# American Institute of Mathematical Sciences

November  2013, 18(9): 2427-2439. doi: 10.3934/dcdsb.2013.18.2427

## Advantages for controls imposed in a proper subset

 1 School of Mathematics and Statistics, Wuhan University, Wuhan, 430072, China 2 School of Mathematical Sciences, Fudan University, KLMNS, Shanghai, 200433, China

Received  October 2012 Revised  August 2013 Published  September 2013

In this paper, we study some time optimal control problems for heat equations on $\Omega\times \mathbb{R}^+$. Two properties under consideration are the existence and the bang-bang properties of time optimal controls. It is proved that those two properties hold when controls are imposed on most proper subsets of $\Omega$; while they do not stand when controls are active on the whole $\Omega$. Besides, a new property for eigenfunctions of $-\Delta$ with Dirichlet boundary condition is revealed.
Citation: Gengsheng Wang, Yashan Xu. Advantages for controls imposed in a proper subset. Discrete & Continuous Dynamical Systems - B, 2013, 18 (9) : 2427-2439. doi: 10.3934/dcdsb.2013.18.2427
##### References:
 [1] L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, (1998). [2] H. O. Fattorini, Time-optimal control of solutions of operational differential equations., Journal of the Society for Industrial and Applied Mathematics Series A, 2 (1964), 54. doi: 10.1137/0302005. [3] H. O. Fattorini, "Infinite-dimensional Optimization and Control Theory,", Encyclopedia of Mathematics and its Applications, (1999). [4] Qing Han and Fanghua Lin, "Elliptic Partial Differential Equations,", Courant Lecture Notes in Mathematics, (1997). [5] Steven G. Krantz, "Function Theory of Several Complex Variables,", Pure and Applied Mathematics, (1982). [6] F. H. Lin, A uniqueness theorem for parabolic equations,, Communications on Pure and Applied Mathematics, 43 (1990), 127. doi: 10.1002/cpa.3160430105. [7] Qi L and Gengsheng Wang, On the existence of time optimal controls with constraints of the rectangular type for heat equations,, SIAM Journal on Control and Optimization, 49 (2011), 1124. doi: 10.1137/10081277X. [8] A. M. Micheletti, Perturbazione dello spettro dell'operatore di Laplace, in relazione ad una variazione del campo, (Italian), Ann. Scuola Norm. Sup. Pisa, 26 (1972), 151. [9] Victor J. Mizel and Thomas I. Seidman, An abstract bang-bang principle and time-optimal boundary control of the heat equation,, SIAM Journal on Control and Optimization, 35 (1997), 1204. doi: 10.1137/S0363012996265470. [10] Jaime H. Ortega and Enrique Zuazua, Generic simplicity of the spectrum and stabilization for a plate equation,, SIAM Journal on Control and Optimization, 39 (2000), 1585. doi: 10.1137/S0363012900358483. [11] Kim Dang Phung and Gengsheng Wang, An observability estimate for parabolic equations from a measurable set in time and its applications,, Journal of the European Mathematical Society, 15 (2013), 681. doi: 10.4171/JEMS/371. [12] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes,", Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt Interscience Publishers John Wiley & Sons, (1962). [13] E. D. Sontag, "Mathematical Control Theory: Deterministic Finite-Dimensional Systems,", Second edition, (1998). [14] K. Uhlenbeck, Generic properties of eigenfunctions,, American Journal of Mathematics, 98 (1976), 1059. doi: 10.2307/2374041. [15] Gengsheng Wang, $L^\infty$-null controllability for the heat equation and its consequences for the time optimal control problem,, SIAM Journal on Control and Optimization, 47 (2008), 1701. doi: 10.1137/060678191.

show all references

##### References:
 [1] L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, (1998). [2] H. O. Fattorini, Time-optimal control of solutions of operational differential equations., Journal of the Society for Industrial and Applied Mathematics Series A, 2 (1964), 54. doi: 10.1137/0302005. [3] H. O. Fattorini, "Infinite-dimensional Optimization and Control Theory,", Encyclopedia of Mathematics and its Applications, (1999). [4] Qing Han and Fanghua Lin, "Elliptic Partial Differential Equations,", Courant Lecture Notes in Mathematics, (1997). [5] Steven G. Krantz, "Function Theory of Several Complex Variables,", Pure and Applied Mathematics, (1982). [6] F. H. Lin, A uniqueness theorem for parabolic equations,, Communications on Pure and Applied Mathematics, 43 (1990), 127. doi: 10.1002/cpa.3160430105. [7] Qi L and Gengsheng Wang, On the existence of time optimal controls with constraints of the rectangular type for heat equations,, SIAM Journal on Control and Optimization, 49 (2011), 1124. doi: 10.1137/10081277X. [8] A. M. Micheletti, Perturbazione dello spettro dell'operatore di Laplace, in relazione ad una variazione del campo, (Italian), Ann. Scuola Norm. Sup. Pisa, 26 (1972), 151. [9] Victor J. Mizel and Thomas I. Seidman, An abstract bang-bang principle and time-optimal boundary control of the heat equation,, SIAM Journal on Control and Optimization, 35 (1997), 1204. doi: 10.1137/S0363012996265470. [10] Jaime H. Ortega and Enrique Zuazua, Generic simplicity of the spectrum and stabilization for a plate equation,, SIAM Journal on Control and Optimization, 39 (2000), 1585. doi: 10.1137/S0363012900358483. [11] Kim Dang Phung and Gengsheng Wang, An observability estimate for parabolic equations from a measurable set in time and its applications,, Journal of the European Mathematical Society, 15 (2013), 681. doi: 10.4171/JEMS/371. [12] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, "The Mathematical Theory of Optimal Processes,", Translated from the Russian by K. N. Trirogoff; edited by L. W. Neustadt Interscience Publishers John Wiley & Sons, (1962). [13] E. D. Sontag, "Mathematical Control Theory: Deterministic Finite-Dimensional Systems,", Second edition, (1998). [14] K. Uhlenbeck, Generic properties of eigenfunctions,, American Journal of Mathematics, 98 (1976), 1059. doi: 10.2307/2374041. [15] Gengsheng Wang, $L^\infty$-null controllability for the heat equation and its consequences for the time optimal control problem,, SIAM Journal on Control and Optimization, 47 (2008), 1701. doi: 10.1137/060678191.
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