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Russia Nizhni Novgorod
Section  Mathematics 
Title  On the application of Gaussian functions for discretization of optimal control problems 
Author(s)  Chernov A.V.^{ab} 
Affiliations  Nizhni Novgorod State Technical University^{a}, Nizhni Novgorod State University^{b} 
Abstract  On the example of well known problem of a road construction we study the opportunities of numerical solution for lumped optimal control problems by the method of control parametrization with the help of a linear combination of $\mu$ Gaussian functions. Recall that a Gaussian function (named also as quadratic exponent) is one defined as follows $\varphi(x)=\dfrac{1}{\sigma\sqrt{2\pi}}\exp\left[\dfrac{(xm)^2}{2\sigma^2}\right]$. The method is based on reduction of an original infinite dimensional optimization problem to finite dimensional minimization problem of a cost functional with respect to control approximation parameters. This paper is guided by the former author's research concerned the opportunities of approximation of one variable functions on a finite segment by a linear combination of $\mu$ Gaussian functions, and is to be regarded as its direct continuation. First of all, we prove an assertion concerning approximation on any finite segment for mother wavelet Mexican hat by a linear combination of two Gaussian functions. Hence, we obtain theoretical justification of the opportunity of an effective approximation for one variable functions on any finite segment with the help of linear combinations of Gaussian functions. After that, we give a comparison by quality of the approximation under study with the approximation in the style of Kotelnikov by means of numerical experiments. Then we give the road construction problem formulation and also the results of numerical solution for this problem which demonstrate obviously the advantages of our approach, in particular, a stability of numerical solution with respect to evaluation error of the approximation parameters for an optimal control, even with usage of small count of such parameters. 
Keywords  control parametrization technique, lumped problem of optimal control, approximation by quadratic exponents, Gaussian function 
UDC  517.518, 517.977.56 
MSC  41A30, 49M25, 49N90 
DOI  10.20537/vm170406 
Received  29 August 2017 
Language  Russian 
Citation  Chernov A.V. On the application of Gaussian functions for discretization of optimal control problems, Vestnik Udmurtskogo Universiteta. Matematika. Mekhanika. Komp'yuternye Nauki, 2017, vol. 27, issue 4, pp. 558575. 
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