American Institute of Mathematical Sciences

Advanced
September  2015, 35(9): 4367-4384. doi: 10.3934/dcds.2015.35.4367

Optimal control of dynamical systems with polynomial impulses

Elena Goncharova 1, and  Maxim Staritsyn 1,

1. 

Institute for System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences, Irkutsk, Russian Federation, Russian Federation

Received  April 2014 Revised  September 2014 Published  April 2015

The paper is devoted to the $BV$-relaxation of a dynamical system, whose right-hand side is a $p$th degree polynomial with rational powers of control under a uniform bound on its $L_p$-norm, and coefficients containing usual measurable bounded control.
    Under natural convexity assumptions, we give an explicit representation of generalized solutions to the control system by a measure differential equation. The main results concern an optimal impulsive control problem for the relaxed system: We establish the existence of a minimizer, and give necessary optimality conditions in the form of a Maximum Principle.
Keywords: Maximum Principle., measure differential equations, Optimal control, impulsive control, $BV$-relaxation.
Mathematics Subject Classification: Primary: 49N25, 49K99; Secondary: 49J9.
Citation: Elena Goncharova, Maxim Staritsyn. Optimal control of dynamical systems with polynomial impulses. Discrete & Continuous Dynamical Systems, 2015, 35 (9) : 4367-4384. doi: 10.3934/dcds.2015.35.4367
References:
[1]

J. Math. Sci., 165 (2010), 654-688. doi: 10.1007/s10958-010-9834-z.  Google Scholar

[2]

SIAM J. Control Optim., 43 (2005), 1812-1843. doi: 10.1137/S0363012903430068.  Google Scholar

[3]

Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar

[4]

Discr. Cont. Dynam. Syst., 20 (2008), 1-35. doi: 10.3934/dcds.2008.20.1.  Google Scholar

[5]

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Mem., 2 (1993), 30pp.  Google Scholar

[6]

Arch. Ration. Mech. Anal., 196 (2010), 97-141. doi: 10.1007/s00205-009-0237-6.  Google Scholar

[7]

SIAM J. Control Optim., 31 (1993), 1205-1220. doi: 10.1137/0331057.  Google Scholar

[8]

J. Optim. Theory Appl., 81 (1994), 435-457. doi: 10.1007/BF02193094.  Google Scholar

[9]

IMACS Ann. Comput. Appl. Math., 8 (1990), 103-109.  Google Scholar

[10]

Comput. Math. Math. Phys., 49 (2009), 942-957. doi: 10.1134/S0965542509060050.  Google Scholar

[11]

Autom. Remote Control, 17 (1972), 14-21. Google Scholar

[12]

North-Holland, Amsterdam, 1979.  Google Scholar

[13]

J. Math. Sci., 139 (2006), 7087-7150. doi: 10.1007/s10958-006-0408-z.  Google Scholar

[14]

SIAM J. Control Optim., 34 (1996), 1420-1440. doi: 10.1137/S0363012994263214.  Google Scholar

[15]

Kluwer Academic / Plenum Publishers, New York, 2001. Google Scholar

[16]

Differential Integral Equations, 8 (1995), 269-288.  Google Scholar

[17]

(2014) [published online as arXiv:1406.7655v1]. Google Scholar

[18]

SIAM J. Control Optim., 48 (2009), 415-437. doi: 10.1137/08071805X.  Google Scholar

[19]

Indiana Univ. Math. J., 49 (2000), 1043-1077. doi: 10.1512/iumj.2000.49.1736.  Google Scholar

[20]

J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1965), 191-205. doi: 10.1137/0303016.  Google Scholar

[21]

SIAM J. Control Optim., 26 (1988), 205-229. doi: 10.1137/0326013.  Google Scholar

[22]

J. Math. Anal. Appl., 4 (1962), 111-128. doi: 10.1016/0022-247X(62)90033-1.  Google Scholar

[23]

Academic Press, New York, 1972.  Google Scholar

[24]

J. SIAM Control Ser. A, 3 (1965), 424-438. doi: 10.1137/0303028.  Google Scholar

[25]

Kluwer Academic Publishers, Dorderecht, 1997. doi: 10.1007/978-94-015-8893-5.  Google Scholar

show all references

References:
[1]

J. Math. Sci., 165 (2010), 654-688. doi: 10.1007/s10958-010-9834-z.  Google Scholar

[2]

SIAM J. Control Optim., 43 (2005), 1812-1843. doi: 10.1137/S0363012903430068.  Google Scholar

[3]

Springer-Verlag, Berlin, 1984. doi: 10.1007/978-3-642-69512-4.  Google Scholar

[4]

Discr. Cont. Dynam. Syst., 20 (2008), 1-35. doi: 10.3934/dcds.2008.20.1.  Google Scholar

[5]

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Mem., 2 (1993), 30pp.  Google Scholar

[6]

Arch. Ration. Mech. Anal., 196 (2010), 97-141. doi: 10.1007/s00205-009-0237-6.  Google Scholar

[7]

SIAM J. Control Optim., 31 (1993), 1205-1220. doi: 10.1137/0331057.  Google Scholar

[8]

J. Optim. Theory Appl., 81 (1994), 435-457. doi: 10.1007/BF02193094.  Google Scholar

[9]

IMACS Ann. Comput. Appl. Math., 8 (1990), 103-109.  Google Scholar

[10]

Comput. Math. Math. Phys., 49 (2009), 942-957. doi: 10.1134/S0965542509060050.  Google Scholar

[11]

Autom. Remote Control, 17 (1972), 14-21. Google Scholar

[12]

North-Holland, Amsterdam, 1979.  Google Scholar

[13]

J. Math. Sci., 139 (2006), 7087-7150. doi: 10.1007/s10958-006-0408-z.  Google Scholar

[14]

SIAM J. Control Optim., 34 (1996), 1420-1440. doi: 10.1137/S0363012994263214.  Google Scholar

[15]

Kluwer Academic / Plenum Publishers, New York, 2001. Google Scholar

[16]

Differential Integral Equations, 8 (1995), 269-288.  Google Scholar

[17]

(2014) [published online as arXiv:1406.7655v1]. Google Scholar

[18]

SIAM J. Control Optim., 48 (2009), 415-437. doi: 10.1137/08071805X.  Google Scholar

[19]

Indiana Univ. Math. J., 49 (2000), 1043-1077. doi: 10.1512/iumj.2000.49.1736.  Google Scholar

[20]

J. Soc. Indust. Appl. Math. Ser. A Control, 3 (1965), 191-205. doi: 10.1137/0303016.  Google Scholar

[21]

SIAM J. Control Optim., 26 (1988), 205-229. doi: 10.1137/0326013.  Google Scholar

[22]

J. Math. Anal. Appl., 4 (1962), 111-128. doi: 10.1016/0022-247X(62)90033-1.  Google Scholar

[23]

Academic Press, New York, 1972.  Google Scholar

[24]

J. SIAM Control Ser. A, 3 (1965), 424-438. doi: 10.1137/0303028.  Google Scholar

[25]

Kluwer Academic Publishers, Dorderecht, 1997. doi: 10.1007/978-94-015-8893-5.  Google Scholar

[1]

Shi'an Wang, N. U. Ahmed. Optimal control and stabilization of building maintenance units based on minimum principle. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1713-1727. doi: 10.3934/jimo.2020041
[2]

Qi Lü, Xu Zhang. A concise introduction to control theory for stochastic partial differential equations. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021020
[3]

Guirong Jiang, Qishao Lu. The dynamics of a Prey-Predator model with impulsive state feedback control. Discrete & Continuous Dynamical Systems - B, 2006, 6 (6) : 1301-1320. doi: 10.3934/dcdsb.2006.6.1301
[4]

Tobias Geiger, Daniel Wachsmuth, Gerd Wachsmuth. Optimal control of ODEs with state suprema. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021012
[5]

Shanjian Tang, Fu Zhang. Path-dependent optimal stochastic control and viscosity solution of associated Bellman equations. Discrete & Continuous Dynamical Systems, 2015, 35 (11) : 5521-5553. doi: 10.3934/dcds.2015.35.5521
[6]

Diana Keller. Optimal control of a linear stochastic Schrödinger equation. Conference Publications, 2013, 2013 (special) : 437-446. doi: 10.3934/proc.2013.2013.437
[7]

Lorenzo Freddi. Optimal control of the transmission rate in compartmental epidemics. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021007
[8]

Marzia Bisi, Maria Groppi, Giorgio Martalò, Romina Travaglini. Optimal control of leachate recirculation for anaerobic processes in landfills. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2957-2976. doi: 10.3934/dcdsb.2020215
[9]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213
[10]

Wensheng Yin, Jinde Cao, Guoqiang Zheng. Further results on stabilization of stochastic differential equations with delayed feedback control under $ G $-expectation framework. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021072
[11]

Paula A. González-Parra, Sunmi Lee, Leticia Velázquez, Carlos Castillo-Chavez. A note on the use of optimal control on a discrete time model of influenza dynamics. Mathematical Biosciences & Engineering, 2011, 8 (1) : 183-197. doi: 10.3934/mbe.2011.8.183
[12]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399
[13]

Xiaohong Li, Mingxin Sun, Zhaohua Gong, Enmin Feng. Multistage optimal control for microbial fed-batch fermentation process. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021040
[14]

John T. Betts, Stephen Campbell, Claire Digirolamo. Examination of solving optimal control problems with delays using GPOPS-Ⅱ. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 283-305. doi: 10.3934/naco.2020026
[15]

Livia Betz, Irwin Yousept. Optimal control of elliptic variational inequalities with bounded and unbounded operators. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021009
[16]

Christian Meyer, Stephan Walther. Optimal control of perfect plasticity part I: Stress tracking. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021022
[17]

Changjun Yu, Lei Yuan, Shuxuan Su. A new gradient computational formula for optimal control problems with time-delay. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021076
[18]

Muberra Allahverdi, Harun Aydilek, Asiye Aydilek, Ali Allahverdi. A better dominance relation and heuristics for Two-Machine No-Wait Flowshops with Maximum Lateness Performance Measure. Journal of Industrial & Management Optimization, 2021, 17 (4) : 1973-1991. doi: 10.3934/jimo.2020054
[19]

Vladimir Gaitsgory, Ilya Shvartsman. Linear programming estimates for Cesàro and Abel limits of optimal values in optimal control problems. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021102
[20]

Zengyun Wang, Jinde Cao, Zuowei Cai, Lihong Huang. Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2677-2692. doi: 10.3934/dcdsb.2020200

2019 Impact Factor: 1.338

Tools

Metrics

  • PDF downloads (56)
  • HTML views (0)
  • Cited by (3)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences