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Zero‐order asymptotic solution of a class of singularly perturbed linear‐quadratic problems with weak controls in a critical case
Author(s) -
Kurina Galina,
Thi Hoai Nguyen
Publication year - 2019
Publication title -
optimal control applications and methods
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.458
H-Index - 44
eISSN - 1099-1514
pISSN - 0143-2087
DOI - 10.1002/oca.2514
Subject(s) - mathematics , zero (linguistics) , power series , mathematical analysis , quadratic equation , series (stratigraphy) , asymptotic expansion , optimal control , method of matched asymptotic expansions , boundary (topology) , state (computer science) , boundary value problem , mathematical optimization , paleontology , philosophy , linguistics , geometry , biology , algorithm
Summary We consider a linear‐quadratic optimal control problem where the second power of a small parameter stands in front of the derivative and a control in a state equation and in front of a quadratic form with respect to control in a performance index; moreover, in the state equation, the nonhomogeneity has the first order of the power of a small parameter, and a matrix in front of the state variable is singular if the small parameter is equal to zero. Using immediate substituting a postulated asymptotic expansion of a solution, containing a regular series and four boundary layer functions series, into the problem condition, we obtain problems for finding asymptotic terms of the zero order for the optimal control and the first order for the optimal trajectory. An illustrative example is given.