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Generalized splines in R^n and optimal control

Rui C. Rodrigues·Delfim F. M. Torres
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Abstract

We have found an inconsistency in our previous version of the paper "Generalized splines in R^n and optimal control". We give a new-time-dependent definition of spline curves in R^n which results from solving a non-autonomous linear quadratic optimal control problem (P) where the matrix B(t) is assumed to be rectangular with maximum rank. Nevertheless, our results are only valid if B(t) is a square (nonsingular) matrix. This was pointed out to us by Andrey Sarychev. We have proceeded with the necessary corrections. %%%%%%%%%%%%%%%%%% We give a new time-dependent definition of spline curves in R^n, which extends a recent definition of vector-valued splines introduced by Rodrigues and Silva Leite for the time-independent case. Previous results are based on a variational approach, with lengthy arguments, which do not cover the non-autonomous situation. We show that the previous results are a consequence of the Pontryagin maximum principle, and are easily generalized using the methods of optimal control. Main result asserts that vector-valued splines are related to the Pontryagin extremals of a non-autonomous linear-quadratic optimal control problem.

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Rui C. Rodrigues, Delfim F. M. Torres. 2005-06-14. Generalized splines in R^n and optimal control. https://arxiv.org/abs/math/0405200

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