Research On Solutions Of Sylvester Tensor Equations

Sylvester tensor equations are widely used in control system,image processing,model reduction,fluid mechanics,and so on.Among these applications,solutions of Sylvester tensor equations play an important role in the automatic control theory and the process of solving high-dimensional linear partial differential equations.In the thesis,we will focus on the problems related to solutions of Sylvester tensor equations.Firstly,study the existence of solutions of the general Sylvester tensor equation.Using the theory of the Jordan canonical form and the block tensor,give the necessary and sufficient conditions for the existence of solutions of the general Sylvester tensor equation.Based on the conditions for the solvability,obtain explicit solutions of the general Sylvester tensor equation.Secondly,discuss the uniqueness of the solution of the Lyapunov tensor equation.Due to the related theory of the positive definite tensor,generalize the Lyapunov's alge-braic theorem and the Stein's theorem to the Lyapunov tensor equation.Besides,under the existence and uniqueness of the solution of the Lyapunov tensor equation,express the explicit solution of the Lyapunov tensor equation by the tensor integral and the tensor series.Thirdly,analyze the sensitivity of the solution of the Lyapunov tensor equation.Based on the theory of the tensor norm,give the perturbation bounds in the Frobenius norm and the l~mspectral norm for the solution of the Lyapunov tensor equation.For both of them,compare the perturbation bound in the Frobenius norm and the Shi-Wei-Ling perturbation bound,and explain the relationship between the perturbation bound in the l~m spectral norm and the damping coefficient of a linear time-invariant system.Finally,research the dimension of the solution space of the 2nd-order generalized T-Sylvester tensor equation.According to the related theory of the generalized sigular value decomposition,give dimensional formulae of the solution space of the 2nd-order generalized T-Sylvester tensor equation.In particular,apply the main results to the theory of the connectivity of the graph,and then connect the 2nd-order generalized T-Sylvester tensor equation and the connectivity of the graph.