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Mat. Sb., 2000, Volume 191, Number 6, Pages 69–100 (Mi msb484)  

This article is cited in 19 scientific papers (total in 20 papers)

On the problem of control synthesis: the Pontryagin alternating integral and the Hamilton–Jacobi equation

A. B. Kurzhanskii, N. B. Melnikov

M. V. Lomonosov Moscow State University

Abstract: This paper deals with the problem of control synthesis under unknown, but bounded disturbances for a system with linear structure and hard (geometric) bounds on the control and the disturbance inputs. It emphasizes the role of set-valued methods and, in particular, of the Pontryagin multivalued alternating integral in the corresponding solution schemes. Close ties with the Hamilton–Jacobi techniques are discussed.
This paper also discusses an approach producing effective numerical solutions on the basis of appropriate ellipsoidal techniques. It presents a framework for going over from the abstract theory to numerically realizable ellipsoidal representations.


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English version:
Sbornik: Mathematics, 2000, 191:6, 849–881

Bibliographic databases:

UDC: 517.977
MSC: Primary 93C15, 93B50, 49J53; Secondary 49Lxx
Received: 24.06.1999

Citation: A. B. Kurzhanskii, N. B. Melnikov, “On the problem of control synthesis: the Pontryagin alternating integral and the Hamilton–Jacobi equation”, Mat. Sb., 191:6 (2000), 69–100; Sb. Math., 191:6 (2000), 849–881

Citation in format AMSBIB
\by A.~B.~Kurzhanskii, N.~B.~Melnikov
\paper On the problem of control synthesis: the Pontryagin alternating integral and the~Hamilton--Jacobi equation
\jour Mat. Sb.
\yr 2000
\vol 191
\issue 6
\pages 69--100
\jour Sb. Math.
\yr 2000
\vol 191
\issue 6
\pages 849--881

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    This publication is cited in the following articles:
    1. Dar'in A.N., Kurzhanski A.B., “Nonlinear control synthesis under two types of constraints”, Differ. Equ., 37:11 (2001), 1549–1558  mathnet  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    2. Kurzhanski A.B., Varaiya P., “On reachability under uncertainty”, SIAM J. Control Optim., 41:1 (2002), 181–216  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    3. Daryin A., “Nonlinear synthesis for uncertain systems with diverse types of constraints”, Nonlinear Control Systems 2001, Ifac Symposia Series, 2002, 1511–1516  mathscinet  isi
    4. Kurzhanski A.B., Varaiya P., “Reachability under uncertainty”, Proc. of the 41st IEEE Conference on Decision AND Control, 1–4 (2002), 1982–1987  crossref  isi
    5. Dar'in A.N., Kurzhanskii A.B., “Control under indeterminacy and double constraints”, Differ. Equ., 39:11 (2003), 1554–1567  mathnet  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    6. Vostrikov I.V., Dar'in A.N., Kurzhanskii A.B., “On the damping of a ladder-type vibration system subjected to uncertain perturbations”, Differ. Equ., 42:11 (2006), 1524–1535  mathnet  crossref  mathscinet  zmath  isi  elib  scopus  scopus  scopus
    7. Daryin A.N., Kurzhanski A.B., Vostrikov I.V., “Reachability approaches and ellipsoidal techniques for closed-loop control of oscillating systems under uncertainty”, 45th IEEE Conference on Decision and Control, 2006, 6385–6390  crossref  mathscinet  isi
    8. Dias S., Smirnov G., “High Precision Attitude Control via Differential Game Techniques”, Numerical Analysis and Applied Mathematics, AIP Conference Proceedings, 1281, 2010, 469–472  crossref  adsnasa  isi  scopus  scopus  scopus
    9. Iskanadjiev I., “Duality of the Alternating Integral for Quasi-Linear Differential Games”, Nonlinear Anal.-Model Control, 17:2 (2012), 169–181  mathscinet  zmath  isi  elib
    10. Gagarinov P.V., “Vychislenie alternirovannykh trubok dostizhimosti dlya lineinykh upravlyaemykh sistem pri neopredelennosti”, Vestnik moskovskogo universiteta. seriya 15: vychislitelnaya matematika i kibernetika, 4 (2012), 17–25  mathscinet  elib
    11. A. Azamov, I. M. Iskanadjiev, “Pontryagin’s Alternating Integral for Differential Inclusions with Counteraction”, Contributions to Game Theory and Management, 5 (2012), 33–44  mathnet
    12. Iskanadjiev I.M., “On the Pontryagin Lower Alternating Integral”, J. Automat. Inf. Sci., 45:2 (2013), 33–40  crossref  isi  scopus  scopus  scopus
    13. A. A. Chikrii, A. A. Belousov, “O lineinykh differentsialnykh igrakh s vypuklymi integralnymi ogranicheniyami”, Tr. IMM UrO RAN, 19, no. 4, 2013, 308–319  mathnet  mathscinet  elib
    14. A. A. Chikrii, G. Ts. Chikrii, “Matrix resolving functions in game dynamic problems”, Proc. Steklov Inst. Math. (Suppl.), 291, suppl. 1 (2015), 56–65  mathnet  crossref  mathscinet  isi  elib
    15. “Alexander Borisovich Kurzhanski. On the occasion of his 75th birthday”, Proc. Steklov Inst. Math. (Suppl.), 292, suppl. 1 (2016), 1–13  mathnet  crossref  mathscinet  isi
    16. L. A. Vlasenko, A. A. Chikrii, “On a differential game in a distributed system”, Proc. Steklov Inst. Math. (Suppl.), 292, suppl. 1 (2016), 276–285  mathnet  crossref  mathscinet  isi  elib
    17. E. K. Kostousova, “On the polyhedral method of solving problems of control strategy synthesis”, Proc. Steklov Inst. Math. (Suppl.), 292, suppl. 1 (2016), 140–155  mathnet  crossref  mathscinet  isi  elib
    18. Kostousova E.K., “On Control Synthesis For Uncertain Differential Systems Using a Polyhedral Technique”, Large-Scale Scientific Computing, LSSC 2013, Lecture Notes in Computer Science, 8353, eds. Lirkov I., Margenov S., Wasniewski J., Springer-Verlag Berlin, 2014, 98–106  crossref  mathscinet  isi  scopus  scopus  scopus
    19. Iskanadjiev I.M., “Lower Pontryagin'S Alternating Integral For Differential Inclusions”, Cybern. Syst. Anal., 51:5 (2015), 782–791  crossref  mathscinet  zmath  isi  scopus  scopus  scopus
    20. A. A. Chikrii, “Verkhnyaya i nizhnyaya razreshayuschie funktsii v igrovykh zadachakh dinamiki”, Tr. IMM UrO RAN, 23:1 (2017), 293–305  mathnet  crossref  elib
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