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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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
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Abstract	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.
AbstractList	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.[PUBLICATION ABSTRACT] 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.
Author	Sager, Sebastian Hante, Falk M.
Author_xml	– sequence: 1 givenname: Falk M. surname: Hante fullname: Hante, Falk M. email: hante@math.fau.de organization: Department of Mathematics, University of Erlangen-Nuremberg – sequence: 2 givenname: Sebastian surname: Sager fullname: Sager, Sebastian organization: Institute of Mathematical Optimization, Otto-von-Guericke University
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Title	Relaxation methods for mixed-integer optimal control of partial differential equations
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