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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. [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.
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