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 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.

[1]  Peter I. Kogut,et al.  Optimal Control Problems for Partial Differential Equations on Reticulated Domains , 2011 .

[2]  Falk M. Hante,et al.  Modeling and Analysis of Modal Switching in Networked Transport Systems , 2009 .

[3]  H. Frankowska,et al.  A priori estimates for operational differential inclusions , 1990 .

[4]  Johannes P. Schlöder,et al.  An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization: Part II: Software aspects and applications , 2003, Comput. Chem. Eng..

[5]  M. Pachter,et al.  Optimal control of partial differential equations , 1980 .

[6]  S. Krein,et al.  Linear Differential Equations in Banach Space , 1972 .

[7]  Michael Hinze,et al.  Discrete Concepts in PDE Constrained Optimization , 2009 .

[8]  Jiongmin Yong,et al.  Optimal Control Theory for Infinite Dimensional Systems , 1994 .

[9]  J.A.M. Kuipers,et al.  Safety analysis of switching between reductive and oxidative conditions in a reaction coupling reverse flow reactor , 2000 .

[10]  Günter Leugering,et al.  An augmented BV setting for feedback switching control , 2010, J. Syst. Sci. Complex..

[11]  F. Tröltzsch.  Optimal Control of Partial Differential Equations: Theory, Methods and Applications , 2010 .

[12]  S. Mitter,et al.  Representation and Control of Infinite Dimensional Systems , 1992 .

[13]  Günter Leugering,et al.  Optimal Boundary Control of Convention-Reaction Transport Systems with Binary Control Functions , 2009, HSCC.

[14]  H. Bock,et al.  A Multiple Shooting Algorithm for Direct Solution of Optimal Control Problems , 1984 .

[15]  Orest V. Iftime,et al.  Optimal control of switched distributed parameter systems with spatially scheduled actuators , 2009, Autom..

[16]  Johannes P. Schlöder,et al.  An efficient multiple shooting based reduced SQP strategy for large-scale dynamic process optimization. Part 1: theoretical aspects , 2003, Comput. Chem. Eng..

[17]  Moritz Diehl,et al.  The integer approximation error in mixed-integer optimal control , 2012, Math. Program..

[18]  Valentina E. Balas,et al.  On the Switching Control , 2009 .

[19]  Kazimierz Malanowski,et al.  Lipschitz Stability Of Solutions To Parametric Optimal Control Problems For Parabolic Equations , 1998 .

[20]  乔花玲,et al.  关于Semigroups of Linear Operators and Applications to Partial Differential Equations的两个注解 , 2003 .

[21]  Martin Gugat,et al.  Optimal switching boundary control of a string to rest in finite time , 2008 .

[22]  Sebastian Engell,et al.  Optimisation and control of chromatography , 2005, Comput. Chem. Eng..

[23]  G. Marriott,et al.  Optically switchable chelates: optical control and sensing of metal ions. , 2008, The Journal of organic chemistry.

[24]  Stefan Wendl,et al.  Optimal Control of Partial Differential Equations , 2021, Applied Mathematical Sciences.

[25]  Stefan Ulbrich,et al.  Optimization with PDE Constraints , 2008, Mathematical modelling.

[26]  K. Hoffmann,et al.  Optimal Control of Partial Differential Equations , 1991 .

[27]  Christian Kirches,et al.  Combinatorial integral approximation , 2011, Math. Methods Oper. Res..

[28]  Fredi Tröltzsch,et al.  Lipschitz Stability of Solutions to Parametric Optimal Control Problems for Parabolic Equations , 1999 .

[29]  Peter N. Brown,et al.  Decay to Uniform States in Ecological Interactions , 1980 .

[30]  Alain Bensoussan,et al.  Representation and Control of Infinite Dimensional Systems (Systems & Control: Foundations & Applications) , 2006 .

[31]  G. Pianigiani,et al.  Evolution inclusions in non separable Banach spaces , 1999 .

[32]  L. Biegler,et al.  Nonlinear Programming Superstructure for Optimal Dynamic Operations of Simulated Moving Bed Processes , 2006 .

[33]  S. Sager.  Reformulations and algorithms for the optimization of switching decisions in nonlinear optimal control , 2009 .

[34]  Sebastian Sager,et al.  Numerical methods for mixed-integer optimal control problems , 2006 .

[35]  Gerhard Reinelt,et al.  Direct methods with maximal lower bound for mixed-integer optimal control problems , 2009, Math. Program..

[36]  E. S. Noussair,et al.  Differential equations in Banach spaces , 1973, Bulletin of the Australian Mathematical Society.