Quadratic conditions for nonsingular extremals in optimal control (a theoretical treatment)
type: Item
instance of: scholarly article
title: Quadratic conditions for nonsingular extremals in optimal control (a theoretical treatment)
zbMATH Open document ID: 0916.49016
author: Nikolai P. Osmolovskii
published in: Russian Journal of Mathematical Physics
publication date: 1999-01-26T00:00:00Z
review text: The subject of this paper is an optimal control problem dealing with the dynamic controlled object given by: \[ \dot{x}=f(x, u, t), \] \[ g(x, u, t)=0,\quad h(x, u, t) \leq 0, \] \[ (x(t_{0}), x(t_{1})) \in P,\qquad (x, u, t) \in Q \] where \(P\) and \(Q\) are open sets, \(x, u, f, g, h\) are the vector functions fulfilling the inequality \(d(g) \leq d(u)\). There is considered the following control optimal problem on a fixed time interval \([t_{0}, t_{1}]\) \[ \text{minimum }\{ J(x, u)=(F_0(x(t_{0}), x(t_{1})) \} \] subjected to the constraints \[ F(x(t_{0}), x(t_{1})) \leq 0, \quad G(x(t_{0}), x(t_{1})) = 0 \] where \(F\) and \(G\) are also vector functions fulfilling the same inequality as above, i.e., \(d(g) \leq d(u)\). Necessary conditions and quadratic sufficient conditions have been stated for different types of minima in the canonical optimal control problem. Jacobi-type conditions have been used in an examination of the definiteness of the quadratic form. In the time-optimal control problem for a linear systems the general quadratic conditions have been reduced to a quite simple form in a finite-dimensional space. The work establishes a survey of results of investigations published previously by the same author.
Mathematics Subject Classification ID: 49K15
Mathematics Subject Classification ID: 49J15
Mathematics Subject Classification ID: 93C15
zbMATH DE Number: 1206245
zbMATH Keywords: optimal control problem
zbMATH Keywords: necessary and sufficient conditions for optimal solution
zbMATH Keywords: control systems
reviewed by: Władysław Hejmo
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1206245