• Previous Article
    Variational problems of Herglotz type with time delay: DuBois--Reymond condition and Noether's first theorem
  • DCDS Home
  • This Issue
  • Next Article
    Adaptive time--mesh refinement in optimal control problems with state constraints
September  2015, 35(9): 4573-4592. doi: 10.3934/dcds.2015.35.4573

When are minimizing controls also minimizing relaxed controls?

1. 

Imperial College London, Electrical and Electronical Engineering Department, South Kensington Campus, London SW7 2AZ, United Kingdom, United Kingdom

Received  May 2014 Published  April 2015

Relaxation refers to the procedure of enlarging the domain of a variational problem or the search space for the solution of a set of equations, to guarantee the existence of solutions. In optimal control theory relaxation involves replacing the set of permissible velocities in the dynamic constraint by its convex hull. Usually the infimum cost is the same for the original optimal control problem and its relaxation. But it is possible that the relaxed infimum cost is strictly less than the infimum cost. It is important to identify such situations, because then we can no longer study the infimum cost by solving the relaxed problem and evaluating the cost of the relaxed minimizer. Following on from earlier work by Warga, we explore the relation between the existence of an infimum gap and abnormality of necessary conditions (i.e. they are valid with the cost multiplier set to zero). Two kinds of theorems are proved. One asserts that a local minimizer, which is not also a relaxed minimizer, satisfies an abnormal form of the Pontryagin Maximum Principle. The other asserts that a local relaxed minimizer that is not also a minimizer satisfies an abnormal form of the relaxed Pontryagin Maximum Principle.
Citation: Michele Palladino, Richard B. Vinter. When are minimizing controls also minimizing relaxed controls?. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4573-4592. doi: 10.3934/dcds.2015.35.4573
References:
[1]

F. H. Clarke, The maximum principle under minimal hypotheses,, SIAM J. Control and Optim., 14 (1976), 1078.  doi: 10.1137/0314067.  Google Scholar

[2]

F. H. Clarke, Optimization and Nonsmooth Analysis,, Wiley-Interscience, (1983).   Google Scholar

[3]

F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory,, Graduate Texts in Mathematics Vol. 178, (1998).   Google Scholar

[4]

A. L. Dontchev and R. T. Rockafellar, Implicit Functions and Solution Mappings: A View from Variational Analysis,, Monographs in Mathematics, (2009).  doi: 10.1007/978-0-387-87821-8.  Google Scholar

[5]

A. Ioffe, Euler lagrange and hamiltonian formalisms in dynamic optimization,, Trans. Amer. Math. Soc., 349 (1997), 2871.  doi: 10.1090/S0002-9947-97-01795-9.  Google Scholar

[6]

A. Ioffe, Optimal control of differential inclusions: New developments and open problems,, Proc. 41 IEEE Conference on Decision and Control, 3 (2002), 3127.  doi: 10.1109/CDC.2002.1184349.  Google Scholar

[7]

M. Palladino and R. B. Vinter, Minimizers that are not also relaxed minimizers,, SIAM J. Control and Optim., 52 (2014), 2164.  doi: 10.1137/130909627.  Google Scholar

[8]

T. T. Rockafellar and J.-B. Wets, Variational Analysis,, Grundlehren er Mathematischen Wissenshaft, (1998).  doi: 10.1007/978-3-642-02431-3.  Google Scholar

[9]

R. B. Vinter, Optimal Control,, Birkhäuser, (2000).   Google Scholar

[10]

J. Warga, Normal control problems have no minimizing strictly original solutions,, Bulletin of the Amer. Math. Soc., 77 (1971), 625.  doi: 10.1090/S0002-9904-1971-12779-9.  Google Scholar

[11]

J. Warga, Optimal Control of Differential and Functional Equations,, Academic Press, (1972).   Google Scholar

[12]

J. Warga, Controllability, extremality, and abnormality in nonsmooth optimal control,, J. Optim. Theory and Applic., 41 (1983), 239.  doi: 10.1007/BF00934445.  Google Scholar

[13]

J. Warga, Optimization and controllability without differentiability assumptions,, SIAM J. Control and Optim., 21 (1983), 837.  doi: 10.1137/0321051.  Google Scholar

show all references

References:
[1]

F. H. Clarke, The maximum principle under minimal hypotheses,, SIAM J. Control and Optim., 14 (1976), 1078.  doi: 10.1137/0314067.  Google Scholar

[2]

F. H. Clarke, Optimization and Nonsmooth Analysis,, Wiley-Interscience, (1983).   Google Scholar

[3]

F. H. Clarke, Y. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory,, Graduate Texts in Mathematics Vol. 178, (1998).   Google Scholar

[4]

A. L. Dontchev and R. T. Rockafellar, Implicit Functions and Solution Mappings: A View from Variational Analysis,, Monographs in Mathematics, (2009).  doi: 10.1007/978-0-387-87821-8.  Google Scholar

[5]

A. Ioffe, Euler lagrange and hamiltonian formalisms in dynamic optimization,, Trans. Amer. Math. Soc., 349 (1997), 2871.  doi: 10.1090/S0002-9947-97-01795-9.  Google Scholar

[6]

A. Ioffe, Optimal control of differential inclusions: New developments and open problems,, Proc. 41 IEEE Conference on Decision and Control, 3 (2002), 3127.  doi: 10.1109/CDC.2002.1184349.  Google Scholar

[7]

M. Palladino and R. B. Vinter, Minimizers that are not also relaxed minimizers,, SIAM J. Control and Optim., 52 (2014), 2164.  doi: 10.1137/130909627.  Google Scholar

[8]

T. T. Rockafellar and J.-B. Wets, Variational Analysis,, Grundlehren er Mathematischen Wissenshaft, (1998).  doi: 10.1007/978-3-642-02431-3.  Google Scholar

[9]

R. B. Vinter, Optimal Control,, Birkhäuser, (2000).   Google Scholar

[10]

J. Warga, Normal control problems have no minimizing strictly original solutions,, Bulletin of the Amer. Math. Soc., 77 (1971), 625.  doi: 10.1090/S0002-9904-1971-12779-9.  Google Scholar

[11]

J. Warga, Optimal Control of Differential and Functional Equations,, Academic Press, (1972).   Google Scholar

[12]

J. Warga, Controllability, extremality, and abnormality in nonsmooth optimal control,, J. Optim. Theory and Applic., 41 (1983), 239.  doi: 10.1007/BF00934445.  Google Scholar

[13]

J. Warga, Optimization and controllability without differentiability assumptions,, SIAM J. Control and Optim., 21 (1983), 837.  doi: 10.1137/0321051.  Google Scholar

[1]

Stefan Doboszczak, Manil T. Mohan, Sivaguru S. Sritharan. Pontryagin maximum principle for the optimal control of linearized compressible navier-stokes equations with state constraints. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020110

[2]

Hai Huang, Xianlong Fu. Optimal control problems for a neutral integro-differential system with infinite delay. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020107

[3]

Manil T. Mohan. First order necessary conditions of optimality for the two dimensional tidal dynamics system. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020045

[4]

Hong Niu, Zhijiang Feng, Qijin Xiao, Yajun Zhang. A PID control method based on optimal control strategy. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 117-126. doi: 10.3934/naco.2020019

[5]

Lorenzo Zambotti. A brief and personal history of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 471-487. doi: 10.3934/dcds.2020264

[6]

Fabio Camilli, Giulia Cavagnari, Raul De Maio, Benedetto Piccoli. Superposition principle and schemes for measure differential equations. Kinetic & Related Models, , () : -. doi: 10.3934/krm.2020050

[7]

Lars Grüne, Matthias A. Müller, Christopher M. Kellett, Steven R. Weller. Strict dissipativity for discrete time discounted optimal control problems. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020046

[8]

Yueyang Zheng, Jingtao Shi. A stackelberg game of backward stochastic differential equations with partial information. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020047

[9]

Stefan Ruschel, Serhiy Yanchuk. The spectrum of delay differential equations with multiple hierarchical large delays. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 151-175. doi: 10.3934/dcdss.2020321

[10]

Pengyu Chen. Non-autonomous stochastic evolution equations with nonlinear noise and nonlocal conditions governed by noncompact evolution families. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020383

[11]

Mengni Li. Global regularity for a class of Monge-Ampère type equations with nonzero boundary conditions. Communications on Pure & Applied Analysis, 2021, 20 (1) : 301-317. doi: 10.3934/cpaa.2020267

[12]

Youming Guo, Tingting Li. Optimal control strategies for an online game addiction model with low and high risk exposure. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020347

[13]

Pierluigi Colli, Gianni Gilardi, Jürgen Sprekels. Deep quench approximation and optimal control of general Cahn–Hilliard systems with fractional operators and double obstacle potentials. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 243-271. doi: 10.3934/dcdss.2020213

[14]

Thabet Abdeljawad, Mohammad Esmael Samei. Applying quantum calculus for the existence of solution of $ q $-integro-differential equations with three criteria. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020440

[15]

Fathalla A. Rihan, Hebatallah J. Alsakaji. Stochastic delay differential equations of three-species prey-predator system with cooperation among prey species. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020468

[16]

Bernard Bonnard, Jérémy Rouot. Geometric optimal techniques to control the muscular force response to functional electrical stimulation using a non-isometric force-fatigue model. Journal of Geometric Mechanics, 2020  doi: 10.3934/jgm.2020032

[17]

Xuemei Chen, Julia Dobrosotskaya. Inpainting via sparse recovery with directional constraints. Mathematical Foundations of Computing, 2020, 3 (4) : 229-247. doi: 10.3934/mfc.2020025

[18]

Zuliang Lu, Fei Huang, Xiankui Wu, Lin Li, Shang Liu. Convergence and quasi-optimality of $ L^2- $norms based an adaptive finite element method for nonlinear optimal control problems. Electronic Research Archive, 2020, 28 (4) : 1459-1486. doi: 10.3934/era.2020077

[19]

João Marcos do Ó, Bruno Ribeiro, Bernhard Ruf. Hamiltonian elliptic systems in dimension two with arbitrary and double exponential growth conditions. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 277-296. doi: 10.3934/dcds.2020138

[20]

Soniya Singh, Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of second order impulsive systems with state-dependent delay in Banach spaces. Evolution Equations & Control Theory, 2020  doi: 10.3934/eect.2020103

2019 Impact Factor: 1.338

Metrics

  • PDF downloads (65)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]