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Some Results On Nonsmooth Systems including Max-type Functions

Year 2024, Volume: 16 Issue: 1, 147 - 153, 30.06.2024

Serkan İlter , Hülya Duru

https://doi.org/10.47000/tjmcs.1330667

Abstract

In this paper, we consider optimal control systems with continuous-time which is governed by systems
of ordinary differential equation including max-type functions. We derive some properties concerning nonsmooth concepts, and a special form of adjoint condition involved in the maximum principle giving necessary conditions of optimality for optimal control problems of these systems.

Keywords

Optimal control problems nonsmooth analysis max-type functions

References

  • Agarwal, R., Hristova, S., Strict stability in terms of two measures for impulsive differential equations with ‘supremum’, Appl. Anal., 91(7)(2012), 1379–1392.
  • Aliprantis, C.D., Border, K.C., Infinite Dimensional Analysis A Hitchhiker’s Guide, 3nd ed., Springer-Verlag, Berlin, 2007.
  • Bainov, D., Hristova, S., Differential Equations with Maxima, Taylor and Francis, 2011.
  • Bantsur, N.R., Trofimchuk, O.P., On the existence and stability of periodic and almost periodic solutions of quasilinear equations with maxima, Ukrainian Mathematical Journal, 50(6)(1998), 747–754.
  • Brogliato, B., Nonsmooth Mechanics: Models, Dynamics and Control, Springer-Verlag, London, 1999.
  • Clarke, F.H., Generalized gradients and applications, Trans. Amer. Math. Soc., 205(1975), 247–262.
  • Clarke, F.H., The Maximum Principle under minimal hypotheses, SIAM J. Control and Optimization, 14(6)(1976), 1078–1091.
  • Clarke, F.H., Optimization and Nonsmooth Analysis, SIAM Classics in Applied Mathematics, Philadelphia, 1990.
  • Demyanov, V.F., Rubinov, A.M., Quasidifferential Calculus, Optimization Software, Inc. Publications Division, New York, 1986.
  • Geiger, T., Wachsmuth, D., Wachsmuth, G. Optimal control of ODEs with state suprema, Mathematical Control and Related Fields, 11(3)(2021), 555–578.
  • Giorgi, G., Guerraggio, A., Thierfelder, J., Mathematics of Optimization: Smooth and Nonsmooth Case, Elsevier Science, 2004.
  • Hristova, S., Golev, A., Stefanova, K., Approximate method for boundary value problems of anti-periodic type for differential equations with maxima, Boundary Value Problems, 1(2013), 12.
  • Ilter, S., Weak maximum principle for optimal control problems of nonsmooth systems, Applied Mathematics and Computation, 218(3)(2011), 805–808.
  • Jankovski, T., Systems of differential equations with maxima, Doklady Akademii Nauk, 8(1997), 57–60.
  • McClamroch, N.H., State Models of Dynamic Systems, Springer-Verlag, New York-Heidelberg-Berlin, 1980.
  • Otrocol, D., Systems of functional differential equations with maxima, of mixed type, Electron. J. Qual. Theory Differ. Equ., 5(2014), 1–9.
  • Popov, E., Automatic Regulation and Control, Nauka, Moscow, 1966.Roberts, A.W., Varberg, D.E., Convex Functions, Academic Press, New York-London, 1973.
  • Roberts, A.W., Varberg, D.E., Convex Functions, Academic Press, New York-London, 1973.
  • Rockafellar, R.T., Convex Analysis, Princeton University, New Jersey, 1970.
  • Shpakovich, V.P., Muntyan, V.I., Method of averaging for differential equations with maxima, Ukrainian Mathematical Journal, 39(5)(1987), 543–545.
  • Vinter, R.B., Optimal Control, Birkh¨auser, Boston, 2000.

Year 2024, Volume: 16 Issue: 1, 147 - 153, 30.06.2024

Serkan İlter , Hülya Duru

https://doi.org/10.47000/tjmcs.1330667

Abstract

Thanks

Ön izleme aşamasında, "Ek Bilgiler" bölümünde kendi ülkenizden iki, yurt dışından iki hakemi aşağıdaki şekilde önermelisiniz/ uyarısı yapılmaktadır, ancak Ek Bilgiler bölümde hakem önerilerini yazabileceğim uygun alt-bölüm bulamamam sebebiyle bu bölümde belirtmek istedim: bu önerilerimiz aşağıdaki gibidir. Teşekkürler, İyi çalışmalar. Özkan DEĞER, Türkiye, İstanbul Üniversitesi, ozdeger@istanbul.edu.tr; Gülçin DİNÇ YALÇIN, Türkiye, Eskişehir Teknik Üniversitesi, gdinc@eskisehir.edu.tr; Ali Hakan TOR, Türkiye, Abdullah Gül Üniversitesi, hakan.tor@agu.edu.tr; Savin TREANTA, Romania, Polytechnic University of Bucharest, Romanya, savin_treanta@yahoo.com; Basil KOKKİNİS, Greece, National Technical University of Athens, bkok@math.ntua.gr

References

  • Agarwal, R., Hristova, S., Strict stability in terms of two measures for impulsive differential equations with ‘supremum’, Appl. Anal., 91(7)(2012), 1379–1392.
  • Aliprantis, C.D., Border, K.C., Infinite Dimensional Analysis A Hitchhiker’s Guide, 3nd ed., Springer-Verlag, Berlin, 2007.
  • Bainov, D., Hristova, S., Differential Equations with Maxima, Taylor and Francis, 2011.
  • Bantsur, N.R., Trofimchuk, O.P., On the existence and stability of periodic and almost periodic solutions of quasilinear equations with maxima, Ukrainian Mathematical Journal, 50(6)(1998), 747–754.
  • Brogliato, B., Nonsmooth Mechanics: Models, Dynamics and Control, Springer-Verlag, London, 1999.
  • Clarke, F.H., Generalized gradients and applications, Trans. Amer. Math. Soc., 205(1975), 247–262.
  • Clarke, F.H., The Maximum Principle under minimal hypotheses, SIAM J. Control and Optimization, 14(6)(1976), 1078–1091.
  • Clarke, F.H., Optimization and Nonsmooth Analysis, SIAM Classics in Applied Mathematics, Philadelphia, 1990.
  • Demyanov, V.F., Rubinov, A.M., Quasidifferential Calculus, Optimization Software, Inc. Publications Division, New York, 1986.
  • Geiger, T., Wachsmuth, D., Wachsmuth, G. Optimal control of ODEs with state suprema, Mathematical Control and Related Fields, 11(3)(2021), 555–578.
  • Giorgi, G., Guerraggio, A., Thierfelder, J., Mathematics of Optimization: Smooth and Nonsmooth Case, Elsevier Science, 2004.
  • Hristova, S., Golev, A., Stefanova, K., Approximate method for boundary value problems of anti-periodic type for differential equations with maxima, Boundary Value Problems, 1(2013), 12.
  • Ilter, S., Weak maximum principle for optimal control problems of nonsmooth systems, Applied Mathematics and Computation, 218(3)(2011), 805–808.
  • Jankovski, T., Systems of differential equations with maxima, Doklady Akademii Nauk, 8(1997), 57–60.
  • McClamroch, N.H., State Models of Dynamic Systems, Springer-Verlag, New York-Heidelberg-Berlin, 1980.
  • Otrocol, D., Systems of functional differential equations with maxima, of mixed type, Electron. J. Qual. Theory Differ. Equ., 5(2014), 1–9.
  • Popov, E., Automatic Regulation and Control, Nauka, Moscow, 1966.Roberts, A.W., Varberg, D.E., Convex Functions, Academic Press, New York-London, 1973.
  • Roberts, A.W., Varberg, D.E., Convex Functions, Academic Press, New York-London, 1973.
  • Rockafellar, R.T., Convex Analysis, Princeton University, New Jersey, 1970.
  • Shpakovich, V.P., Muntyan, V.I., Method of averaging for differential equations with maxima, Ukrainian Mathematical Journal, 39(5)(1987), 543–545.
  • Vinter, R.B., Optimal Control, Birkh¨auser, Boston, 2000.
There are 21 citations in total.

Details

Primary Language English
Subjects Calculus of Variations, Mathematical Aspects of Systems Theory and Control Theory
Journal Section Articles
Authors

Serkan İlter 0000-0002-7847-5124

Hülya Duru 0000-0003-0689-460X

Publication Date June 30, 2024
Published in Issue Year 2024 Volume: 16 Issue: 1

Cite

APA İlter, S., & Duru, H. (2024). Some Results On Nonsmooth Systems including Max-type Functions. Turkish Journal of Mathematics and Computer Science, 16(1), 147-153. https://doi.org/10.47000/tjmcs.1330667
AMA İlter S, Duru H. Some Results On Nonsmooth Systems including Max-type Functions. TJMCS. June 2024;16(1):147-153. doi:10.47000/tjmcs.1330667
Chicago İlter, Serkan, and Hülya Duru. “Some Results On Nonsmooth Systems Including Max-Type Functions”. Turkish Journal of Mathematics and Computer Science 16, no. 1 (June 2024): 147-53. https://doi.org/10.47000/tjmcs.1330667.
EndNote İlter S, Duru H (June 1, 2024) Some Results On Nonsmooth Systems including Max-type Functions. Turkish Journal of Mathematics and Computer Science 16 1 147–153.
IEEE S. İlter and H. Duru, “Some Results On Nonsmooth Systems including Max-type Functions”, TJMCS, vol. 16, no. 1, pp. 147–153, 2024, doi: 10.47000/tjmcs.1330667.
ISNAD İlter, Serkan - Duru, Hülya. “Some Results On Nonsmooth Systems Including Max-Type Functions”. Turkish Journal of Mathematics and Computer Science 16/1 (June2024), 147-153. https://doi.org/10.47000/tjmcs.1330667.
JAMA İlter S, Duru H. Some Results On Nonsmooth Systems including Max-type Functions. TJMCS. 2024;16:147–153.
MLA İlter, Serkan and Hülya Duru. “Some Results On Nonsmooth Systems Including Max-Type Functions”. Turkish Journal of Mathematics and Computer Science, vol. 16, no. 1, 2024, pp. 147-53, doi:10.47000/tjmcs.1330667.
Vancouver İlter S, Duru H. Some Results On Nonsmooth Systems including Max-type Functions. TJMCS. 2024;16(1):147-53.