The Derivation of Structural Properties of LMg Splines by Necessary Condition of Optimal Control

and Applied Analysis 3 Transversality conditions: λ (0) = − ∂ ∂x (0) [θ 1 F 1 (x (0))] , λ (1) = ∂ ∂x (1) [θ N F N (x (1))] ,


Introduction
Spline smoothing is widely used in many application areas as a basic tool of applied mathematics and a class of powerful and flexible modeling techniques.The operator spline function theory provides new descriptions and numerical methods for some optimal control problems [1][2][3].Splines defined in terms of one linear differential operator are called L-g splines.The structural and continuity properties of these splines were discussed in detail in [4,5].They play an important role in the approximation of functions and optimal control.Splines defined in terms of two differential operators were first considered by de Figueiredo [6] and were called LM-g splines.He defined LM-g splines by separating Green's function and showed their applications in the estimation of stochastic processes, minimum-energy controls, digital signal processing, and system modeling.However, it was difficult to discuss the structural and continuity properties of LM-g splines by the method of separating Green's function.Soon afterwards, Weinert et al. [7] redefined LM-g splines with certain integrodifferential operators in terms of the system inverses.In [7], these splines were indicated as sample functions of a certain type of autoregressive-moving average (ARMA) stochastic processes, so they were also called ARMA splines.This stochastic correspondence was used to develop the recursive algorithms for ARMA splines, interpolating splines, and smoothing splines [5,8].Zhang and Fang [9] established a new recursive algorithm for LM-g splines by using system inverses and reproducing kernels, and based on this, a recursive algorithm for the minimum-energy control of a class of linear systems was also obtained.
Although LM-g splines have been applied extensively in many fields and some recursive algorithms for which these splines have been developed, a systematic research on the structural and continuity properties of these splines has not been done.It is a quite difficult task to discuss the structural and continuity properties of abstract operator splines.Opfer and Oberle [10] and Fredenhagen et al. [11] presented a derivation of cubic splines with obstacles by the optimal control criteria.Afterwards, this method was used to deduce the structural and continuity properties of generalized interpolating splines with obstacles [12].Takahashi and Martin [13] considered and made the generalized splines by solving an optimal control problem for linear systems.Egerstedt and Martin [2] showed how splines arose naturally in the theory of linear control systems from the latest methods and applications and by using the tools of optimization over vector spaces, demonstrated how splines were a consequence of the optimization problem, and revealed many natural relations among control theory, numerical analysis, and statistics.Alhanaty and Bercovier [14] introduced optimal control methods over reparametrization for curve and surface design.Zhang and Liu [15,16] considered the structural and continuity characteristics of the interpolating splines defined by
For the convenience of discussing the structural and continuity properties of LM-g spline () and its applications, we will restrict attention to a rather broad class of constraint functionals {  }  =1 , called extended Hermite-Birkhoff (EHB) functionals, which have the form where From (1), we define The operators { *  }  =1 are called the partial adjoint operator of . * =  *  is called the adjoint operator of .

Structural Properties of LM-g Spline
According to the optimal control theory of linear systems, the minimization problem ( 9) is to find a control function () which is piecewise smooth with certain conditions to minimize the objective function with the constraints From the necessary conditions of the optimal control with state constraints [11,12], we know that there exist piecewise continuously differentiable Lagrange multipliers () = ( 1 (),  2 (), . . .,   ()) and constants  1 ,  2 , . . .,   , such that, for the Hamiltonian function and the state constraints (14), the following properties hold.
Hence, we complete the proof of Theorem 1.
Theorem 2. Let () be the L-g spline with the EHB functional interpolation constraints (10).Then () has the following properties: From the discussion above, we know if we can find the solution  * () of the optimal control problem of system (34)-(35) which is determined by one linear differential operator  and subjected to the same constraint conditions as system ( 12)-( 14), the solution ũ * () of the optimal control problem of system (12)-( 14) which is determined by two linear differential operators ,  can be obtained; that is, ũ * () =  *  * ().
In order to obtain the LM-g spline () by the corresponding L-g spline from (39), it is necessary to consider the explicit expression of  * .From ( 8) and (33), we can obtain  * as follows.(41)

Conclusion and Prospect
According to [5][6][7], we define the LM-g spline as an integraldifferential operator spline; this integral-differential operator is determined by the inversion of system.The correspondence between LM-g spline interpolating and state-restricted optimal control problems has been used to deduce the structural and continuity properties of LM-g splines.This correspondence is also used to establish the meaningful relationship between LM-g splines and L-g splines.
For our results in this paper, we believe that there are some topics worthy of further consideration.First, under some appropriate conditions, the results can be generalized to variable-coefficient operators  and .Second, the structural and continuity properties of LM-g splines with state constraints can be deduced by optimal control.Finally, the applications of our results in optimal control and digital signal processing need further investigation.