# American Institute of Mathematical Sciences

• Previous Article
Adaptive time--mesh refinement in optimal control problems with state constraints
• DCDS Home
• This Issue
• Next Article
Variational problems of Herglotz type with time delay: DuBois--Reymond condition and Noether's first theorem
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. [2] F. H. Clarke, Optimization and Nonsmooth Analysis,, Wiley-Interscience, (1983). [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). [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. [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. [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. [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. [8] T. T. Rockafellar and J.-B. Wets, Variational Analysis,, Grundlehren er Mathematischen Wissenshaft, (1998). doi: 10.1007/978-3-642-02431-3. [9] R. B. Vinter, Optimal Control,, Birkhäuser, (2000). [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. [11] J. Warga, Optimal Control of Differential and Functional Equations,, Academic Press, (1972). [12] J. Warga, Controllability, extremality, and abnormality in nonsmooth optimal control,, J. Optim. Theory and Applic., 41 (1983), 239. doi: 10.1007/BF00934445. [13] J. Warga, Optimization and controllability without differentiability assumptions,, SIAM J. Control and Optim., 21 (1983), 837. doi: 10.1137/0321051.

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. [2] F. H. Clarke, Optimization and Nonsmooth Analysis,, Wiley-Interscience, (1983). [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). [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. [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. [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. [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. [8] T. T. Rockafellar and J.-B. Wets, Variational Analysis,, Grundlehren er Mathematischen Wissenshaft, (1998). doi: 10.1007/978-3-642-02431-3. [9] R. B. Vinter, Optimal Control,, Birkhäuser, (2000). [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. [11] J. Warga, Optimal Control of Differential and Functional Equations,, Academic Press, (1972). [12] J. Warga, Controllability, extremality, and abnormality in nonsmooth optimal control,, J. Optim. Theory and Applic., 41 (1983), 239. doi: 10.1007/BF00934445. [13] J. Warga, Optimization and controllability without differentiability assumptions,, SIAM J. Control and Optim., 21 (1983), 837. doi: 10.1137/0321051.
 [1] Jan-Hendrik Webert, Philip E. Gill, Sven-Joachim Kimmerle, Matthias Gerdts. A study of structure-exploiting SQP algorithms for an optimal control problem with coupled hyperbolic and ordinary differential equation constraints. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1259-1282. doi: 10.3934/dcdss.2018071 [2] Francis Clarke. A general theorem on necessary conditions in optimal control. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 485-503. doi: 10.3934/dcds.2011.29.485 [3] Andrei V. Dmitruk, Nikolai P. Osmolovskii. Necessary conditions for a weak minimum in optimal control problems with integral equations on a variable time interval. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4323-4343. doi: 10.3934/dcds.2015.35.4323 [4] Hongwei Lou, Jiongmin Yong. Second-order necessary conditions for optimal control of semilinear elliptic equations with leading term containing controls. Mathematical Control & Related Fields, 2018, 8 (1) : 57-88. doi: 10.3934/mcrf.2018003 [5] Andrei V. Dmitruk, Nikolai P. Osmolovski. Necessary conditions for a weak minimum in a general optimal control problem with integral equations on a variable time interval. Mathematical Control & Related Fields, 2017, 7 (4) : 507-535. doi: 10.3934/mcrf.2017019 [6] Elimhan N. Mahmudov. Optimal control of second order delay-discrete and delay-differential inclusions with state constraints. Evolution Equations & Control Theory, 2018, 7 (3) : 501-529. doi: 10.3934/eect.2018024 [7] Lukáš Adam, Jiří Outrata. On optimal control of a sweeping process coupled with an ordinary differential equation. Discrete & Continuous Dynamical Systems - B, 2014, 19 (9) : 2709-2738. doi: 10.3934/dcdsb.2014.19.2709 [8] Sofia O. Lopes, Fernando A. C. C. Fontes, Maria do Rosário de Pinho. On constraint qualifications for nondegenerate necessary conditions of optimality applied to optimal control problems. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 559-575. doi: 10.3934/dcds.2011.29.559 [9] Hongwei Lou, Weihan Wang. Optimal blowup/quenching time for controlled autonomous ordinary differential equations. Mathematical Control & Related Fields, 2015, 5 (3) : 517-527. doi: 10.3934/mcrf.2015.5.517 [10] Huaiqiang Yu, Bin Liu. Pontryagin's principle for local solutions of optimal control governed by the 2D Navier-Stokes equations with mixed control-state constraints. Mathematical Control & Related Fields, 2012, 2 (1) : 61-80. doi: 10.3934/mcrf.2012.2.61 [11] Jianxiong Ye, An Li. Necessary optimality conditions for nonautonomous optimal control problems and its applications to bilevel optimal control. Journal of Industrial & Management Optimization, 2017, 13 (5) : 1-21. doi: 10.3934/jimo.2018101 [12] Luís Tiago Paiva, Fernando A. C. C. Fontes. Adaptive time--mesh refinement in optimal control problems with state constraints. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 4553-4572. doi: 10.3934/dcds.2015.35.4553 [13] Theodore Tachim-Medjo. Optimal control of a two-phase flow model with state constraints. Mathematical Control & Related Fields, 2016, 6 (2) : 335-362. doi: 10.3934/mcrf.2016006 [14] Stepan Sorokin, Maxim Staritsyn. Feedback necessary optimality conditions for a class of terminally constrained state-linear variational problems inspired by impulsive control. Numerical Algebra, Control & Optimization, 2017, 7 (2) : 201-210. doi: 10.3934/naco.2017014 [15] Janosch Rieger. The Euler scheme for state constrained ordinary differential inclusions. Discrete & Continuous Dynamical Systems - B, 2016, 21 (8) : 2729-2744. doi: 10.3934/dcdsb.2016070 [16] Hongwei Lou. Second-order necessary/sufficient conditions for optimal control problems in the absence of linear structure. Discrete & Continuous Dynamical Systems - B, 2010, 14 (4) : 1445-1464. doi: 10.3934/dcdsb.2010.14.1445 [17] Haiyan Zhang. A necessary condition for mean-field type stochastic differential equations with correlated state and observation noises. Journal of Industrial & Management Optimization, 2016, 12 (4) : 1287-1301. doi: 10.3934/jimo.2016.12.1287 [18] Matthias Gerdts, Martin Kunkel. A nonsmooth Newton's method for discretized optimal control problems with state and control constraints. Journal of Industrial & Management Optimization, 2008, 4 (2) : 247-270. doi: 10.3934/jimo.2008.4.247 [19] Piernicola Bettiol. State constrained $L^\infty$ optimal control problems interpreted as differential games. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3989-4017. doi: 10.3934/dcds.2015.35.3989 [20] Shahlar F. Maharramov. Necessary optimality conditions for switching control problems. Journal of Industrial & Management Optimization, 2010, 6 (1) : 47-55. doi: 10.3934/jimo.2010.6.47

2017 Impact Factor: 1.179