# American Institute of Mathematical Sciences

November  2007, 8(4): 925-941. doi: 10.3934/dcdsb.2007.8.925

## On the existence of time optimal controls for linear evolution equations

 1 Yangtze Center of Mathematics, Sichuan University, Chengdu 610064, China 2 School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China 3 Academy of Mathematics and Systems Sciences, Chinese Academy of Sciences, Beijing 100190, China

Received  November 2006 Revised  May 2007 Published  August 2007

This work concerns with the existence of the time optimal controls for some linear evolution equations without the a priori assumption on the existence of admissible controls. Both global and local existence results are presented. Some necessary conditions, sufficient conditions, and necessary and sufficient conditions for the existence of time optimal controls are derived by establishing the relationship between controllability and time optimal control problems.
Citation: Kim Dang Phung, Gengsheng Wang, Xu Zhang. On the existence of time optimal controls for linear evolution equations. Discrete & Continuous Dynamical Systems - B, 2007, 8 (4) : 925-941. doi: 10.3934/dcdsb.2007.8.925
 [1] Christian Clason, Barbara Kaltenbacher. Avoiding degeneracy in the Westervelt equation by state constrained optimal control. Evolution Equations & Control Theory, 2013, 2 (2) : 281-300. doi: 10.3934/eect.2013.2.281 [2] Ulisse Stefanelli, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of a rate-independent evolution equation via viscous regularization. Discrete & Continuous Dynamical Systems - S, 2017, 10 (6) : 1467-1485. doi: 10.3934/dcdss.2017076 [3] Kazimierz Malanowski, Helmut Maurer. Sensitivity analysis for state constrained optimal control problems. Discrete & Continuous Dynamical Systems - A, 1998, 4 (2) : 241-272. doi: 10.3934/dcds.1998.4.241 [4] Jiongmin Yong. Time-inconsistent optimal control problems and the equilibrium HJB equation. Mathematical Control & Related Fields, 2012, 2 (3) : 271-329. doi: 10.3934/mcrf.2012.2.271 [5] David González-Sánchez, Onésimo Hernández-Lerma. On the Euler equation approach to discrete--time nonstationary optimal control problems. Journal of Dynamics & Games, 2014, 1 (1) : 57-78. doi: 10.3934/jdg.2014.1.57 [6] Qing Tang. On an optimal control problem of time-fractional advection-diffusion equation. Discrete & Continuous Dynamical Systems - B, 2020, 25 (2) : 761-779. doi: 10.3934/dcdsb.2019266 [7] Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control & Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305 [8] Mohamed Ouzahra. Controllability of the semilinear wave equation governed by a multiplicative control. Evolution Equations & Control Theory, 2019, 8 (4) : 669-686. doi: 10.3934/eect.2019039 [9] Bin Li, Kok Lay Teo, Cheng-Chew Lim, Guang Ren Duan. An optimal PID controller design for nonlinear constrained optimal control problems. Discrete & Continuous Dynamical Systems - B, 2011, 16 (4) : 1101-1117. doi: 10.3934/dcdsb.2011.16.1101 [10] M. Arisawa, P.-L. Lions. Continuity of admissible trajectories for state constraints control problems. Discrete & Continuous Dynamical Systems - A, 1996, 2 (3) : 297-305. doi: 10.3934/dcds.1996.2.297 [11] Lili Chang, Wei Gong, Guiquan Sun, Ningning Yan. PDE-constrained optimal control approach for the approximation of an inverse Cauchy problem. Inverse Problems & Imaging, 2015, 9 (3) : 791-814. doi: 10.3934/ipi.2015.9.791 [12] Piernicola Bettiol. State constrained $L^\infty$ optimal control problems interpreted as differential games. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3989-4017. doi: 10.3934/dcds.2015.35.3989 [13] Hee-Dae Kwon, Jeehyun Lee, Sung-Dae Yang. Eigenseries solutions to optimal control problem and controllability problems on hyperbolic PDEs. Discrete & Continuous Dynamical Systems - B, 2010, 13 (2) : 305-325. doi: 10.3934/dcdsb.2010.13.305 [14] Piermarco Cannarsa, Cristina Pignotti, Carlo Sinestrari. Semiconcavity for optimal control problems with exit time. Discrete & Continuous Dynamical Systems - A, 2000, 6 (4) : 975-997. doi: 10.3934/dcds.2000.6.975 [15] Jérome Lohéac, Jean-François Scheid. Time optimal control for a nonholonomic system with state constraint. Mathematical Control & Related Fields, 2013, 3 (2) : 185-208. doi: 10.3934/mcrf.2013.3.185 [16] Piermarco Cannarsa, Carlo Sinestrari. On a class of nonlinear time optimal control problems. Discrete & Continuous Dynamical Systems - A, 1995, 1 (2) : 285-300. doi: 10.3934/dcds.1995.1.285 [17] Gengsheng Wang, Guojie Zheng. The optimal control to restore the periodic property of a linear evolution system with small perturbation. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1621-1639. doi: 10.3934/dcdsb.2010.14.1621 [18] Elimhan N. Mahmudov. Optimal control of evolution differential inclusions with polynomial linear differential operators. Evolution Equations & Control Theory, 2019, 8 (3) : 603-619. doi: 10.3934/eect.2019028 [19] Leszek Gasiński. Optimal control problem of Bolza-type for evolution hemivariational inequality. Conference Publications, 2003, 2003 (Special) : 320-326. doi: 10.3934/proc.2003.2003.320 [20] Leszek Gasiński, Nikolaos S. Papageorgiou. Relaxation of optimal control problems driven by nonlinear evolution equations. Evolution Equations & Control Theory, 2020, 9 (4) : 1027-1040. doi: 10.3934/eect.2020050

2019 Impact Factor: 1.27