Xin Yu’s research while affiliated with Zhejiang University and other places

In this paper, we investigate the stabilization for infinite dimensional switched linear systems comprising two unstable subsystems. From the algebraic structure of the state space, two switching strategies are designed to stabilize the infinite dimensional switched linear systems. Under these strategies and suitable assumptions, the systems with time-varying disturbances possess a good robust stability. In addition, we can also obtain an arbitrarily low switching frequency for the switched systems.

In this paper we study the priori error analysis for semidiscrete Galerkin finite element approximation of the impulsive optimal control problem governed by linear heat equation with control constraint. We first derive Pontryagin's maximum principle and construct the semidiscrete finite element approximation scheme for our impulsive optimal control problem. Then, the error estimates for optimal control u, the related state y and the adjoint state φ will be obtained. Finally, an efficient algorithm for our numerical experiment is designed, which verifies the theoretical results obtained in this paper.

This paper is devoted to the study of semidiscrete finite element approximation of time optimal control problem governed by semilinear heat equation with nonsmooth initial data. When the control is acted locally, an error estimate for the optimal time is obtained by making use of the approximation results for the equations. Moreover, with the help of Pontryagin’s maximum principle and the unique continuation property for the heat equation, a better error estimate for the optimal time can be derived when the control is acted globally into the state equation.

This paper is devoted to the study of numerical computation for a kind of time optimal control problem for the tubular reactor system. This kind of time optimal control problem is aimed at delaying the initiation time τ of the active control as late as possible, such that the state governed by this controlled system can reach the target set at a given ending time T . To compute the time optimal control problem, we firstly approximate the original problem by finite element method and get a new approximation time optimal control problem governed by ordinary differential equations. Then, through the control parameterization method and time-scaling transformation, the approximation problem becomes an optimal parameter selection problem. Finally, we use Sequential Quadratic Program algorithm to solve the optimal parameter selection problem. A numerical simulation is given for illustration.

This paper is devoted to study the abstract functional differential equation (FDE) of the following form (u) over dot(t) - Au(t) + Phi u(t), where A generates a C-regularized semigroup, which is the generalization of C-0-semigroup and can be applied to deal with many important differential operators that the C-0-semigroup can not be used to. We first show that the C-well-posedness of a FDE is equivalent to the r-well-posedness of an abstract Cauchy problem in a product Banach space, where the operator r is related with the operator C and will be defined in the following text. Then, by making use of a perturbation result of C-regularized semigroup, a sufficient condition is provided for the C-well-posedness of FDEs. Moreover, an illustrative application to partial differential equation (PDE) with delay is given in the last section.

This paper is devoted to study the abstract functional differential equation (FDE) of the following form where A generates a C-regularized semigroup, which is the generalization of C0-semigroup and can be applied to deal with many important differential operators that the C0-semigroup can not be used to. We first show that the C-well-posedness of a FDE is equivalent to the ℓ-well-posedness of an abstract Cauchy problem in a product Banach space, where the operator ℓ is related with the operator ℓ and will be defined in the following text. Then, by making use of a perturbation result of ℓ-regularized semigroup, a suffcient condition is provided for the C-well-posedness of FDEs. Moreover, an illustrative application to partial differential equation (PDE) with delay is given in the last section.

This paper deals with the numerical approximation for the time optimal control problem governed by the Benjamin-Bona-Mahony (BBM) equation, which is an unspecified terminal time problem. Firstly, by projecting the original problem with the finite element method (FEM), another approximate problem governed by a system of ordinary differential equations will be obtained. Then, the parameterisation method for the optimal time and the control function will be carried out and the unspecified terminal time problem can be reduced to an optimal parameter selection problem with a fixed time horizon [0, 1]. This optimal parameter selection problem is a standard nonlinear mathematical programming problem and can be solved by sequential quadratic programming (SQP) algorithm. Finally, some numerical simulation studies will be given to illustrate the effectiveness of our numerical approximation method for the time optimal control problem governed by the BBM equation.

This paper deals with the boundary stabilization problem for the 1D Fisher’s partial differential equation (PDE) defined on a bounded interval, which is a nonlinear unstable distributed parameter system. The stabilizing controller derived using the backstepping technique for linear parabolic PDEs (associated with Fisher’s PDE) can be extended to the nonlinear case. In the work, we can prove that the boundary controller based on the linear backstepping synthesis approach can make the closed-loop nonlinear Fisher’s system exponentially stable (with any desired decay rate) locally in both L2(0,1)L2(0,1) and H1(0,1)H1(0,1), respectively. The effectiveness of the control synthesis approach is illustrated by a numerical simulation.