American Institute of Mathematical Sciences

Advanced
May  2005, 5(2): 239-264. doi: 10.3934/dcdsb.2005.5.239

Optimal control of finite horizon type for a multidimensional delayed switching system

Fabio Bagagiolo 1,

1. 

Dipartimento di Matematica, Universitá di Trento, Via Sommarive, 14, 38050 Povo di Trento (TN)

Received  August 2003 Revised  June 2004 Published  February 2005

We consider a finite horizon optimal control problem for an ODE system, with trajectories presenting a delayed two-values switching along a fixed direction. In particular the system exhibits hysteresis. Due to the presence of the switching component of the trajectories, several definitions of value functions are possible. None of these value functions is in general continuous. We prove that, under general hypotheses, the "least value function", i.e. the value function of the more relaxed problem, is the unique lower semicontinuous viscosity solution of two suitably coupled Hamilton-Jacobi-Bellman equations. Such a coupling involves boundary conditions in the viscosity sense.
Keywords: exit time, discontinuous viscosity solutions., Delayed switchings, optimal control.
Mathematics Subject Classification: 47J40, 49J15, 49L2.
Citation: Fabio Bagagiolo. Optimal control of finite horizon type for a multidimensional delayed switching system. Discrete & Continuous Dynamical Systems - B, 2005, 5 (2) : 239-264. doi: 10.3934/dcdsb.2005.5.239
[1]

Piermarco Cannarsa, Cristina Pignotti, Carlo Sinestrari. Semiconcavity for optimal control problems with exit time. Discrete & Continuous Dynamical Systems, 2000, 6 (4) : 975-997. doi: 10.3934/dcds.2000.6.975
[2]

M. Motta, C. Sartori. Exit time problems for nonlinear unbounded control systems. Discrete & Continuous Dynamical Systems, 1999, 5 (1) : 137-156. doi: 10.3934/dcds.1999.5.137
[3]

Luong V. Nguyen. A note on optimality conditions for optimal exit time problems. Mathematical Control & Related Fields, 2015, 5 (2) : 291-303. doi: 10.3934/mcrf.2015.5.291
[4]

Ling Yun Wang, Wei Hua Gui, Kok Lay Teo, Ryan Loxton, Chun Hua Yang. Time delayed optimal control problems with multiple characteristic time points: Computation and industrial applications. Journal of Industrial & Management Optimization, 2009, 5 (4) : 705-718. doi: 10.3934/jimo.2009.5.705
[5]

Filipe Rodrigues, Cristiana J. Silva, Delfim F. M. Torres, Helmut Maurer. Optimal control of a delayed HIV model. Discrete & Continuous Dynamical Systems - B, 2018, 23 (1) : 443-458. doi: 10.3934/dcdsb.2018030
[6]

Shu Zhang, Jian Xu. Time-varying delayed feedback control for an internet congestion control model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (2) : 653-668. doi: 10.3934/dcdsb.2011.16.653
[7]

Wenjie Li, Lihong Huang, Jinchen Ji. Globally exponentially stable periodic solution in a general delayed predator-prey model under discontinuous prey control strategy. Discrete & Continuous Dynamical Systems - B, 2020, 25 (7) : 2639-2664. doi: 10.3934/dcdsb.2020026
[8]

Ana P. Lemos-Paião, Cristiana J. Silva, Delfim F. M. Torres. A sufficient optimality condition for delayed state-linear optimal control problems. Discrete & Continuous Dynamical Systems - B, 2019, 24 (5) : 2293-2313. doi: 10.3934/dcdsb.2019096
[9]

Changjun Yu, Shuxuan Su, Yanqin Bai. On the optimal control problems with characteristic time control constraints. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021021
[10]

Edward Hooton, Pavel Kravetc, Dmitrii Rachinskii. Restrictions to the use of time-delayed feedback control in symmetric settings. Discrete & Continuous Dynamical Systems - B, 2018, 23 (2) : 543-556. doi: 10.3934/dcdsb.2017207
[11]

Isabelle Schneider, Matthias Bosewitz. Eliminating restrictions of time-delayed feedback control using equivariance. Discrete & Continuous Dynamical Systems, 2016, 36 (1) : 451-467. doi: 10.3934/dcds.2016.36.451
[12]

Mihai Bostan, Gawtum Namah. Time periodic viscosity solutions of Hamilton-Jacobi equations. Communications on Pure & Applied Analysis, 2007, 6 (2) : 389-410. doi: 10.3934/cpaa.2007.6.389
[13]

Boris Andreianov, Kenneth H. Karlsen, Nils H. Risebro. On vanishing viscosity approximation of conservation laws with discontinuous flux. Networks & Heterogeneous Media, 2010, 5 (3) : 617-633. doi: 10.3934/nhm.2010.5.617
[14]

Jérome Lohéac, Jean-François Scheid. Time optimal control for a nonholonomic system with state constraint. Mathematical Control & Related Fields, 2013, 3 (2) : 185-208. doi: 10.3934/mcrf.2013.3.185
[15]

Piermarco Cannarsa, Carlo Sinestrari. On a class of nonlinear time optimal control problems. Discrete & Continuous Dynamical Systems, 1995, 1 (2) : 285-300. doi: 10.3934/dcds.1995.1.285
[16]

Weijiu Liu. Asymptotic behavior of solutions of time-delayed Burgers' equation. Discrete & Continuous Dynamical Systems - B, 2002, 2 (1) : 47-56. doi: 10.3934/dcdsb.2002.2.47
[17]

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
[18]

Steven Richardson, Song Wang. The viscosity approximation to the Hamilton-Jacobi-Bellman equation in optimal feedback control: Upper bounds for extended domains. Journal of Industrial & Management Optimization, 2010, 6 (1) : 161-175. doi: 10.3934/jimo.2010.6.161
[19]

Tianliang Hou, Yanping Chen. Superconvergence for elliptic optimal control problems discretized by RT1 mixed finite elements and linear discontinuous elements. Journal of Industrial & Management Optimization, 2013, 9 (3) : 631-642. doi: 10.3934/jimo.2013.9.631
[20]

Konstantinos Chrysafinos, Efthimios N. Karatzas. Symmetric error estimates for discontinuous Galerkin approximations for an optimal control problem associated to semilinear parabolic PDE's. Discrete & Continuous Dynamical Systems - B, 2012, 17 (5) : 1473-1506. doi: 10.3934/dcdsb.2012.17.1473

2020 Impact Factor: 1.327

Tools

Metrics

  • PDF downloads (34)
  • HTML views (0)
  • Cited by (4)

Other articles
by authors

[Back to Top]

Copyright © 2021 American Institute of Mathematical Sciences