April  2017, 13(2): 623-631. doi: 10.3934/jimo.2016036

Parametric solutions to the regulator-conjugate matrix equations

1. 

Institute of Data and Knowledge Engineering, Henan University, Kaifeng 475004, China

2. 

Institute of electric power, North China University of Water Resources and Electric Power, Zhengzhou 450011, China

3. 

Center for Control Theory and Guidance Technology, Harbin Institute of Technology, Harbin 150001, China

* Corresponding author: Lingling Lv

Received  August 2015 Revised  December 2015 Published  May 2016

Fund Project: This work is supported by the Programs of National Natural Science Foundation of China (Nos. 61402149,11501200, U1604148), Innovative Talents of Higher Learning Institutions of Henan (No. 17HASTIT023), Program for Innovative Research Team in University of Henan Province (No. 16IRTSTHN017).

The problem of solving regulator-conjugate matrix equations is considered in this paper. The regulator-conjugate matrix equations are a class of nonhomogeneous equations. Utilizing several complex matrix operations and the concepts of controllability-like and observability-like matrices, a special solution to this problem is constructed, which includes solving an ordinary algebraic matrix. Combined with our recent results on Sylvester-conjugate matrix equations, complete solutions to regulator-conjugate matrix equations can be obtained by superposition principle. The correctness and effectiveness are verified by a numerical example.

Citation: Lei Zhang, Anfu Zhu, Aiguo Wu, Lingling Lv. Parametric solutions to the regulator-conjugate matrix equations. Journal of Industrial & Management Optimization, 2017, 13 (2) : 623-631. doi: 10.3934/jimo.2016036
References:
[1]

P. BennerJ. R. Li and T. Penzl, Numerical solution of large scale Lyapunov equations, Riccati equations, and linear quadratic optimal control problems, Numerical Linear Algebra with Applications, 15 (2008), 755-777.  doi: 10.1002/nla.622.  Google Scholar

[2]

J. BevisF. Hall and R. Hartwig, The matrix equation $A\bar{X}-XB=C$ and its special cases, SIAM Journal on Matrix Analysis and Applications, 60 (2010), 95-111.   Google Scholar

[3]

Y. Hong and R. Horn, A canonical form for matrices under consimilarity, Linear Algebra and its Applications, 102 (1988), 143-168.  doi: 10.1016/0024-3795(88)90324-2.  Google Scholar

[4]

T. Jiang and M. Wei, On solutions of the matrix equations $X-AXB=C$ and $X-A\overline{X} B=C$, Linear Algebra and its Applications, 367 (2003), 225-233.  doi: 10.1016/S0024-3795(02)00633-X.  Google Scholar

[5]

X. Jiang and Y. Zhang, A smoothing-type algorithm for absolute value equations, Journal of Industrial & Management Optimization, 9 (2013), 789-798.  doi: 10.3934/jimo.2013.9.789.  Google Scholar

[6]

A. WuG. FengG. Duan and and W. Wu, Closed-form solutions to Sylvester-conjugate matrix equations, Computers & Mathematics with Applications, 60 (2010), 95-111.  doi: 10.1016/j.camwa.2010.04.035.  Google Scholar

[7]

A. Wu and G. Duan, Solution to the generalised Sylvester matrix equation AV+ BW= EVF, IET Control Theory & Applications, 1 (2007), 402-408.  doi: 10.1049/iet-cta:20050390.  Google Scholar

[8]

A. WuL. LvG. Duan and W. Liu, Parametric solutions to Sylvester-conjugate matrix equations, Computers & Mathematics with Applications, 62 (2011), 3317-3325.  doi: 10.1016/j.camwa.2011.08.034.  Google Scholar

[9]

A. WuG. Duan and H. Yu, On solutions of the matrix equations $XF-AX= C$ and $XF-A\bar{X}= C $, Applied Mathematics and Computation, 183 (2006), 932-941.  doi: 10.1016/j.amc.2006.06.039.  Google Scholar

[10]

C. YangJ. Liu and Y. Liu, Solutions of the generalized Sylvester matrix equation and the application in eigenstructure assignment, Asian Journal of Control, 14 (2012), 1669-1675.  doi: 10.1002/asjc.448.  Google Scholar

[11]

K. F. C. YiuK. L. Mak and K. L. Teo, Airfoil design via optimal control theory, Journal of Industrial & Management Optimization, 1 (2005), 133-148.  doi: 10.3934/jimo.2005.1.133.  Google Scholar

[12]

B. Zhou and G. Duan, A new solution to the generalized Sylvester matrix equation AV-EVF= BW, Systems & Control Letters, 55 (2006), 193-198.  doi: 10.1016/j.sysconle.2005.07.002.  Google Scholar

[13]

B. ZhouG. Duan and Z. Li, A Stein matrix equation approach for computing coprime matrix fraction description, IET Control Theory & Applications, 3 (2009), 691-700.  doi: 10.1049/iet-cta.2008.0128.  Google Scholar

show all references

References:
[1]

P. BennerJ. R. Li and T. Penzl, Numerical solution of large scale Lyapunov equations, Riccati equations, and linear quadratic optimal control problems, Numerical Linear Algebra with Applications, 15 (2008), 755-777.  doi: 10.1002/nla.622.  Google Scholar

[2]

J. BevisF. Hall and R. Hartwig, The matrix equation $A\bar{X}-XB=C$ and its special cases, SIAM Journal on Matrix Analysis and Applications, 60 (2010), 95-111.   Google Scholar

[3]

Y. Hong and R. Horn, A canonical form for matrices under consimilarity, Linear Algebra and its Applications, 102 (1988), 143-168.  doi: 10.1016/0024-3795(88)90324-2.  Google Scholar

[4]

T. Jiang and M. Wei, On solutions of the matrix equations $X-AXB=C$ and $X-A\overline{X} B=C$, Linear Algebra and its Applications, 367 (2003), 225-233.  doi: 10.1016/S0024-3795(02)00633-X.  Google Scholar

[5]

X. Jiang and Y. Zhang, A smoothing-type algorithm for absolute value equations, Journal of Industrial & Management Optimization, 9 (2013), 789-798.  doi: 10.3934/jimo.2013.9.789.  Google Scholar

[6]

A. WuG. FengG. Duan and and W. Wu, Closed-form solutions to Sylvester-conjugate matrix equations, Computers & Mathematics with Applications, 60 (2010), 95-111.  doi: 10.1016/j.camwa.2010.04.035.  Google Scholar

[7]

A. Wu and G. Duan, Solution to the generalised Sylvester matrix equation AV+ BW= EVF, IET Control Theory & Applications, 1 (2007), 402-408.  doi: 10.1049/iet-cta:20050390.  Google Scholar

[8]

A. WuL. LvG. Duan and W. Liu, Parametric solutions to Sylvester-conjugate matrix equations, Computers & Mathematics with Applications, 62 (2011), 3317-3325.  doi: 10.1016/j.camwa.2011.08.034.  Google Scholar

[9]

A. WuG. Duan and H. Yu, On solutions of the matrix equations $XF-AX= C$ and $XF-A\bar{X}= C $, Applied Mathematics and Computation, 183 (2006), 932-941.  doi: 10.1016/j.amc.2006.06.039.  Google Scholar

[10]

C. YangJ. Liu and Y. Liu, Solutions of the generalized Sylvester matrix equation and the application in eigenstructure assignment, Asian Journal of Control, 14 (2012), 1669-1675.  doi: 10.1002/asjc.448.  Google Scholar

[11]

K. F. C. YiuK. L. Mak and K. L. Teo, Airfoil design via optimal control theory, Journal of Industrial & Management Optimization, 1 (2005), 133-148.  doi: 10.3934/jimo.2005.1.133.  Google Scholar

[12]

B. Zhou and G. Duan, A new solution to the generalized Sylvester matrix equation AV-EVF= BW, Systems & Control Letters, 55 (2006), 193-198.  doi: 10.1016/j.sysconle.2005.07.002.  Google Scholar

[13]

B. ZhouG. Duan and Z. Li, A Stein matrix equation approach for computing coprime matrix fraction description, IET Control Theory & Applications, 3 (2009), 691-700.  doi: 10.1049/iet-cta.2008.0128.  Google Scholar

[1]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

[2]

Min Li. A three term Polak-Ribière-Polyak conjugate gradient method close to the memoryless BFGS quasi-Newton method. Journal of Industrial & Management Optimization, 2020, 16 (1) : 245-260. doi: 10.3934/jimo.2018149

[3]

Nikolaos Roidos. Expanding solutions of quasilinear parabolic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021026

[4]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

[5]

Yimin Zhang, Youjun Wang, Yaotian Shen. Solutions for quasilinear Schrödinger equations with critical Sobolev-Hardy exponents. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1037-1054. doi: 10.3934/cpaa.2011.10.1037

[6]

Arunima Bhattacharya, Micah Warren. $ C^{2, \alpha} $ estimates for solutions to almost Linear elliptic equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021024

[7]

Yahui Niu. A Hopf type lemma and the symmetry of solutions for a class of Kirchhoff equations. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021027

[8]

Pengyu Chen. Periodic solutions to non-autonomous evolution equations with multi-delays. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2921-2939. doi: 10.3934/dcdsb.2020211

[9]

Lidan Wang, Lihe Wang, Chunqin Zhou. Classification of positive solutions for fully nonlinear elliptic equations in unbounded cylinders. Communications on Pure & Applied Analysis, 2021, 20 (3) : 1241-1261. doi: 10.3934/cpaa.2021019

[10]

Tomasz Kosmala, Markus Riedle. Variational solutions of stochastic partial differential equations with cylindrical Lévy noise. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2879-2898. doi: 10.3934/dcdsb.2020209

[11]

Shuang Wang, Dingbian Qian. Periodic solutions of p-Laplacian equations via rotation numbers. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021060

[12]

Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115

[13]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027

[14]

Jianping Gao, Shangjiang Guo, Wenxian Shen. Persistence and time periodic positive solutions of doubly nonlocal Fisher-KPP equations in time periodic and space heterogeneous media. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2645-2676. doi: 10.3934/dcdsb.2020199

[15]

Guodong Wang, Bijun Zuo. Energy equality for weak solutions to the 3D magnetohydrodynamic equations in a bounded domain. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021078

[16]

Qiwei Wu, Liping Luan. Large-time behavior of solutions to unipolar Euler-Poisson equations with time-dependent damping. Communications on Pure & Applied Analysis, 2021, 20 (3) : 995-1023. doi: 10.3934/cpaa.2021003

[17]

Yongqiang Fu, Xiaoju Zhang. Global existence and asymptotic behavior of weak solutions for time-space fractional Kirchhoff-type diffusion equations. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021091

[18]

Tôn Việt Tạ. Strict solutions to stochastic semilinear evolution equations in M-type 2 Banach spaces. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021050

[19]

Pengyu Chen, Xuping Zhang, Zhitao Zhang. Asymptotic behavior of time periodic solutions for extended Fisher-Kolmogorov equations with delays. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021103

[20]

Asato Mukai, Yukihiro Seki. Refined construction of type II blow-up solutions for semilinear heat equations with Joseph–Lundgren supercritical nonlinearity. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021060

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (76)
  • HTML views (370)
  • Cited by (1)

Other articles
by authors

[Back to Top]