Constrained retarded nonlinear optimal control problems in Banach state space
type: Item
instance of: scholarly article
title: Constrained retarded nonlinear optimal control problems in Banach state space
zbMATH Open document ID: 0864.49019
DOI: 10.1007/BF01182624
published in: Applied Mathematics and Optimization
publication date: 1997-06-22T00:00:00Z
review text: The author considers the functional differential equation \[ x'(t)= f\bigl(x (\cdot), u(t),t\bigr) \tag{1} \] where \(f:C(0,1;X)\times U\times [0,1]\to X\) \((X\) a Banach space, \(C(0,1;X)\) the space of continuous \(X\)-valued functions furnished with the supremum norm, \(U\) a topological space). The function \(f\) is nonanticipating in the sense that, for a given \(t\), it only depends on past values of \(x(t)\). An optimal control problem is assembled as follows. Let \(Y,Z\) be Banach spaces, \(r:X\to Z\), \(g:C(0,1;X) \times U\times [0,1]\to Y\) another nonanticipating function, \(L\) a one-dimensional subspace of \(Z\), \(b\in Z\), \(\varphi: Z\to \mathbb{R}\) a linear functional, \(C\) a subset of \(Y\). The problem is to minimize the cost functional \(\varphi(r(x(1)))\) among the solutions of (1) satisfying an initial condition and \(r(x(1)) \in b+L\), \(g(x(\cdot), u(t),t)\in C\), where the first inclusion takes care of the target condition and the second of the state constraints, which may be of functional type. The apparent restriction that the cost functional should depend only on the final point of the trajectory is lifted (as in classical control theory) by introducing an additional ``cost coordinate''. The author shows necessary conditions of the type of Pontryagin's maximum principle including target conditions both of instantaneous type (in particular, point targets) and of functional type; as expected, if there is not much slack in the target conditions one must rely on controllability properties of the linearized system to guarantee nontriviality of the costate.
reviewed by: Hector O. Fattorini
Mathematics Subject Classification ID: 49K25
Mathematics Subject Classification ID: 34K35
Mathematics Subject Classification ID: 93C30
zbMATH DE Number: 920738
zbMATH Keywords: functional differential equation
zbMATH Keywords: optimal control
zbMATH Keywords: nonanticipating function
zbMATH Keywords: cost functional
zbMATH Keywords: necessary conditions
zbMATH Keywords: Pontryagin's maximum principle
author: Atle Seierstad
MaRDI profile type: Publication
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full work available at URL: https://doi.org/10.1007/bf01182624
OpenAlex ID: W2079826826
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description: scientific article; zbMATH DE number 920738