# American Institute of Mathematical Sciences

September 2018, 8(3&4): 1081-1095. doi: 10.3934/mcrf.2018046

## 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

* Corresponding author: Jie Zhong (jiezhongmath@gmail.com)

Dedicated to Professor Jiongmin Yong on the Occasion of His 60th Birthday

Received  September 2017 Revised  February 2018 Published  September 2018

Fund Project: The first author is supported in part by the National Natural Science Foundation of China, China Postdoctoral Science Foundation and Central South University Postdoctoral Science Foundation

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.

Citation: Donghui Yang, Jie Zhong. Optimal actuator location of the minimum norm controls for stochastic heat equations. Mathematical Control & Related Fields, 2018, 8 (3&4) : 1081-1095. doi: 10.3934/mcrf.2018046
##### 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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##### 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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