Small Time Asymptotics on the Diagonal for Hörmander’s Type Hypoelliptic Operators

We compute the small time asymptotics of the fundamental solution of Hörmander’s type hypoelliptic operators with drift, on the diagonal at a point x 0 . We show that the order of the asymptotics depends on the controllability of an associated control problem and of its approximating system. If the...

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Bibliographic Details
Published inJournal of dynamical and control systems Vol. 23; no. 1; pp. 111 - 143
Main Author Paoli, Elisa
Format Journal Article
LanguageEnglish
Published New York Springer US 01.01.2017
Springer Nature B.V
Subjects
Approximation
Asymptotic properties
Calculus of Variations and Optimal Control; Optimization
Control
Control systems
Controllability
Dynamical Systems
Dynamical Systems and Ergodic Theory
Lie groups
Mathematics
Mathematics and Statistics
Nonlinear programming
Operators (mathematics)
Stability
Systems Theory
Vibration
Control problem with drift
Fundamental solution
49J20
Hypoelliptic operators
Small time asymptocics
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Summary:We compute the small time asymptotics of the fundamental solution of Hörmander’s type hypoelliptic operators with drift, on the diagonal at a point x 0 . We show that the order of the asymptotics depends on the controllability of an associated control problem and of its approximating system. If the control problem of the approximating system is controllable at x 0 , then so is also the original control problem, and in this case we show that the fundamental solution blows up as t − N / 2 , where N is a number determined by the Lie algebra at x 0 of the fields, that define the hypoelliptic operator.
ISSN:1079-2724
1573-8698
DOI:10.1007/s10883-016-9321-z