Linear Quadratic Optimal Control for Systems Governed by First-Order Hyperbolic Partial Differential Equations

This paper focuses on linear-quadratic(LQ)optimal control for a class of systems governed by first-order hyperbolic partial differential equations(PDEs).Different from most of the previous works,an approach of discretization-then-continuousization is proposed in this paper to cope with the infinite-dimensional nature of PDE systems.The contributions of this paper consist of the following aspects:(1)The differential Riccati equations and the solvability condition of the LQ optimal control problems are obtained via the discretization-then-continuousization method.(2)A numerical calculation way of the differential Riccati equations and a practical design way of the optimal controller are proposed.Meanwhile,the relationship between the optimal costate and the optimal state is established by solving a set of forward and backward partial difference equations(FBPDEs).(3)The correctness of the method used in this paper is verified by a complementary continuous method and the comparative analysis with the existing operator results is presented.It is shown that the proposed results not only contain the classic results of the standard LQ control problem of systems governed by ordinary differential equations as a special case,but also support the existing operator results and give a more convenient form of computation.

[ℓ_(p)]_(e.r) Euler-Riesz Difference Sequence Spaces

Bas¸ar and Braha[1],introduced the sequence spaces˘ℓ_(∞),c˘and c˘0 of EulerCesaro bounded,convergent and null difference sequences and studied their some´properties.Then,in[2],we introduced the sequence spaces[ℓ_(∞)]_(e.r),[c]_(e.r)and[c_(0)]_(e.r)of Euler-Riesz bounded,convergent and null difference sequences by using the composition of the Euler mean E1 and Riesz mean Rq with backward difference operator∆.The main purpose of this study is to introduce the sequence space[ℓ_(p)]_(e.r)of Euler-Riesz p−absolutely convergent series,where 1≤p<∞,difference sequences by using the composition of the Euler mean E1 and Riesz mean Rq with backward difference operator∆.Furthermore,the inclusionℓ_(p)⊂[ℓ_(p)]_(e.r)hold,the basis of the sequence space[ℓ_(p)]_(e.r)is constucted andα−,β−andγ−duals of the space are determined.Finally,the classes of matrix transformations from the[ℓ_(p)]_(e.r)Euler-Riesz difference sequence space to the spacesℓ_(∞),c and c0 are characterized.We devote the final section of the paper to examine some geometric properties of the space[ℓ_(p)]_(e.r).

Semi-Implicit Interior Penalty Discontinuous Galerkin Methods for Viscous Compressible Flows

We deal with the numerical solution of the Navier-Stokes equations describing a motion of viscous compressible fluids.In order to obtain a sufficiently stable higher order scheme with respect to the time and space coordinates,we develop a combination of the discontinuous Galerkin finite element(DGFE)method for the space discretization and the backward difference formulae(BDF)for the time discretization.Since the resulting discrete problem leads to a system of nonlinear algebraic equations at each time step,we employ suitable linearizations of inviscid as well as viscous fluxes which give a linear algebraic problem at each time step.Finally,the resulting BDF-DGFE scheme is applied to steady as well as unsteady flows and achieved results are compared with reference data.

ANALYSIS OF A FULLY DISCRETE FINITE ELEMENT METHOD FOR PARABOLIC INTERFACE PROBLEMS WITH NONSMOOTH INITIAL DATA

This article concerns numerical approximation of a parabolic interface problem with general L 2 initial value.The problem is discretized by a finite element method with a quasi-uniform triangulation of the domain fitting the interface,with piecewise linear approximation to the interface.The semi-discrete finite element problem is furthermore discretized in time by the k-step backward difference formula with k=1,...,6.To maintain high-order convergence in time for possibly nonsmooth L 2 initial value,we modify the standard backward difference formula at the first k−1 time levels by using a method recently developed for fractional evolution equations.An error bound of O(t−k nτk+t−1 n h 2|log h|)is established for the fully discrete finite element method for general L 2 initial data.

A SECOND ORDER UNCONDITIONALLY CONVERGENT FINITE ELEMENT METHOD FOR THE THERMAL EQUATION WITH JOULE HEATING PROBLEM

In this paper,we study the finite element approximation for nonlinear thermal equation.Because the nonlinearity of the equation,our theoretical analysis is based on the error of temporal and spatial discretization.We consider a fully discrete second order backward difference formula based on a finite element method to approximate the temperature and electric potential,and establish optimal L^(2) error estimates for the fully discrete finite element solution without any restriction on the time-step size.The discrete solution is bounded in infinite norm.Finally,several numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method.

A MFE method combined with L1-approximation for a nonlinear time-fractional coupled diffusion system

In this paper,a nonlinear time-fractional coupled diffusion system is solved by using a mixed finite element(MFE)method in space combined with L1-approximation and implicit second-order backward difference scheme in time.The stability for nonlinear fully discrete finite element scheme is analyzed and a priori error estimates are derived.Finally,some numerical tests are shown to verify our theoretical analysis.

A Comparison of Semi-Lagrangian and Lagrange-Galerkin hp-FEM Methods in Convection-Diffusion Problems

We perform a comparison in terms of accuracy and CPU time between second order BDF semi-Lagrangian and Lagrange-Galerkin schemes in combination with high order finite element method.The numerical results show that for polynomials of degree 2 semi-Lagrangian schemes are faster than Lagrange-Galerkin schemes for the same number of degrees of freedom,however,for the same level of accuracy both methods are about the same in terms of CPU time.For polynomials of degree larger than 2,Lagrange-Galerkin schemes behave better than semi-Lagrangian schemes in terms of both accuracy and CPU time;specially,for polynomials of degree 8 or larger.Also,we have performed tests on the parallelization of these schemes and the speedup obtained is quasi-optimal even with more than 100 processors.

SOME NEW DISCRETE INEQUALITIES OF OPIAL WITH TWO SEQUENCES

In this paper, we establish some new discrete inequalities of Opial-type with two sequences by making use of some classical inequalities. These results contain as special cases improvements of results given in the literature, and these improvements are new even in the important discrete case.