Geodesic
Feedback minimum principle for quasi-optimal processes of terminally-constrained control problems
The Bulletin of Irkutsk State University. Series Mathematics, Tome 19 (2017), pp. 113-128 Cet article a éte moissonné depuis la source Math-Net.Ru

Voir la notice de l'article

In a series of previous works, we derived nonlocal necessary optimality conditions for free-endpoint problems. These conditions strengthen the Maximum Principle and are unified under the name “feedback minimum principle”. The present paper is aimed at extending these optimality conditions to terminally constrained problems. We propose a scheme of the proof of this generalization based on a “lift” of the constraints by means of a modified Lagrange function with a quadratic penalty. Implementation of this scheme employs necessary optimality conditions for quasi-optimal processes in approximating optimal control problems. In view of this, in the first part of the work, the feedback minimum principle is extended to quasi-optimal free-endpoint processes (i.e. strengthen the so-called $\varepsilon$-perturbed Maximum Principle). In the second part, this result is used to derive the approximate feedback minimum principle for a smooth terminally constrained problem. In an extended interpretation, the final assertion looks rather natural: If the constraints of the original problem are lifted by a sequence of relaxed approximating problems with the property of global convergence, then the global minimum at a feasible point of the original problem is admitted if and only if, for all $\varepsilon>0$, this point is $\varepsilon$-optimal, for all approximating problems of a sufficiently large index. In respect of optimal control with terminal constraints, the feedback $\varepsilon$-principle serves exactly for realization of the formulated assertion.
Keywords: perturbed Maximum Principle, feedback controls, terminal constraints, modified Lagrangians. 
@article{IIGUM_2017_19_a8,
     author = {V. A. Dykhta},
     title = {Feedback minimum principle for quasi-optimal processes of terminally-constrained control problems},
     journal = {The Bulletin of Irkutsk State University. Series Mathematics},
     pages = {113--128},
     year = {2017},
     volume = {19},
     language = {ru},
     url = {http://geodesic.mathdoc.fr/item/IIGUM_2017_19_a8/}
}
TY  - JOUR
AU  - V. A. Dykhta
TI  - Feedback minimum principle for quasi-optimal processes of terminally-constrained control problems
JO  - The Bulletin of Irkutsk State University. Series Mathematics
PY  - 2017
SP  - 113
EP  - 128
VL  - 19
UR  - http://geodesic.mathdoc.fr/item/IIGUM_2017_19_a8/
LA  - ru
ID  - IIGUM_2017_19_a8
ER  - 
%0 Journal Article
%A V. A. Dykhta
%T Feedback minimum principle for quasi-optimal processes of terminally-constrained control problems
%J The Bulletin of Irkutsk State University. Series Mathematics
%D 2017
%P 113-128
%V 19
%U http://geodesic.mathdoc.fr/item/IIGUM_2017_19_a8/
%G ru
%F IIGUM_2017_19_a8
V. A. Dykhta. Feedback minimum principle for quasi-optimal processes of terminally-constrained control problems. The Bulletin of Irkutsk State University. Series Mathematics, Tome 19 (2017), pp. 113-128. http://geodesic.mathdoc.fr/item/IIGUM_2017_19_a8/

[1] Arutyunov A. V., Optimality Conditions: Abnormal and Degenerate Problems, Mathematics and Its Applications, 526, Kluwer Academic Publishers, Dordrecht–Boston–London, 2000, 300 pp. | MR | MR | Zbl

[2] Bertsecas D. P., Constrained Optimization and Lagrange Multiplier Methods, Athena Scientific, 1996, 410 pp. | MR

[3] Gamkrelidze R. V., Principles of Optimal Control Theory, Mathematical Concepts and Methods in Science and Engineering, 7, New York Plenum Press, 1978, 175 pp. | MR | MR | Zbl

[4] Dmitruk A. V., Osmolovskii N. P., “On the Proof of Pontryagin's Maximum Principle by Means of Needle Variations”, Journal of Mathematical Sciences, 218:5 (2016), 581–598 | DOI | MR | Zbl

[5] Dykhta V. A., “Variational Optimality Conditions with Feedback Descent Controls that Strengthen the Maximum Principle”, The Bulletin of Irkutsk State University, 8 (2014), 86–103 | Zbl

[6] Dykhta V. A., “Variational Necessary Optimality Conditions with Feedback Descent Controls for Optimal Control Problems”, Doklady Mathematics, 91:3 (2015), 394–396 | DOI | DOI | MR | Zbl

[7] Dykhta V. A., “Positional Strengthenings of the Maximum Principle and Sufficient Optimality Conditions”, Proceedings of the Steklov Institute of Mathematics, 293, suppl. 1 (2016), S43–S57 | DOI | MR | Zbl | Zbl

[8] Dykhta V. A., Samsonyuk O. N., Hamilton–Jacobi inequalities and variational optimality conditions, ISU, Irkutsk, 2015, 150 pp.

[9] Clarke F., Optimization and nonsmooth analysis, Universite de Montreal, Montreal, 1989, 312 pp. | MR | MR | Zbl

[10] Krasovskii N. N., Subbotin A. I., Game-Theoretical Control Problems, Springer Series in Soviet Mathematics, Springer-Verlag, New York, 1988, 517 pp. | DOI | MR | MR | Zbl

[11] Mordukhovich B. S., Approximation Methods in Problems of Optimization and Control, Nauka, M., 1988, 360 pp. | MR

[12] Plotnikov V. I., Sumin M. I., “Construction of minimizing sequences”, Differ. Uravn., 19:4 (1983), 581–588

[13] Polyak B. T., Introduction to Optimization, Optimization Software, New York, 1987, 464 pp. | MR | MR

[14] F. Clarke, Necessary condition in dynamic optimization, Memoirs of the Amer. Math. Soc., 13, no. 816, 2005, 113 pp. | MR

[15] F. H. Clarke, C. Nour, “Nonconvex duality in optimal control”, SIAM J. Control Optim., 43 (2005), 2036–2048 | DOI | MR | Zbl

[16] I. Ekeland, “Nonconvex minimization problems”, Bull. Amer. Math. Soc., 1:3 (1979), 443–474 | DOI | MR | Zbl