| Spline interpolation constitutes a class of piecewise polynomial approximation that is commonly used when approximating many of the functions that arise in actual physical processes. Spline approximations of functions are preferred to most approximation and interpolation methods because of their inherent smoothness properties.; Generally, classical polynomial splines tend to exhibit undesirable undulations. In this work, we discuss a technique, based on control principles, for eliminating these undulations and increasing the smoothness properties of the spline interpolants. We give a generalization of the classical polynomial splines and show that this generalization is, in fact, a family of splines that covers the broad spectrum of polynomial, trigonometric and exponential splines. A particular element in this family is determined by the appropriate control data.; In their paper Splines and Control Theory, Zhang, Tomlinson, and Martin show the relationship between control theory and spline approximation by studying the minimum energy problem, namely:; minimize{dollar}{dollar}J(u)=intsbsp{lcub}0{rcub}{lcub}T{rcub} usp2(s)ds{dollar}{dollar}subject to{dollar}{dollar}{lcub}dover dt{rcub}vec x(t)=Avec x(t)+vec bu(t), tinlbrack0,Trbrack.{dollar}{dollar}However, this approach does not eliminate the undulations associated with classical polynomial splines. In an attempt to overcome this problem, we use a modified cost function:{dollar}{dollar}J(u)=intsbOmegasumsbsp{lcub}k=0{rcub}{lcub}n{rcub}betasbsp{lcub}k{rcub}{lcub}2{rcub}vert usp{lcub}(k){rcub}vertsp2dt{dollar}{dollar}Here, {dollar}vec xinIRsp{lcub}m{rcub}, vec yinIRsp{lcub}p{rcub}, vec uin{lcub}bf L{rcub}sp2(Omega,IRsp{lcub}l{rcub}), tinOmegasubsetIR,{dollar} and {dollar}{lcub}bf A{rcub}in{lcub}cal L{rcub}(IRsp{lcub}m{rcub},IRsp{lcub}m{rcub}), {lcub}bf B{rcub}in{lcub}cal L{rcub}(IRsp{lcub}l{rcub},IRsp{lcub}m{rcub}) {lcub}rm and{rcub} {lcub}bf C{rcub}in{lcub}cal L{rcub}(IRsp{lcub}m{rcub},IRsp{lcub}p{rcub}).{dollar} The vectors {dollar}vec x{dollar} and {dollar}vec u{dollar} are the state and control vectors of the system, respectively. A, B, and C are, respectively, the state, control and observation matrices.; Our goal is to find the control {dollar}vec u{dollar} that will drive the system from one point to the other in the state space {dollar}IRsp{lcub}m{rcub}{dollar} and at the same time minimizes the cost function J(u). The problem will be formulated in the sobolev space {dollar}Hsp{lcub}n{rcub}(Omega).{dollar} We will also establish controllability conditions of the system and then apply the results to spline approximation.; Basically, we intend to use optimal control theory to develop methodology for spline approximations. As an illustration, suppose that the write-head of a computer is required to move from a certain position, {dollar}vec x(tsb0),{dollar} to another position, {dollar}vec x(T),{dollar} then some control u(t) is needed to drive the write-head from the initial state, x(0), to the final state, x(T). Thus, the write-head must go through a certain set of points at given times. A spline curve can be fitted through these data points and then a control that takes the system through this trajectory determined. We will find the set of controls{dollar}{dollar}usb{lcub}i{rcub}(t){lcub}:{rcub} vec x(tsb{lcub}i-1{rcub}){lcub}buildrel{lcub}usb{lcub}i{rcub}(t){rcub}over{lcub}to{rcub}{rcub}vec x(tsb{lcub}i{rcub}){dollar}{dollar}that achieves this while minimizing the functional J(u). Then, by applying the appropriate smoothness requirements of u(t) at the endpoints of each subinterval, {dollar}lbrack tsb{lcub}i-1{rcub},tsb{lcub}i{rcub}rbrack,{dollar} we will obtain and characterize the class of spline functions. Numerical and curve-fitting examples are given to illustrate the advantages of this technique over the classical approach. Convergence properties of the interpolant are als... |
