Relaxation methods for mixed-integer optimal control of partial differential equations

We consider integer-restricted optimal control of systems governed by abstract semilinear evolution equations. This includes the problem of optimal control design for certain distributed parameter systems endowed with multiple actuators, where the task is to minimize costs associated with the dynami...

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Bibliographic Details
Published in:Computational optimization and applications Vol. 55; no. 1; pp. 197 - 225
Main Authors: Hante, Falk M., Sager, Sebastian
Format: Journal Article
Language:English
Published: Boston Springer US 01.05.2013
Springer Nature B.V
Subjects:
Actuators
Approximation
Banach spaces
Chromatography
Control algorithms
Control equipment
Convex and Discrete Geometry
Differential equations
Dynamical systems
Dynamics
Integer programming
Management Science
Mathematical analysis
Mathematics
Mathematics and Statistics
Methods
Numerical analysis
Operations Research
Operations Research/Decision Theory
Optimal control
Optimization
Ordinary differential equations
Partial differential equations
Statistics
Studies
Abstract evolution systems
Integerprogramming
Relaxation methods
Partial differential equations
Optimal control
ISSN:0926-6003, 1573-2894
Online Access:Get full text
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Summary:We consider integer-restricted optimal control of systems governed by abstract semilinear evolution equations. This includes the problem of optimal control design for certain distributed parameter systems endowed with multiple actuators, where the task is to minimize costs associated with the dynamics of the system by choosing, for each instant in time, one of the actuators together with ordinary controls. We consider relaxation techniques that are already used successfully for mixed-integer optimal control of ordinary differential equations. Our analysis yields sufficient conditions such that the optimal value and the optimal state of the relaxed problem can be approximated with arbitrary precision by a control satisfying the integer restrictions. The results are obtained by semigroup theory methods. The approach is constructive and gives rise to a numerical method. We supplement the analysis with numerical experiments.
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ISSN:0926-6003
1573-2894
DOI:10.1007/s10589-012-9518-3