American Institute of Mathematical Sciences

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

Advantages for controls imposed in a proper subset

Gengsheng Wang 1, and  Yashan Xu 2,

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.
Keywords: bang-bang property, heat equations, property of eigenfunctions of the Laplacian., Time optimal control.
Mathematics Subject Classification: Primary: 49J20; Secondary: 35P0.
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).   Google Scholar

[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.  Google Scholar

[3]

H. O. Fattorini, "Infinite-dimensional Optimization and Control Theory,", Encyclopedia of Mathematics and its Applications, (1999).   Google Scholar

[4]

Qing Han and Fanghua Lin, "Elliptic Partial Differential Equations,", Courant Lecture Notes in Mathematics, (1997).   Google Scholar

[5]

Steven G. Krantz, "Function Theory of Several Complex Variables,", Pure and Applied Mathematics, (1982).   Google Scholar

[6]

F. H. Lin, A uniqueness theorem for parabolic equations,, Communications on Pure and Applied Mathematics, 43 (1990), 127.  doi: 10.1002/cpa.3160430105.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[13]

E. D. Sontag, "Mathematical Control Theory: Deterministic Finite-Dimensional Systems,", Second edition, (1998).   Google Scholar

[14]

K. Uhlenbeck, Generic properties of eigenfunctions,, American Journal of Mathematics, 98 (1976), 1059.  doi: 10.2307/2374041.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

L. C. Evans, "Partial Differential Equations,", Graduate Studies in Mathematics, (1998).   Google Scholar

[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.  Google Scholar

[3]

H. O. Fattorini, "Infinite-dimensional Optimization and Control Theory,", Encyclopedia of Mathematics and its Applications, (1999).   Google Scholar

[4]

Qing Han and Fanghua Lin, "Elliptic Partial Differential Equations,", Courant Lecture Notes in Mathematics, (1997).   Google Scholar

[5]

Steven G. Krantz, "Function Theory of Several Complex Variables,", Pure and Applied Mathematics, (1982).   Google Scholar

[6]

F. H. Lin, A uniqueness theorem for parabolic equations,, Communications on Pure and Applied Mathematics, 43 (1990), 127.  doi: 10.1002/cpa.3160430105.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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).   Google Scholar

[13]

E. D. Sontag, "Mathematical Control Theory: Deterministic Finite-Dimensional Systems,", Second edition, (1998).   Google Scholar

[14]

K. Uhlenbeck, Generic properties of eigenfunctions,, American Journal of Mathematics, 98 (1976), 1059.  doi: 10.2307/2374041.  Google Scholar

[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.  Google Scholar

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