Implicit parametrizations and applications in optimization and control
type: Item
instance of: scholarly article
title: Implicit parametrizations and applications in optimization and control
zbMATH Open document ID: 1456.49008
published in: Mathematical Control and Related Fields
publication date: 2021-01-13T00:00:00Z
review text: The subject is the characterization (with numerical applications in mind) of the manifold \(V\) of solutions of the nonlinear system \[ F_j(x_1, x_2, \dots, x_d) = 0 \quad (1 \le j \le l )\quad l \le d - 1 \tag{1} \] in the vicinity of \(x^0 = (x^0_1, x^0_2 , \dots ,x^0_d) \in V,\) under the Jacobian assumption \[ \frac{\partial (F_1, F_2, \dots, F_l)}{\partial (x_1, x_2, \dots, x_l)} \ne 0 \quad \hbox{in} \ x^0 = (x^0_1, x^0_2 , \dots ,x^0_d) \, . \] The first step involves the underdetermined linear system \[ v(x) \cdot \nabla F_j(x) = 0 \quad (1 \le j \le l) \] which is used to obtain bases \((v_1(x), v_2(x), \dots, v_{d - l}(x))\) for the tangent spaces of \(V.\) Next, the chain of differential equations \begin{align*} \frac{\partial y_1(t_1)}{\partial t_1} &= v_1(y_1(t_1)), y_1(0) = x^0 \cr \frac{\partial y_2(t_1, t_2)}{\partial t_2} &= v_2(y_2(t_1, t_2)),\quad y_2(t_1, 0) = y(t_1) \cr & \hskip 2em \dots \dots \dots \dots \cr \frac{\partial y_{d - l}(t_1, t_2, \dots, t_{d - l})}{ \partial t_{d - l}} &= v_{d - l}(y_{d - l}(t_1, t_2, \dots, t_{d - l})) \, , \cr & \hskip 2.7em y_{d - l}(t_1, \dots , t_{d - l - 1}, 0) = y_{d - l - 1}(t_1, t_2, \dots , t_{d - l - 1}) \end{align*} is set up, thus constructing a parametrization of \(V\) which may be considered as an explicit form of the implicit function theorem. The result is used to construct an algorithm for the solution of the problem of minimizing a function \(g(x_0, x_2, \dots , x_d)\) subject to (1). Some generalizations are covered, such as the case where (1) includes inequality constraints and/or regularity is relaxed. In the last section the results are applied to the control problem of minimizing \(l(x(0), x(1))\) among the trajectories of the system \(x'(t) = f(t, x(t), u(t))\) subject to the state-control constraint \(h(x(t), u(t)) = 0.\) There are several numerical implementation of the algorithms and the author notes that computations can be carried out using standard Matlab routines.
Mathematics Subject Classification ID: 49J21
Mathematics Subject Classification ID: 90C30
Mathematics Subject Classification ID: 26B10
Mathematics Subject Classification ID: 34A12
Mathematics Subject Classification ID: 49K15
Mathematics Subject Classification ID: 49K21
Mathematics Subject Classification ID: 49M37
zbMATH DE Number: 7293640
zbMATH Keywords: local parametrizations
zbMATH Keywords: nonlinear programming
zbMATH Keywords: Lagrange multipliers
zbMATH Keywords: implicit function theorem: optimal control with constraints
zbMATH Keywords: numerical algorithms
author: Dan Tiba
reviewed by: Hector O. Fattorini
describes a project that uses: libMC
MaRDI profile type: Publication
full work available at URL: mcrf.2020006
OpenAlex ID: W2995048147
Wikidata QID: Q126418220
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DOI: 10.3934/MCRF.2020006
description: scientific article; zbMATH DE number 7293640