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Time-inconsistent recursive zero-sum stochastic differential games
Optimal actuator location of the minimum norm controls for stochastic heat equations
1. | School of Mathematics and Statistics, School of Information Science and Engineering, Central South University, Changsha 410075, China |
2. | Department of Mathematics, California State University, Los Angeles, Los Angeles, CA 90032, USA |
In this paper, we study the approximate null controllability for the stochastic heat equation with the control acting on a measurable subset, and the optimal actuator location of the minimum norm controls. We formulate a relaxed optimization problem for both actuator location and its corresponding minimum norm control into a two-person zero sum game problem and develop a sufficient and necessary condition for the optimal solution via Nash equilibrium. At last, we prove that the relaxed optimal solution is an optimal actuator location for the classical problem.
References:
[1] |
G. Allaire, A. Münch and F. Periago,
Long time behavior of a two-phase optimal design for the heat equation, SIAM Journal on Control and Optimization, 48 (2010), 5333-5356.
doi: 10.1137/090780481. |
[2] |
J. Apraiz, L. Escauriaza, G. Wang and C. Zhang,
Observability inequalities and measurable sets, Journal of the European Mathematical Society, 16 (2014), 2433-2475.
doi: 10.4171/JEMS/490. |
[3] |
J.-P. Aubin,
Applied Functional Analysis, John Wiley & Sons, New York-Chichester-Brisbane, 1979. |
[4] |
V. Barbu, A. Rǎşcanu and G. Tessitore,
Carleman estimates and controllability of linear stochastic heat equations, Applied Mathematics & Optimization, 47 (2003), 97-120.
doi: 10.1007/s00245-002-0757-z. |
[5] |
N. Darivandi, K. Morris and A. Khajepour, An algorithm for LQ optimal actuator location,
Smart Materials and Structures, 22 (2013), 035001. |
[6] |
K. Du and Q. Meng,
A revisit to-theory of super-parabolic backward stochastic partial differential equations in rd, Stochastic Processes and their Applications, 120 (2010), 1996-2015.
doi: 10.1016/j.spa.2010.06.001. |
[7] |
X. Fu and X. Liu,
Controllability and observability of some stochastic complex ginzburg-landau equations, SIAM Journal on Control and Optimization, 55 (2017), 1102-1127.
doi: 10.1137/15M1039961. |
[8] |
P. Gao, M. Chen and Y. Li,
Observability estimates and null controllability for forward and backward linear stochastic kuramoto-sivashinsky equations, SIAM Journal on Control and Optimization, 53 (2015), 475-500.
doi: 10.1137/130943820. |
[9] |
B.-Z. Guo, Y. Xu and D.-H. Yang,
Optimal actuator location of minimum norm controls for heat equation with general controlled domain, Journal of Differential Equations, 261 (2016), 3588-3614.
doi: 10.1016/j.jde.2016.05.037. |
[10] |
B.-Z. Guo and D.-H. Yang,
Optimal actuator location for time and norm optimal control of null controllable heat equation, Mathematics of Control, Signals, and Systems, 27 (2015), 23-48.
doi: 10.1007/s00498-014-0133-y. |
[11] |
Y. Hu and S. Peng,
Adapted solution of a backward semilinear stochastic evolution equation, Stochastic Analysis and Applications, 9 (1991), 445-459.
doi: 10.1080/07362999108809250. |
[12] |
X. Liu,
Controllability of some coupled stochastic parabolic systems with fractional order spatial differential operators by one control in the drift, SIAM Journal on Control and Optimization, 52 (2014), 836-860.
doi: 10.1137/130926791. |
[13] |
Q. Lü,
Some results on the controllability of forward stochastic heat equations with control on the drift, Journal of Functional Analysis, 260 (2011), 832-851.
doi: 10.1016/j.jfa.2010.10.018. |
[14] |
Q. Lü, J. Yong and X. Zhang,
Representation of itô integrals by lebesgue/bochner integrals, Journal of the European Mathematical Society, 14 (2012), 1795-1823.
doi: 10.4171/JEMS/347. |
[15] |
A. Münch,
Optimal design of the support of the control for the 2-d wave equation: a numerical method, Int. J. Numer. Anal. Model, 5 (2008), 331-351.
|
[16] |
A. Münch,
Optimal location of the support of the control for the 1-d wave equation: Numerical investigations, Computational Optimization and Applications, 42 (2009), 443-470.
doi: 10.1007/s10589-007-9133-x. |
[17] |
K. D. Phung and G. Wang,
An observability estimate for parabolic equations from a measurable set in time and its applications, J. Eur. Math. Soc, 15 (2013), 681-703.
doi: 10.4171/JEMS/371. |
[18] |
Y. Privat, E. Trélat and E. Zuazua,
Optimal shape and location of sensors for parabolic equations with random initial data, Archive for Rational Mechanics and Analysis, 216 (2015), 921-981.
doi: 10.1007/s00205-014-0823-0. |
[19] |
S. Tang and X. Zhang,
Null controllability for forward and backward stochastic parabolic equations, SIAM Journal on Control and Optimization, 48 (2009), 2191-2216.
doi: 10.1137/050641508. |
[20] |
D. Tiba,
Finite element approximation for shape optimization problems with neumann and mixed boundary conditions, SIAM Journal on Control and Optimization, 49 (2011), 1064-1077.
doi: 10.1137/100783236. |
[21] |
D. Yang and J. Zhong,
Observability inequality of backward stochastic heat equations for measurable sets and its applications, SIAM Journal on Control and Optimization, 54 (2016), 1157-1175.
doi: 10.1137/15M1033289. |
[22] |
E. Zuazua,
Controllability of partial differential equations and its semi-discrete approximations, Discrete and Continuous Dynamical Systems, 8 (2002), 469-513.
doi: 10.3934/dcds.2002.8.469. |
[23] |
E. Zuazua,
Controllability and observability of partial differential equations: Some results and open problems, Handbook of Differential Equations: Evolutionary Equations, 3 (2007), 527-621.
doi: 10.1016/S1874-5717(07)80010-7. |
show all references
References:
[1] |
G. Allaire, A. Münch and F. Periago,
Long time behavior of a two-phase optimal design for the heat equation, SIAM Journal on Control and Optimization, 48 (2010), 5333-5356.
doi: 10.1137/090780481. |
[2] |
J. Apraiz, L. Escauriaza, G. Wang and C. Zhang,
Observability inequalities and measurable sets, Journal of the European Mathematical Society, 16 (2014), 2433-2475.
doi: 10.4171/JEMS/490. |
[3] |
J.-P. Aubin,
Applied Functional Analysis, John Wiley & Sons, New York-Chichester-Brisbane, 1979. |
[4] |
V. Barbu, A. Rǎşcanu and G. Tessitore,
Carleman estimates and controllability of linear stochastic heat equations, Applied Mathematics & Optimization, 47 (2003), 97-120.
doi: 10.1007/s00245-002-0757-z. |
[5] |
N. Darivandi, K. Morris and A. Khajepour, An algorithm for LQ optimal actuator location,
Smart Materials and Structures, 22 (2013), 035001. |
[6] |
K. Du and Q. Meng,
A revisit to-theory of super-parabolic backward stochastic partial differential equations in rd, Stochastic Processes and their Applications, 120 (2010), 1996-2015.
doi: 10.1016/j.spa.2010.06.001. |
[7] |
X. Fu and X. Liu,
Controllability and observability of some stochastic complex ginzburg-landau equations, SIAM Journal on Control and Optimization, 55 (2017), 1102-1127.
doi: 10.1137/15M1039961. |
[8] |
P. Gao, M. Chen and Y. Li,
Observability estimates and null controllability for forward and backward linear stochastic kuramoto-sivashinsky equations, SIAM Journal on Control and Optimization, 53 (2015), 475-500.
doi: 10.1137/130943820. |
[9] |
B.-Z. Guo, Y. Xu and D.-H. Yang,
Optimal actuator location of minimum norm controls for heat equation with general controlled domain, Journal of Differential Equations, 261 (2016), 3588-3614.
doi: 10.1016/j.jde.2016.05.037. |
[10] |
B.-Z. Guo and D.-H. Yang,
Optimal actuator location for time and norm optimal control of null controllable heat equation, Mathematics of Control, Signals, and Systems, 27 (2015), 23-48.
doi: 10.1007/s00498-014-0133-y. |
[11] |
Y. Hu and S. Peng,
Adapted solution of a backward semilinear stochastic evolution equation, Stochastic Analysis and Applications, 9 (1991), 445-459.
doi: 10.1080/07362999108809250. |
[12] |
X. Liu,
Controllability of some coupled stochastic parabolic systems with fractional order spatial differential operators by one control in the drift, SIAM Journal on Control and Optimization, 52 (2014), 836-860.
doi: 10.1137/130926791. |
[13] |
Q. Lü,
Some results on the controllability of forward stochastic heat equations with control on the drift, Journal of Functional Analysis, 260 (2011), 832-851.
doi: 10.1016/j.jfa.2010.10.018. |
[14] |
Q. Lü, J. Yong and X. Zhang,
Representation of itô integrals by lebesgue/bochner integrals, Journal of the European Mathematical Society, 14 (2012), 1795-1823.
doi: 10.4171/JEMS/347. |
[15] |
A. Münch,
Optimal design of the support of the control for the 2-d wave equation: a numerical method, Int. J. Numer. Anal. Model, 5 (2008), 331-351.
|
[16] |
A. Münch,
Optimal location of the support of the control for the 1-d wave equation: Numerical investigations, Computational Optimization and Applications, 42 (2009), 443-470.
doi: 10.1007/s10589-007-9133-x. |
[17] |
K. D. Phung and G. Wang,
An observability estimate for parabolic equations from a measurable set in time and its applications, J. Eur. Math. Soc, 15 (2013), 681-703.
doi: 10.4171/JEMS/371. |
[18] |
Y. Privat, E. Trélat and E. Zuazua,
Optimal shape and location of sensors for parabolic equations with random initial data, Archive for Rational Mechanics and Analysis, 216 (2015), 921-981.
doi: 10.1007/s00205-014-0823-0. |
[19] |
S. Tang and X. Zhang,
Null controllability for forward and backward stochastic parabolic equations, SIAM Journal on Control and Optimization, 48 (2009), 2191-2216.
doi: 10.1137/050641508. |
[20] |
D. Tiba,
Finite element approximation for shape optimization problems with neumann and mixed boundary conditions, SIAM Journal on Control and Optimization, 49 (2011), 1064-1077.
doi: 10.1137/100783236. |
[21] |
D. Yang and J. Zhong,
Observability inequality of backward stochastic heat equations for measurable sets and its applications, SIAM Journal on Control and Optimization, 54 (2016), 1157-1175.
doi: 10.1137/15M1033289. |
[22] |
E. Zuazua,
Controllability of partial differential equations and its semi-discrete approximations, Discrete and Continuous Dynamical Systems, 8 (2002), 469-513.
doi: 10.3934/dcds.2002.8.469. |
[23] |
E. Zuazua,
Controllability and observability of partial differential equations: Some results and open problems, Handbook of Differential Equations: Evolutionary Equations, 3 (2007), 527-621.
doi: 10.1016/S1874-5717(07)80010-7. |
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