On one variational problem of the piecewise-linear dynamical approximation

Some properties of the discrete variational problem of the dynamic approximation in the complex Euclidean $(L+1)$-dimensional space are studied here. It generalizes familiar problems of the mean square polynomial approximation of the functions given on the finite interval in accordance with their references. In the problem under consideration sequence approximation $\mathbf y=\{y_i\}_0^L$ of the references of the function $y(t)\in L^2[0,T]$, $T=Lh$ on the lattice $I_h$ is achieved by solving homogeneous linear differential equations or difference equations of the given order $n$ with constant but possibly unknown coefficients. Thus, it is shown that in the latter case the approximation problem also includes the identification problem. The analysis of its properties is the main subject of the article. The problem is set to find vector of coefficients $\alpha$ of difference equation $\sum_0^n\widehat y_{i+k}\alpha_i=0$, where $k=\overline{0,L-n}$. Coefficients $\alpha$ and initial conditions of the transient process $\widehat{\mathbf y}$ of this equation are optimized. The optimization purpose is to achieve the best approximation of the dynamic process $\mathbf y\in E$ being considered here. The approximation criterion is a minimum of the quantity $\|\mathbf y-\widehat{\mathbf y}\|^2_E$. The variational problem under study is shown to be reduced to the problem of projecting vector $\mathbf y$ in $E$ on the kernels of the difference operators with unknown coefficients $\alpha\in\omega\subset\mathcal S\subset E^{n+1}$, where $\alpha$ is a direction, $S$ is a sphere or a hyperplane. The problem under study is shown to be related to the problems of the discretization and identifiability. In this case vector coordinates $\mathbf y\in E$ is an exact solution of differential equation on the lattice $I_h$ and $\mathbf y=\widehat{\mathbf y}$. The problem of the variational identification is compared with algebraic methods of identification. The orthogonal complement to the kernels of the difference operators are shown to always have Toeplitz basis. This results in fast projecting algorithms of computation. The problem of finding optimal vector $\widehat\alpha$ is shown to be reduced to the problem of the absolute minimization of the identification functional depending on the direction $\alpha$ in $E^{n+1}$. The iterative procedure of its minimization on a sphere with wide domain and high speed of convergence is presented here. The variational problem considered here can be applied in mathematical modeling for control problem and research purposes. On the finite intervals, for example, it is possible to use piecewise-linear dynamic approximations of the complex dynamic processes with difference and differential equations of the specified type.