of the support of the control for the 2-d wave equation: Numerical investigations, Mathematical Modelling and Numerical Analysis 5(2): 331-351.Search in Google Scholar

Münch, A. and Pazoto, A. (2007). Uniform stabilization of a numerical approximation of the locally damped wave equation, Control, Optimization and Calculus of Variations 13(2): 265-293.10.1051/cocv:2007009Search in Google Scholar

Münch, A., Pedregal, P. and Periago, F. (2006). Optimal design of the damping set for the stabilization of the wave equation, Journal of Differential Equations 231(1): 331-358.10.1016/j.jde.2006.06.009Search in Google Scholar

Münch, A., Pedregal, P. and Periago, F. (2009). Optimal internal stabilization of the linear system of elasticity, Archive Rational Mechanical Analysis, (to appear), DOI: 10.1007/s00205-008-0187-4.10.1007/s00205-008-0187-4Search in Google Scholar

Osher, S. and Fedkiw, R. (1996). Level Set Methods: Evolving Interfaces in Geometry, Fluid Mechanics, Computer Vision, and Materials Science, Cambridge University Press, Cambridge.Search in Google Scholar

Osher, S. and Fedkiw, R. (2003). Level Set Methods and Dynamic Implicit Surfaces, Applied Mathematical Sciences, Vol. 153, Springer-Verlag, New York, NY.Search in Google Scholar

Osher, S. and Sethian, J.A. (1988). Fronts propagating with curvature-dependent speed: Algorithms based on Hamilton-Jacobi formulations, Journal of Computational Physics 79(1): 12-49.10.1016/0021-9991(88)90002-2Search in Google Scholar

Sokołowski, J. and Żochowski, A. (1999). On the topological derivative in shape optimization, SIAM Journal on Control and Optimization 37(4): 1251-1272.10.1137/S0363012997323230Search in Google Scholar

Wang, M.Y., Wang, X. and Guo, D. (2003). A level set method for structural topology optimization, Computer Methods in Applied Mechanics and Engineering 192(1-2): 227-246.10.1016/S0045-7825(02)00559-5Search in Google Scholar

Zolésio, J.-P. and Truchi, C. (1988). Shape stabilization of wave equation, Boundary Control and Boundary Variations (Nice, 1986), Lectures Notes in Computer Science, Vol. 100, Springer, Berlin, pp. 372-398.Search in Google Scholar