| [1] | Barbu V, Korman P. Analysis and Control of Nonlinear Infinite Dimensional Systems, Vol. 190. Amsterdam:Elsevier, 1993 | | [2] | Chen T, Francis B A. Optimal Sampled-Data Control Systems. London: Springer-Verlag, 1995 | | [3] | Duan Y L, Wang L J, Zhang C. Minimal time impulse control of an evolution equation. Journal of Optimization Theory and Applications, 2019, 183(3): 902-919 | | [4] | Fattorini H O. Infinite Dimensional Linear Control Systems:the Time Optimal and Norm Optimal Problems. Amsterdam: Elsevier, 2005 | | [5] | Fenchel W. On conjugate convex functions. Canadian Journal of Mathematics, 1949, 1(1): 73-77 | | [6] | Kunisch K, Wang L J. Time optimal controls of the linear fitzhugh-nagumo equation with pointwise control constraints. Journal of Mathematical Analysis and Applications, 2012, 395(1): 114-130 | | [7] | Kunisch K, Wang L J. Time optimal control of the heat equation with pointwise control constraints. ESAIM: Control, Optimisation and Calculus of Variations, 2013, 19(2): 460-485 | | [8] | Lü Q, Wang G S. On the existence of time optimal controls with constraints of the rectangular type for heat equations. SIAM Journal on Control and Optimization, 2011, 49(3): 1124-1149 | | [9] | Li X J, Yong J M. Optimal Control Theory for Infinite Dimensional Systems. Boston: Birkh?user, 1995 | | [10] | Lions J L. Remarks on approximate controllability. Journal d'Analyse Mathématique, 1992, 59(1): 103-116 | | [11] | Phung K D, Wang G S, Xu Y S. Impulse output rapid stabilization for heat equations. Journal of Differential Equations, 2016, 263(18): 5012-5041 | | [12] | Phung K D, Wang G S, Zhang X. On the existence of time optimal controls for linear evolution equations. Discrete and Continuous Dynamical Systems - Series B, 2012, 4(4): 925-941 | | [13] | Trélat E, Wang G S, Xu Y S. Characterization by observability inequalities of controllability and stabilization properties. Pure and Applied Analysis, 2019, 2(1): 93-122 | | [14] | Tré E, Wang L J, Zhang Y B. Impulse and sampled-data optimal control of heat equations, and error estimates. SIAM Journal on Control and Optimization, 2015, 54(5): 2787-2819 | | [15] | Wang G S. The existence of time optimal control of semilinear parabolic equations. Systems & Control Letters, 2004, 53(3/4): 171-175 | | [16] | Wang G S. L∞-null controllability for the heat equation and its consequences for the time optimal control problem. SIAM Journal on Control and Optimization, 2008, 47(4): 1701-1720 | | [17] | Wang G S, Wang M, Zhang C, Zhang Y B. Observable set, observability, interpolation inequality and spectral inequality for the heat equation in Rn. Journal de Mathématiques Pures et Appliquées, 2019, 126: 144-194 | | [18] | Wang G S, Wang M, Zhang Y B. Observability and unique continuation inequalities for the schr{?}dinger equation. Journal of the European Mathematical Society, 2019, 21(11): 3513-3572 | | [19] | Wang G S, Yang D H, Zhang Y B. Time optimal sampled-data controls for heat equations. Comptes Rendus Mathematique, 2017, 355(12): 1252-1290 | | [20] | Wang L J. Minimal time impulse control problem of semilinear heat equation. Journal of Optimization Theory and Applications, 2021, 188(13): 805-822 | | [21] | Wang L J, Wang G S. The optimal time control of a phase-field system. SIAM Journal on Control and Optimization, 2003, 42(4): 1483-1508 |
|
),Liu Hanbing*(