• Previous Article
    A lattice method for option evaluation with regime-switching asset correlation structure
  • JIMO Home
  • This Issue
  • Next Article
    The Glowinski–Le Tallec splitting method revisited: A general convergence and convergence rate analysis
July  2021, 17(4): 1713-1727. doi: 10.3934/jimo.2020041

Optimal control and stabilization of building maintenance units based on minimum principle

School of EECS, University of Ottawa, 800 King Edward Ave. Ottawa, ON K1N 6N5, Canada

* Corresponding author: Shi'an Wang

Received  August 2019 Revised  October 2019 Published  February 2020

In this paper we present a mathematical model describing the physical dynamics of a building maintenance unit (BMU) equipped with reaction jets. The momentum provided by reaction jets is considered as the control variable. We introduce an objective functional based on the deviation of the BMU from its equilibrium state due to external high-wind forces. Pontryagin minimum principle is then used to determine the optimal control policy so as to minimize possible deviation from the rest state thereby increasing the stability of the BMU and reducing the risk to the workers as well as the public. We present a series of numerical results corresponding to three different scenarios for the formulated optimal control problem. These results show that, under high-wind conditions the BMU can be stabilized and brought to its equilibrium state with appropriate controls in a short period of time. Therefore, it is believed that the dynamic model presented here would be potentially useful for stabilizing building maintenance units thereby reducing the risk to the workers and the general public.

Citation: 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
References:
[1]

N. U. Ahmed, Dynamic Systems and Control with Applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/6262.  Google Scholar

[2]

N. U. Ahmed, Elements of Finite Dimensional Systems and Control Theory, Pitman Monographs and Surveys in Pure and Applied Mathematics, 37, John Wiley & Sons, Inc., New York, 1988.  Google Scholar

[3]

T. Ahmed and N.U. Ahmed, Optimal Control of Antigen-Antibody Interactions for Cancer Immunotherapy, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications & Algorithms, 26 (2019), 135-152.   Google Scholar

[4]

D. Allen, What is building maintenance?, Facilities, 11 (1993), 7-12.  doi: 10.1108/EUM0000000002230.  Google Scholar

[5]

Building Maintenance Units, Report of Alimak Group AB. Available from: https://alimakservice.com/building-maintenance-units/. Google Scholar

[6]

T. R. Chandrupatla, A. D. Belegundu and T. Ramesh, et al., Introduction to Finite Elements in Engineering, Prentice Hall, 2002. Google Scholar

[7]

J. C. P. Cheng, W. Chen and Y. Tan, et al., A BIM-based decision support system framework for predictive maintenance management of building facilities, 16th International Conference on Computing in Civil and Building Engineering, Osaka, Japan, 2016, 711–718. Google Scholar

[8] T. Glad and L. Ljung, Control Theory, CRC Press, London, 2014.  doi: 10.1201/9781315274737.  Google Scholar
[9]

R. M. W. HornerM. A. El-Haram and A. K. Munns, Building maintenance strategy: A new management approach, J. Quality Maintenance Engineering, 3 (1997), 273-280.  doi: 10.1108/13552519710176881.  Google Scholar

[10]

Instability of Building Maintenance Units, WorkSafe Victoria, 2018. Available from: https://www.worksafe.vic.gov.au/safety-alerts/instability-building-maintenance-units. Google Scholar

[11]

C. H. Ko, RFID-based building maintenance system, Automat. Construction, 18 (2009), 275-284.  doi: 10.1016/j.autcon.2008.09.001.  Google Scholar

[12]

H. Lind and H. Muyingo, Building maintenance strategies: Planning under uncertainty, Property Management, 30 (2012), 14-28.  doi: 10.1108/02637471211198152.  Google Scholar

[13]

P. MaryamN.U. Ahmed and M.C.E. Yagoub, Optimum Decision Policy for Replacement of Conventional Energy Sources by Renewable Ones, International Journal of Energy Science, 3 (2013), 311-319.  doi: 10.14355/ijes.2013.0305.03.  Google Scholar

[14]

I. Motawa and A. Almarshad, A knowledge-based BIM system for building maintenance, Automat. Construction, 29 (2013), 173-182.  doi: 10.1016/j.autcon.2012.09.008.  Google Scholar

[15]

K. Ogata and Y. Yang, Modern Control Engineering, Prentice Hall, New Jersey, 2002. Google Scholar

[16]

L. S. Pontryagin, V. G. Boltyanskii and R. V. Gamkrelidze, et al., The Mathematical Theory of Optimal Processes, The Macmillan Co., New York, 1964.  Google Scholar

[17]

I. H. Seeley, Building Maintenance, Building and Surveying Series, Palgrave, London, 1987. doi: 10.1007/978-1-349-18925-0.  Google Scholar

[18]

J. Shen, A. K. Sanyal and N. A. Chaturvedi, et al., Dynamics and control of a 3D pendulum, 43$^rd$ IEEE Conference on Decision and Control, Atlantis, Bahamas, 2004, 323–328. doi: 10.1109/CDC.2004.1428650.  Google Scholar

[19]

M.M. SuruzN.U. Ahmed and M. Chowdhury, Optimum policy for integration of renewable energy sources into the power generation system, Energy Economics, 34 (2012), 558-567.  doi: 10.1016/j.eneco.2011.08.002.  Google Scholar

[20]

K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Pitman Monographs and Surveys in Pure and Applied Mathematics, 55, John Wiley & Sons, Inc., New York, 1991. doi: 20.500.11937/24319.  Google Scholar

[21]

S. Wang and N.U. Ahmed, Dynamic model of urban traffic and optimum management of its flow and congestion, Dynamic Systems and Applications, 26 (2017), 575-588.  doi: 10.12732/dsa.v26i34.12.  Google Scholar

[22]

S. WangN.U. Ahmed and T.H. Yeap, Optimum management of urban traffic flow based on a stochastic dynamic model, IEEE Transactions on Intelligent Transportation Systems, 20 (2019), 4377-4389.  doi: 10.1109/TITS.2018.2884463.  Google Scholar

[23]

X. Wang, Solving optimal control problems with MATLAB: Indirect methods, ISE Dept., NCSU, Raleigh, NC, 2009. Google Scholar

[24]

D. V. Zenkov, On Hamel's equations, Theoret. Appl. Mechanics, 43 (2016), 191-220.  doi: 10.2298/TAM160612011Z.  Google Scholar

[25]

D. V. Zenkov, M. Leok and A. M. Bloch, Hamel's formalism and variational integrators on a sphere, 51st IEEE Conference on Decision and Control, Hawaii, 2012, 7504–7510. doi: 10.1109/CDC.2012.6426779.  Google Scholar

show all references

References:
[1]

N. U. Ahmed, Dynamic Systems and Control with Applications, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2006. doi: 10.1142/6262.  Google Scholar

[2]

N. U. Ahmed, Elements of Finite Dimensional Systems and Control Theory, Pitman Monographs and Surveys in Pure and Applied Mathematics, 37, John Wiley & Sons, Inc., New York, 1988.  Google Scholar

[3]

T. Ahmed and N.U. Ahmed, Optimal Control of Antigen-Antibody Interactions for Cancer Immunotherapy, Dynamics of Continuous, Discrete and Impulsive Systems Series B: Applications & Algorithms, 26 (2019), 135-152.   Google Scholar

[4]

D. Allen, What is building maintenance?, Facilities, 11 (1993), 7-12.  doi: 10.1108/EUM0000000002230.  Google Scholar

[5]

Building Maintenance Units, Report of Alimak Group AB. Available from: https://alimakservice.com/building-maintenance-units/. Google Scholar

[6]

T. R. Chandrupatla, A. D. Belegundu and T. Ramesh, et al., Introduction to Finite Elements in Engineering, Prentice Hall, 2002. Google Scholar

[7]

J. C. P. Cheng, W. Chen and Y. Tan, et al., A BIM-based decision support system framework for predictive maintenance management of building facilities, 16th International Conference on Computing in Civil and Building Engineering, Osaka, Japan, 2016, 711–718. Google Scholar

[8] T. Glad and L. Ljung, Control Theory, CRC Press, London, 2014.  doi: 10.1201/9781315274737.  Google Scholar
[9]

R. M. W. HornerM. A. El-Haram and A. K. Munns, Building maintenance strategy: A new management approach, J. Quality Maintenance Engineering, 3 (1997), 273-280.  doi: 10.1108/13552519710176881.  Google Scholar

[10]

Instability of Building Maintenance Units, WorkSafe Victoria, 2018. Available from: https://www.worksafe.vic.gov.au/safety-alerts/instability-building-maintenance-units. Google Scholar

[11]

C. H. Ko, RFID-based building maintenance system, Automat. Construction, 18 (2009), 275-284.  doi: 10.1016/j.autcon.2008.09.001.  Google Scholar

[12]

H. Lind and H. Muyingo, Building maintenance strategies: Planning under uncertainty, Property Management, 30 (2012), 14-28.  doi: 10.1108/02637471211198152.  Google Scholar

[13]

P. MaryamN.U. Ahmed and M.C.E. Yagoub, Optimum Decision Policy for Replacement of Conventional Energy Sources by Renewable Ones, International Journal of Energy Science, 3 (2013), 311-319.  doi: 10.14355/ijes.2013.0305.03.  Google Scholar

[14]

I. Motawa and A. Almarshad, A knowledge-based BIM system for building maintenance, Automat. Construction, 29 (2013), 173-182.  doi: 10.1016/j.autcon.2012.09.008.  Google Scholar

[15]

K. Ogata and Y. Yang, Modern Control Engineering, Prentice Hall, New Jersey, 2002. Google Scholar

[16]

L. S. Pontryagin, V. G. Boltyanskii and R. V. Gamkrelidze, et al., The Mathematical Theory of Optimal Processes, The Macmillan Co., New York, 1964.  Google Scholar

[17]

I. H. Seeley, Building Maintenance, Building and Surveying Series, Palgrave, London, 1987. doi: 10.1007/978-1-349-18925-0.  Google Scholar

[18]

J. Shen, A. K. Sanyal and N. A. Chaturvedi, et al., Dynamics and control of a 3D pendulum, 43$^rd$ IEEE Conference on Decision and Control, Atlantis, Bahamas, 2004, 323–328. doi: 10.1109/CDC.2004.1428650.  Google Scholar

[19]

M.M. SuruzN.U. Ahmed and M. Chowdhury, Optimum policy for integration of renewable energy sources into the power generation system, Energy Economics, 34 (2012), 558-567.  doi: 10.1016/j.eneco.2011.08.002.  Google Scholar

[20]

K. L. Teo, C. J. Goh and K. H. Wong, A Unified Computational Approach to Optimal Control Problems, Pitman Monographs and Surveys in Pure and Applied Mathematics, 55, John Wiley & Sons, Inc., New York, 1991. doi: 20.500.11937/24319.  Google Scholar

[21]

S. Wang and N.U. Ahmed, Dynamic model of urban traffic and optimum management of its flow and congestion, Dynamic Systems and Applications, 26 (2017), 575-588.  doi: 10.12732/dsa.v26i34.12.  Google Scholar

[22]

S. WangN.U. Ahmed and T.H. Yeap, Optimum management of urban traffic flow based on a stochastic dynamic model, IEEE Transactions on Intelligent Transportation Systems, 20 (2019), 4377-4389.  doi: 10.1109/TITS.2018.2884463.  Google Scholar

[23]

X. Wang, Solving optimal control problems with MATLAB: Indirect methods, ISE Dept., NCSU, Raleigh, NC, 2009. Google Scholar

[24]

D. V. Zenkov, On Hamel's equations, Theoret. Appl. Mechanics, 43 (2016), 191-220.  doi: 10.2298/TAM160612011Z.  Google Scholar

[25]

D. V. Zenkov, M. Leok and A. M. Bloch, Hamel's formalism and variational integrators on a sphere, 51st IEEE Conference on Decision and Control, Hawaii, 2012, 7504–7510. doi: 10.1109/CDC.2012.6426779.  Google Scholar

Figure 1.  The schematic of a BMU
Figure 2.  The schematic of the BMU body
Figure 3.  Simulation results of scenario 1
Figure 4.  Simulation results corresponding to $ U_{1} $ in scenario 2
Figure 5.  Simulation results corresponding to $ U_{2} $ in scenario 2
Figure 6.  Simulation results of case 1 in scenario 3
Figure 7.  Simulation results of case 2 in scenario 3
Table 1.  Definition of Notations
Notation Description
$ x_{1} = \omega_{x} $ First component of the angular velocity
$ x_{2} = \omega_{y} $ Second component of the angular velocity
$ x_{3} = \gamma_{x} $ First component of the unit vertical vector $ \gamma $
$ x_{4} = \gamma_{y} $ Second component of the unit vertical vector $ \gamma $
$ x_{5} = \gamma_{z} $ Third component of the unit vertical vector $ \gamma $
$ \psi $ Costate vector (adjoint state)
$ C $ Constant inertia matrix
$ I=[0,T] $ Total operating period in seconds
$ U $ Control (decision) constraint set
$ {\mathcal U}_{ad} $ Set of admissible controls
$ \underline{u} $ Lower bound of the control variable
$ \overline{u} $ Upper bound of the control variable
$ J(u) $ Objective (cost) functional
$ \ell $ Integrand of running cost
$ \Phi $ Terminal cost
$ H $ Hamiltonian function
$ V(x) $ Lyapunov function candidate
$ x^{d} $ Desired state during the operating period
$ \bar{x} $ Target state at the terminal time
$ x^{e} $ Equilibrium state of the system
$ \langle a,\; b \rangle $ Scalar product of vectors $ a $ and $ b $
$ a \times b $ Cross product of vectors $ a $ and $ b $
$ x^{T} $ Transpose of vector $ x $
Notation Description
$ x_{1} = \omega_{x} $ First component of the angular velocity
$ x_{2} = \omega_{y} $ Second component of the angular velocity
$ x_{3} = \gamma_{x} $ First component of the unit vertical vector $ \gamma $
$ x_{4} = \gamma_{y} $ Second component of the unit vertical vector $ \gamma $
$ x_{5} = \gamma_{z} $ Third component of the unit vertical vector $ \gamma $
$ \psi $ Costate vector (adjoint state)
$ C $ Constant inertia matrix
$ I=[0,T] $ Total operating period in seconds
$ U $ Control (decision) constraint set
$ {\mathcal U}_{ad} $ Set of admissible controls
$ \underline{u} $ Lower bound of the control variable
$ \overline{u} $ Upper bound of the control variable
$ J(u) $ Objective (cost) functional
$ \ell $ Integrand of running cost
$ \Phi $ Terminal cost
$ H $ Hamiltonian function
$ V(x) $ Lyapunov function candidate
$ x^{d} $ Desired state during the operating period
$ \bar{x} $ Target state at the terminal time
$ x^{e} $ Equilibrium state of the system
$ \langle a,\; b \rangle $ Scalar product of vectors $ a $ and $ b $
$ a \times b $ Cross product of vectors $ a $ and $ b $
$ x^{T} $ Transpose of vector $ x $
Algorithm 1: Computational Algorithm $ (\bullet) $
Require:
Choose the appropriate initial state $ x(0) $;
Set the length of time horizon $ T \in R^{+} $ and the number of subintervals (of equal length) $ N \in \mathbb{Z}^{+} $;
Set step size $ \epsilon $, stopping criterion $ \tau $, maximum number of iterations $ K $ and control bounds $ \underline{u}, \; \overline{u} $.
Ensure:
Optimal cost $ J^o $;
Optimal state trajectory $ x^{o} $;
Optimal control trajectory $ u^o $.

1: Subdivide equally the time horizon $ I = [0, T] $ into $ N $ subintervals and assume the control function is piecewise-constant. That is, $ u^{n}(t) = u^{n}(t_{i}) $, for $ t \in [t_{i}, t_{i+1}),\; i = 0, 1, \cdots, N-1 $, where $ u^{n}(t),\; t \in I $ is the control (decision) policy at the $ n $th iteration (starting from $ n = 0 $).
2: Integrate the state equations from 0 to $ T $ with initial state $ x(0) = x_{0} $ and the assumed controls $ u^{(n)} \equiv u^{n}(t),\; t \in I $, store the obtained state trajectory $ x^{(n)} $ and the control vector $ u^{(n)} $.
3: Use $ x^{(n)} $ and $ u^{(n)} $ to integrate the adjoint equations backward in time starting from the costate $ \psi^{(n)}(T) $ at the terminal time. The terminal costate is given by $ \psi^{(n)}(T) = \Phi_{x}(x^{(n)}(T)) $ where $ \Phi $ is the terminal cost.
4: Use the triple {$ u^{(n)},\; x^{(n)},\; \psi^{(n)} $} to compute the gradient $ g_{n}(t) = \frac{\partial H}{\partial u^{(n)}}(x^{(n)},\; u^{(n)},\; \psi^{(n)}) = H_{u}(x^{(n)},\; u^{(n)},\; \psi^{(n)}) $ and store this vector.
5: Compute the cost functional $ J^{(n)}(u) $ using equation (19) and store this value.
6: If $ \| g_{n} \| < \tau $ then set
$ u^{o} = u^{(n)} $, $ J^{o} = J^{(n)} $
return
Otherwise, go to Step 7.
7: Construct the control policy for the next iteration as
$ u^{(n+1)}(t) = u^{(n)}(t) - \epsilon g_{n}(t),\; t \in I $ by choosing an appropriate $ \epsilon \in (0,1) $ such that $ u^{(n+1)} \in U $. For the chosen $ \epsilon $, if $ u^{(n+1)} > \overline{u} $ set $ u^{(n+1)} = \overline{u} $; if $ u^{(n+1)} < \underline{u} $ set $ u^{(n+1)} = \underline{u} $.
8: If $ n < K $ then set
$ n = n + 1 $, go to Step 2.
Otherwise, display "Stopped before required residual is obtained''.
Algorithm 1: Computational Algorithm $ (\bullet) $
Require:
Choose the appropriate initial state $ x(0) $;
Set the length of time horizon $ T \in R^{+} $ and the number of subintervals (of equal length) $ N \in \mathbb{Z}^{+} $;
Set step size $ \epsilon $, stopping criterion $ \tau $, maximum number of iterations $ K $ and control bounds $ \underline{u}, \; \overline{u} $.
Ensure:
Optimal cost $ J^o $;
Optimal state trajectory $ x^{o} $;
Optimal control trajectory $ u^o $.

1: Subdivide equally the time horizon $ I = [0, T] $ into $ N $ subintervals and assume the control function is piecewise-constant. That is, $ u^{n}(t) = u^{n}(t_{i}) $, for $ t \in [t_{i}, t_{i+1}),\; i = 0, 1, \cdots, N-1 $, where $ u^{n}(t),\; t \in I $ is the control (decision) policy at the $ n $th iteration (starting from $ n = 0 $).
2: Integrate the state equations from 0 to $ T $ with initial state $ x(0) = x_{0} $ and the assumed controls $ u^{(n)} \equiv u^{n}(t),\; t \in I $, store the obtained state trajectory $ x^{(n)} $ and the control vector $ u^{(n)} $.
3: Use $ x^{(n)} $ and $ u^{(n)} $ to integrate the adjoint equations backward in time starting from the costate $ \psi^{(n)}(T) $ at the terminal time. The terminal costate is given by $ \psi^{(n)}(T) = \Phi_{x}(x^{(n)}(T)) $ where $ \Phi $ is the terminal cost.
4: Use the triple {$ u^{(n)},\; x^{(n)},\; \psi^{(n)} $} to compute the gradient $ g_{n}(t) = \frac{\partial H}{\partial u^{(n)}}(x^{(n)},\; u^{(n)},\; \psi^{(n)}) = H_{u}(x^{(n)},\; u^{(n)},\; \psi^{(n)}) $ and store this vector.
5: Compute the cost functional $ J^{(n)}(u) $ using equation (19) and store this value.
6: If $ \| g_{n} \| < \tau $ then set
$ u^{o} = u^{(n)} $, $ J^{o} = J^{(n)} $
return
Otherwise, go to Step 7.
7: Construct the control policy for the next iteration as
$ u^{(n+1)}(t) = u^{(n)}(t) - \epsilon g_{n}(t),\; t \in I $ by choosing an appropriate $ \epsilon \in (0,1) $ such that $ u^{(n+1)} \in U $. For the chosen $ \epsilon $, if $ u^{(n+1)} > \overline{u} $ set $ u^{(n+1)} = \overline{u} $; if $ u^{(n+1)} < \underline{u} $ set $ u^{(n+1)} = \underline{u} $.
8: If $ n < K $ then set
$ n = n + 1 $, go to Step 2.
Otherwise, display "Stopped before required residual is obtained''.
[1]

Baba Issa Camara, Houda Mokrani, Evans K. Afenya. Mathematical modeling of glioma therapy using oncolytic viruses. Mathematical Biosciences & Engineering, 2013, 10 (3) : 565-578. doi: 10.3934/mbe.2013.10.565

[2]

Chih-Chiang Fang. Bayesian decision making in determining optimal leased term and preventive maintenance scheme for leased facilities. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020127

[3]

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

[4]

A. K. Misra, Anupama Sharma, Jia Li. A mathematical model for control of vector borne diseases through media campaigns. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1909-1927. doi: 10.3934/dcdsb.2013.18.1909

[5]

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

[6]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

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

[15]

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

[16]

Jaouad Danane. Optimal control of viral infection model with saturated infection rate. Numerical Algebra, Control & Optimization, 2021, 11 (3) : 363-375. doi: 10.3934/naco.2020031

[17]

Rafael G. L. D'Oliveira, Marcelo Firer. Minimum dimensional Hamming embeddings. Advances in Mathematics of Communications, 2017, 11 (2) : 359-366. doi: 10.3934/amc.2017029

[18]

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

[19]

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

[20]

Andrea Signori. Penalisation of long treatment time and optimal control of a tumour growth model of Cahn–Hilliard type with singular potential. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2519-2542. doi: 10.3934/dcds.2020373

2019 Impact Factor: 1.366

Metrics

  • PDF downloads (109)
  • HTML views (437)
  • Cited by (0)

Other articles
by authors

[Back to Top]