# American Institute of Mathematical Sciences

• Previous Article
The Kato smoothing effect for the nonlinear regularized Schrödinger equation on compact manifolds
• MCRF Home
• This Issue
• Next Article
Controllability of a system of degenerate parabolic equations with non-diagonalizable diffusion matrix
doi: 10.3934/mcrf.2020005

## Switching controls for linear stochastic differential systems

 School of Mathematics, Physics and Data Science, Chongqing University of Science and Technology, Chongqing, 401331, China

* Corresponding author: Yong He

Received  May 2019 Revised  September 2019 Published  November 2019

Fund Project: The author is supported by the science and technology research project of Chongqing Education Commission under grant KJQN201801529.

We analyze the exact controllability problem of switching controls for stochastic control systems endowed with different actuators. The goal is to control the dynamics of the system by switching from an actuator to the other such that, in each instant of time, there are as few active actuators as possible. We prove that, under suitable rank conditions, switching control strategies exist and can be built in a systematic way. The proof is based on building a new functional by the adjoint system whose minimizers are the switching controls.

Citation: Yong He. Switching controls for linear stochastic differential systems. Mathematical Control & Related Fields, doi: 10.3934/mcrf.2020005
##### References:

show all references

##### References:
 [1] Klaus-Jochen Engel, Marjeta Kramar FijavŽ. Exact and positive controllability of boundary control systems. Networks & Heterogeneous Media, 2017, 12 (2) : 319-337. doi: 10.3934/nhm.2017014 [2] Manuel González-Burgos, Sergio Guerrero, Jean Pierre Puel. Local exact controllability to the trajectories of the Boussinesq system via a fictitious control on the divergence equation. Communications on Pure & Applied Analysis, 2009, 8 (1) : 311-333. doi: 10.3934/cpaa.2009.8.311 [3] Yi Zhang, Yuyun Zhao, Tao Xu, Xin Liu. $p$th Moment absolute exponential stability of stochastic control system with Markovian switching. Journal of Industrial & Management Optimization, 2016, 12 (2) : 471-486. doi: 10.3934/jimo.2016.12.471 [4] Qi Lü, Enrique Zuazua. Robust null controllability for heat equations with unknown switching control mode. Discrete & Continuous Dynamical Systems - A, 2014, 34 (10) : 4183-4210. doi: 10.3934/dcds.2014.34.4183 [5] Thomas I. Seidman. Optimal control of a diffusion/reaction/switching system. Evolution Equations & Control Theory, 2013, 2 (4) : 723-731. doi: 10.3934/eect.2013.2.723 [6] Lijuan Wang, Qishu Yan. Optimal control problem for exact synchronization of parabolic system. Mathematical Control & Related Fields, 2019, 9 (3) : 411-424. doi: 10.3934/mcrf.2019019 [7] Belhassen Dehman, Jean-Pierre Raymond. Exact controllability for the Lamé system. Mathematical Control & Related Fields, 2015, 5 (4) : 743-760. doi: 10.3934/mcrf.2015.5.743 [8] Fabio Bagagiolo. Optimal control of finite horizon type for a multidimensional delayed switching system. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 239-264. doi: 10.3934/dcdsb.2005.5.239 [9] Mohammed Aassila. Exact boundary controllability of a coupled system. Discrete & Continuous Dynamical Systems - A, 2000, 6 (3) : 665-672. doi: 10.3934/dcds.2000.6.665 [10] Luca Schenato, Sandro Zampieri. On rendezvous control with randomly switching communication graphs. Networks & Heterogeneous Media, 2007, 2 (4) : 627-646. doi: 10.3934/nhm.2007.2.627 [11] Shahlar F. Maharramov. Necessary optimality conditions for switching control problems. Journal of Industrial & Management Optimization, 2010, 6 (1) : 47-55. doi: 10.3934/jimo.2010.6.47 [12] Yanqing Wang, Donghui Yang, Jiongmin Yong, Zhiyong Yu. Exact controllability of linear stochastic differential equations and related problems. Mathematical Control & Related Fields, 2017, 7 (2) : 305-345. doi: 10.3934/mcrf.2017011 [13] Nicolás Carreño. Local controllability of the $N$-dimensional Boussinesq system with $N-1$ scalar controls in an arbitrary control domain. Mathematical Control & Related Fields, 2012, 2 (4) : 361-382. doi: 10.3934/mcrf.2012.2.361 [14] Yuri B. Gaididei, Carlos Gorria, Rainer Berkemer, Peter L. Christiansen, Atsushi Kawamoto, Mads P. Sørensen, Jens Starke. Stochastic control of traffic patterns. Networks & Heterogeneous Media, 2013, 8 (1) : 261-273. doi: 10.3934/nhm.2013.8.261 [15] Tyrone E. Duncan. Some topics in stochastic control. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1361-1373. doi: 10.3934/dcdsb.2010.14.1361 [16] Bao-Zhu Guo, Liang Zhang. Local exact controllability to positive trajectory for parabolic system of chemotaxis. Mathematical Control & Related Fields, 2016, 6 (1) : 143-165. doi: 10.3934/mcrf.2016.6.143 [17] Bopeng Rao, Laila Toufayli, Ali Wehbe. Stability and controllability of a wave equation with dynamical boundary control. Mathematical Control & Related Fields, 2015, 5 (2) : 305-320. doi: 10.3934/mcrf.2015.5.305 [18] Mohamed Ouzahra. Controllability of the semilinear wave equation governed by a multiplicative control. Evolution Equations & Control Theory, 2019, 8 (4) : 669-686. doi: 10.3934/eect.2019039 [19] Sandra Ricardo, Witold Respondek. When is a control system mechanical?. Journal of Geometric Mechanics, 2010, 2 (3) : 265-302. doi: 10.3934/jgm.2010.2.265 [20] Galina Kurina, Sahlar Meherrem. Decomposition of discrete linear-quadratic optimal control problems for switching systems. Conference Publications, 2015, 2015 (special) : 764-774. doi: 10.3934/proc.2015.0764

2018 Impact Factor: 1.292