American Institute of Mathematical Sciences

Advanced
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

[1]

Karl Kunisch, Lijuan Wang. The bang-bang property of time optimal controls for the Burgers equation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (9) : 3611-3637. doi: 10.3934/dcds.2014.34.3611
[2]

Karl Kunisch, Lijuan Wang. Bang-bang property of time optimal controls of semilinear parabolic equation. Discrete & Continuous Dynamical Systems - A, 2016, 36 (1) : 279-302. doi: 10.3934/dcds.2016.36.279
[3]

Gengsheng Wang, Yubiao Zhang. Decompositions and bang-bang properties. Mathematical Control & Related Fields, 2017, 7 (1) : 73-170. doi: 10.3934/mcrf.2017005
[4]

Walter Alt, Robert Baier, Matthias Gerdts, Frank Lempio. Error bounds for Euler approximation of linear-quadratic control problems with bang-bang solutions. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 547-570. doi: 10.3934/naco.2012.2.547
[5]

Helmut Maurer, Tanya Tarnopolskaya, Neale Fulton. Computation of bang-bang and singular controls in collision avoidance. Journal of Industrial & Management Optimization, 2014, 10 (2) : 443-460. doi: 10.3934/jimo.2014.10.443
[6]

Chunjuan Hou, Yanping Chen, Zuliang Lu. Superconvergence property of finite element methods for parabolic optimal control problems. Journal of Industrial & Management Optimization, 2011, 7 (4) : 927-945. doi: 10.3934/jimo.2011.7.927
[7]

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
[8]

Alexander J. Zaslavski. The turnpike property of discrete-time control problems arising in economic dynamics. Discrete & Continuous Dynamical Systems - B, 2005, 5 (3) : 861-880. doi: 10.3934/dcdsb.2005.5.861
[9]

Bo Su. Doubling property of elliptic equations. Communications on Pure & Applied Analysis, 2008, 7 (1) : 143-147. doi: 10.3934/cpaa.2008.7.143
[10]

Laurent Bourgeois. Quantification of the unique continuation property for the heat equation. Mathematical Control & Related Fields, 2017, 7 (3) : 347-367. doi: 10.3934/mcrf.2017012
[11]

M. Soledad Aronna, J. Frédéric Bonnans, Andrei V. Dmitruk, Pablo A. Lotito. Quadratic order conditions for bang-singular extremals. Numerical Algebra, Control & Optimization, 2012, 2 (3) : 511-546. doi: 10.3934/naco.2012.2.511
[12]

Jan Burczak, P. Kaplický. Evolutionary, symmetric $p$-Laplacian. Interior regularity of time derivatives and its consequences. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2401-2445. doi: 10.3934/cpaa.2016042
[13]

Alfonso C. Casal, Jesús Ildefonso Díaz, José M. Vegas. Finite extinction time property for a delayed linear problem on a manifold without boundary. Conference Publications, 2011, 2011 (Special) : 265-271. doi: 10.3934/proc.2011.2011.265
[14]

Yoshikazu Giga, Hirotoshi Kuroda. A counterexample to finite time stopping property for one-harmonic map flow. Communications on Pure & Applied Analysis, 2015, 14 (1) : 121-125. doi: 10.3934/cpaa.2015.14.121
[15]

Peter E. Kloeden, José Valero. The Kneser property of the weak solutions of the three dimensional Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2010, 28 (1) : 161-179. doi: 10.3934/dcds.2010.28.161
[16]

Shikuan Mao, Yongqin Liu. Decay property for solutions to plate type equations with variable coefficients. Kinetic & Related Models, 2017, 10 (3) : 785-797. doi: 10.3934/krm.2017031
[17]

Agnid Banerjee. A note on the unique continuation property for fully nonlinear elliptic equations. Communications on Pure & Applied Analysis, 2015, 14 (2) : 623-626. doi: 10.3934/cpaa.2015.14.623
[18]

Zdzisław Brzeźniak, Paul André Razafimandimby. Irreducibility and strong Feller property for stochastic evolution equations in Banach spaces. Discrete & Continuous Dynamical Systems - B, 2016, 21 (4) : 1051-1077. doi: 10.3934/dcdsb.2016.21.1051
[19]

Shoichi Hasegawa. Stability and separation property of radial solutions to semilinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (7) : 4127-4136. doi: 10.3934/dcds.2019166
[20]

Kazuhiro Sakai. The oe-property of diffeomorphisms. Discrete & Continuous Dynamical Systems - A, 1998, 4 (3) : 581-591. doi: 10.3934/dcds.1998.4.581

2018 Impact Factor: 1.008

Tools

Metrics

  • PDF downloads (8)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences