Decompositions and bang-bang properties

Gengsheng Wang; Yubiao Zhang

doi:10.3934/mcrf.2017005

Previous Article
MCRF Home
This Issue
Next Article
Control and stabilization of 2 × 2 hyperbolic systems on graphs

March 2017, 7(1): 73-170. doi: 10.3934/mcrf.2017005

Decompositions and bang-bang properties

Gengsheng Wang ^1, and Yubiao Zhang ^2,3,,

1.	School of Mathematics and Statistics, Collaborative Innovation Centre of Mathematics, Wuhan University, Wuhan 430072, China
2.	School of Mathematics and Statistics, Wuhan University, Wuhan 430072, China
3.	Center for Applied Mathematics, Tianjin University, Tianjin 300072, China

* Corresponding author:Yubiao Zhang

Received April 2016 Revised July 2016 Published December 2016

Fund Project: The first author was partially supported by the National Natural Science Foundation of China under grant 11571264. The second author was partially supported by the National Natural Science Foundation of China under grants 11571264 and 11371285.

We study the bang-bang properties of minimal time and minimal norm control problems (where the target set is the origin of the state space and the controlled system is linear and time-invariant) from a new perspective. More precisely, we study how the bang-bang property of each minimal time (or minimal norm) problem depends on a pair of parameters $(M, y_0)$ (or $(T,y_0)$), where $M>0$ is a bound of controls and $y_0$ is the initial state (or $T>0$ is an ending time and $y_0$ is the initial state). The controlled system may have neither the $L^∞$-null controllability nor the backward uniqueness property.

Keywords: Minimal time controls, minimal norm controls, bang-bang properties, decompositions, evolution systems, reachable subspaces.

Mathematics Subject Classification: Primary: 93B03; Secondary: 93C35.

Citation: Gengsheng Wang, Yubiao Zhang. Decompositions and bang-bang properties. Mathematical Control and Related Fields, 2017, 7 (1) : 73-170. doi: 10.3934/mcrf.2017005

References:

[1]	F. Ammar Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, A new relation between the condensation index of complex sequences and the null controllability of parabolic systems, C. R. Math. Acad. Sci. Paris., 351 (2013), 743-746. doi: 10.1016/j.crma.2013.09.014.
[2]	F. Ammar Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Minimal time of controllability of two parabolic equations with disjoint control and coupling domains, C. R. Math. Acad. Sci. Paris, 352 (2014), 391-396. doi: 10.1016/j.crma.2014.03.004.
[3]	F. Ammar Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Minimal time for the null controllability of parabolic systems: the effect of the condensation index of complex sequences, J. Funct. Anal., 267 (2014), 2077-2151. doi: 10.1016/j.jfa.2014.07.024.
[4]	J. Apraiz, L. Escauriaza, G. Wang and C. Zhang, Observability inequalities and measurable sets, J. Eur. Math. Soc., 16 (2014), 2433-2475. doi: 10.4171/JEMS/490.
[5]	N. Arada and J.-P. Raymond, Time optimal problems with Dirichlet boundary controls, Discrete Contin. Dyn. Syst., 9 (2003), 1549-1570. doi: 10.3934/dcds.2003.9.1549.
[6]	V. Barbu, Analysis and Control of Nonlinear Infinite-dimensional Systems, Academic Press, Boston, 1993.
[7]	O. Cârjǎ, On continuity of the minimal time function for distributed control systems, Boll. Un. Mat. Ital. -A, 4 (1985), 293-302.
[8]	J. M. Coron, Control and Nonlinearity, American Mathematical Society, 2007.
[9]	H. O. Fattorini, Infinite Dimensional Linear Control Systems: The Time Optimal and Norm Optimal Problems, ELSEVIER, 2005.
[10]	H. O. Fattorini, Time-optimal control of solutions of operational differential equations, J. SIAM Control -A, 2 (1964), 54-59. doi: 10.1137/0302005.
[11]	H. O. Fattorini, The time-optimal control problem in Banach spaces, Appl. Math. Optim., 1 (1974/75), 163-188. doi: 10.1007/BF01449028.
[12]	H. O. Fattorini, Some remarks on the time optimal control problem in infinite dimension, Calculus of Variations and Optimal Control, 411 (2000), 77-96.
[13]	H. O. Fattorini, Existence of singular extremals and singular functionals in reachable spaces, J. Evol. Equ., 1 (2001), 325-347. doi: 10.1007/PL00001374.
[14]	H. O. Fattorini, Time and norm optimal controls: A survey of recent results and open problems, Acta Math. Sci. -B, 31 (2011), 2203-2218. doi: 10.1016/S0252-9602(11)60394-9.
[15]	W. Gong and N. Yan, Finite element method and its error estimates for the time optimal controls of heat equation, Int. J. Numer. Anal. Model., 13 (2016), 265-279.
[16]	F. Gozzi and P. Loreti, Regularity of the minimum time function and minimal energy problems: the linear case, SIAM J. Control Optim., 37 (1999), 1195-1221. doi: 10.1137/S0363012996312763.
[17]	K. Ito and K. Kunisch, Semismooth Newton methods for time-optimal control for a class of ODEs, SIAM J. Control Optim., 48 (2010), 3997-4013. doi: 10.1137/090753905.
[18]	K. Kunisch and L. Wang, Time optimal control of the heat equation with pointwise control constraints, ESAIM Control Optim. Calc. Var., 19 (2013), 460-485. doi: 10.1051/cocv/2012017.
[19]	K. Kunisch and L. Wang, Time optimal controls of the linear Fitzhugh-Nagumo equation with pointwise control constraints, J. Math. Anal. Appl., 395 (2012), 114-130. doi: 10.1016/j.jmaa.2012.05.028.
[20]	X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.
[21]	P. Lin and G. Wang, Some properties for blowup parabolic equations and their application, J. Math. Pures Appl., 101 (2014), 223-255. doi: 10.1016/j.matpur.2013.06.001.
[22]	J. Lohéac and M. Tucsnak, Maximum principle and bang-bang property of time optimal controls for Schrödinger-type systems, SIAM J. Control Optim., 51 (2013), 4016-4038. doi: 10.1137/120872437.
[23]	J. Lohéac and E. Zuazua, Norm saturating property of time optimal controls for wave-type equations, 2016, <hal-01258878>.
[24]	H. Lou, J. Wen and Y. Xu, Time optimal control problems for some non-smooth systems, Math. Control Relat. Fields, 4 (2014), 289-314. doi: 10.3934/mcrf.2014.4.289.
[25]	Q. Lü, Bang-bang principle of time optimal controls and null controllability of fractional order parabolic equations, Acta Math. Sin. (Engl. Ser.), 26 (2010), 2377-2386. doi: 10.1007/s10114-010-9051-1.
[26]	S. Micu, I. Roventa and M. Tucsnak, Time optimal boundary controls for the heat equation, J. Funct. Anal., 263 (2012), 25-49. doi: 10.1016/j.jfa.2012.04.009.
[27]	V. Mizel and T. Seidman, An abstract bang-bang principle and time-optimal boundary control of the heat equation, SIAM J. Control Optim., 35 (1997), 1204-1216. doi: 10.1137/S0363012996265470.
[28]	A. Münch and E. Zuazua, Numerical approximation of null controls for the heat equation: Ill-posedness and remedies, Inverse Problems, 26 (2010), 085018, 39pp. doi: 10.1088/0266-5611/26/8/085018.
[29]	A. Pazy, Semigroups of Linear Operatots and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
[30]	N. V. Petrov, The Bellman problem for a time-optimality problem, Prikl. Mat. Meh., 34 (1970), 820-826.
[31]	K. D. Phung and G. Wang, An observability estimate for parabolic equations from a measurable set in time and its applications, J. Eur. Math. Soc., 15 (2013), 681-703. doi: 10.4171/JEMS/371.
[32]	K. D. Phung and G. Wang, Quantitative unique continuation for the semilinear heat equation in a convex domain, J. Funct. Anal., 259 (2010), 1230-1247. doi: 10.1016/j.jfa.2010.04.015.
[33]	K. D. Phung, L. Wang and C. Zhang, Bang-bang property for time optimal control of semilinear heat equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 477-499. doi: 10.1016/j.anihpc.2013.04.005.
[34]	K. D. Phung, G. Wang and X. Zhang, On the existence of time optimal controls for linear evolution equations, Discrete Contin. Dyn. Syst. -B, 8 (2007), 925-941. doi: 10.3934/dcdsb.2007.8.925.
[35]	W. Rudin, Real and Complex Analysis, 3^rd edition, McGraw-Hill Companies, 1987.
[36]	E. J. P. G. Schmidt, The "bang-bang" principle for the time-optimal problem in boundary control of the heat equation, SIAM J. Control Optim., 18 (1980), 101-107. doi: 10.1137/0318008.
[37]	C. Silva and E. Trélat, Smooth regularization of bang-bang optimal control problems, IEEE Trans. Automat. Control, 55 (2010), 2488-2499. doi: 10.1109/TAC.2010.2047742.
[38]	E. D. Sontag, Mathematical Control Theory: Deterministic Finite-Dimensional Systems, 2^nd edition, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0577-7.
[39]	M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Verlag AG, 2009. doi: 10.1007/978-3-7643-8994-9.
[40]	G. Wang, L^∞-null controllability for the heat equation and its consequences for the time optimal control problem, SIAM J. Control Optim., 47 (2008), 1701-1720. doi: 10.1137/060678191.
[41]	G. Wang and Y. Xu, Equivalence of three different kinds of optimal control problems for heat equations and its applications, SIAM J. Control Optim., 51 (2013), 848-880. doi: 10.1137/110852449.
[42]	G. Wang and Y. Xu, Advantages for controls imposed in a proper subset, Discrete Contin. Dyn. Syst. -B, 18 (2013), 2427-2439. doi: 10.3934/dcdsb.2013.18.2427.
[43]	G. Wang, Y. Xu and Y. Zhang, Attainable subspaces and the bang-bang property of time optimal controls for heat equations, SIAM J. Control Optim., 53 (2015), 592-621. doi: 10.1137/140966022.
[44]	G. Wang and C. Zhang, Observability inequalities from measurable sets for some evolution equations, preprint, arXiv: 1406.3422v1.
[45]	G. Wang and G. Zheng, An approach to the optimal time for a time optimal control problem of an internally controlled heat equation, SIAM J. Control Optim., 50 (2012), 601-628. doi: 10.1137/100793645.
[46]	G. Wang and E. Zuazua, On the equivalence of minimal time and minimal norm controls for internally controlled heat equations, SIAM J. Control Optim., 50 (2012), 2938-2958. doi: 10.1137/110857398.
[47]	H. Yu, Approximation of time optimal controls for heat equations with perturbations in the system potential, SIAM J. Control Optim., 52 (2014), 1663-1692. doi: 10.1137/120904251.
[48]	C. Zhang, An observability estimate for the heat equation from a product of two measurable sets, J. Math. Anal. Appl., 396 (2012), 7-12. doi: 10.1016/j.jmaa.2012.05.082.
[49]	C. Zhang, The time optimal control with constraints of the rectangular type for linear time-varying ODEs, SIAM J. Control Optim., 51 (2013), 1528-1542. doi: 10.1137/110858999.
[50]	G. Zheng and B. Ma, A time optimal control problem of some linear switching controlled ordinary differential equations, Adv. Difference Equ., 2012 (2012), 1-7. doi: 10.1186/1687-1847-2012-52.

show all references

References:

[1]	F. Ammar Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, A new relation between the condensation index of complex sequences and the null controllability of parabolic systems, C. R. Math. Acad. Sci. Paris., 351 (2013), 743-746. doi: 10.1016/j.crma.2013.09.014.
[2]	F. Ammar Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Minimal time of controllability of two parabolic equations with disjoint control and coupling domains, C. R. Math. Acad. Sci. Paris, 352 (2014), 391-396. doi: 10.1016/j.crma.2014.03.004.
[3]	F. Ammar Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, Minimal time for the null controllability of parabolic systems: the effect of the condensation index of complex sequences, J. Funct. Anal., 267 (2014), 2077-2151. doi: 10.1016/j.jfa.2014.07.024.
[4]	J. Apraiz, L. Escauriaza, G. Wang and C. Zhang, Observability inequalities and measurable sets, J. Eur. Math. Soc., 16 (2014), 2433-2475. doi: 10.4171/JEMS/490.
[5]	N. Arada and J.-P. Raymond, Time optimal problems with Dirichlet boundary controls, Discrete Contin. Dyn. Syst., 9 (2003), 1549-1570. doi: 10.3934/dcds.2003.9.1549.
[6]	V. Barbu, Analysis and Control of Nonlinear Infinite-dimensional Systems, Academic Press, Boston, 1993.
[7]	O. Cârjǎ, On continuity of the minimal time function for distributed control systems, Boll. Un. Mat. Ital. -A, 4 (1985), 293-302.
[8]	J. M. Coron, Control and Nonlinearity, American Mathematical Society, 2007.
[9]	H. O. Fattorini, Infinite Dimensional Linear Control Systems: The Time Optimal and Norm Optimal Problems, ELSEVIER, 2005.
[10]	H. O. Fattorini, Time-optimal control of solutions of operational differential equations, J. SIAM Control -A, 2 (1964), 54-59. doi: 10.1137/0302005.
[11]	H. O. Fattorini, The time-optimal control problem in Banach spaces, Appl. Math. Optim., 1 (1974/75), 163-188. doi: 10.1007/BF01449028.
[12]	H. O. Fattorini, Some remarks on the time optimal control problem in infinite dimension, Calculus of Variations and Optimal Control, 411 (2000), 77-96.
[13]	H. O. Fattorini, Existence of singular extremals and singular functionals in reachable spaces, J. Evol. Equ., 1 (2001), 325-347. doi: 10.1007/PL00001374.
[14]	H. O. Fattorini, Time and norm optimal controls: A survey of recent results and open problems, Acta Math. Sci. -B, 31 (2011), 2203-2218. doi: 10.1016/S0252-9602(11)60394-9.
[15]	W. Gong and N. Yan, Finite element method and its error estimates for the time optimal controls of heat equation, Int. J. Numer. Anal. Model., 13 (2016), 265-279.
[16]	F. Gozzi and P. Loreti, Regularity of the minimum time function and minimal energy problems: the linear case, SIAM J. Control Optim., 37 (1999), 1195-1221. doi: 10.1137/S0363012996312763.
[17]	K. Ito and K. Kunisch, Semismooth Newton methods for time-optimal control for a class of ODEs, SIAM J. Control Optim., 48 (2010), 3997-4013. doi: 10.1137/090753905.
[18]	K. Kunisch and L. Wang, Time optimal control of the heat equation with pointwise control constraints, ESAIM Control Optim. Calc. Var., 19 (2013), 460-485. doi: 10.1051/cocv/2012017.
[19]	K. Kunisch and L. Wang, Time optimal controls of the linear Fitzhugh-Nagumo equation with pointwise control constraints, J. Math. Anal. Appl., 395 (2012), 114-130. doi: 10.1016/j.jmaa.2012.05.028.
[20]	X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.
[21]	P. Lin and G. Wang, Some properties for blowup parabolic equations and their application, J. Math. Pures Appl., 101 (2014), 223-255. doi: 10.1016/j.matpur.2013.06.001.
[22]	J. Lohéac and M. Tucsnak, Maximum principle and bang-bang property of time optimal controls for Schrödinger-type systems, SIAM J. Control Optim., 51 (2013), 4016-4038. doi: 10.1137/120872437.
[23]	J. Lohéac and E. Zuazua, Norm saturating property of time optimal controls for wave-type equations, 2016, <hal-01258878>.
[24]	H. Lou, J. Wen and Y. Xu, Time optimal control problems for some non-smooth systems, Math. Control Relat. Fields, 4 (2014), 289-314. doi: 10.3934/mcrf.2014.4.289.
[25]	Q. Lü, Bang-bang principle of time optimal controls and null controllability of fractional order parabolic equations, Acta Math. Sin. (Engl. Ser.), 26 (2010), 2377-2386. doi: 10.1007/s10114-010-9051-1.
[26]	S. Micu, I. Roventa and M. Tucsnak, Time optimal boundary controls for the heat equation, J. Funct. Anal., 263 (2012), 25-49. doi: 10.1016/j.jfa.2012.04.009.
[27]	V. Mizel and T. Seidman, An abstract bang-bang principle and time-optimal boundary control of the heat equation, SIAM J. Control Optim., 35 (1997), 1204-1216. doi: 10.1137/S0363012996265470.
[28]	A. Münch and E. Zuazua, Numerical approximation of null controls for the heat equation: Ill-posedness and remedies, Inverse Problems, 26 (2010), 085018, 39pp. doi: 10.1088/0266-5611/26/8/085018.
[29]	A. Pazy, Semigroups of Linear Operatots and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.
[30]	N. V. Petrov, The Bellman problem for a time-optimality problem, Prikl. Mat. Meh., 34 (1970), 820-826.
[31]	K. D. Phung and G. Wang, An observability estimate for parabolic equations from a measurable set in time and its applications, J. Eur. Math. Soc., 15 (2013), 681-703. doi: 10.4171/JEMS/371.
[32]	K. D. Phung and G. Wang, Quantitative unique continuation for the semilinear heat equation in a convex domain, J. Funct. Anal., 259 (2010), 1230-1247. doi: 10.1016/j.jfa.2010.04.015.
[33]	K. D. Phung, L. Wang and C. Zhang, Bang-bang property for time optimal control of semilinear heat equation, Ann. Inst. H. Poincaré Anal. Non Linéaire, 31 (2014), 477-499. doi: 10.1016/j.anihpc.2013.04.005.
[34]	K. D. Phung, G. Wang and X. Zhang, On the existence of time optimal controls for linear evolution equations, Discrete Contin. Dyn. Syst. -B, 8 (2007), 925-941. doi: 10.3934/dcdsb.2007.8.925.
[35]	W. Rudin, Real and Complex Analysis, 3^rd edition, McGraw-Hill Companies, 1987.
[36]	E. J. P. G. Schmidt, The "bang-bang" principle for the time-optimal problem in boundary control of the heat equation, SIAM J. Control Optim., 18 (1980), 101-107. doi: 10.1137/0318008.
[37]	C. Silva and E. Trélat, Smooth regularization of bang-bang optimal control problems, IEEE Trans. Automat. Control, 55 (2010), 2488-2499. doi: 10.1109/TAC.2010.2047742.
[38]	E. D. Sontag, Mathematical Control Theory: Deterministic Finite-Dimensional Systems, 2^nd edition, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0577-7.
[39]	M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Verlag AG, 2009. doi: 10.1007/978-3-7643-8994-9.
[40]	G. Wang, L^∞-null controllability for the heat equation and its consequences for the time optimal control problem, SIAM J. Control Optim., 47 (2008), 1701-1720. doi: 10.1137/060678191.
[41]	G. Wang and Y. Xu, Equivalence of three different kinds of optimal control problems for heat equations and its applications, SIAM J. Control Optim., 51 (2013), 848-880. doi: 10.1137/110852449.
[42]	G. Wang and Y. Xu, Advantages for controls imposed in a proper subset, Discrete Contin. Dyn. Syst. -B, 18 (2013), 2427-2439. doi: 10.3934/dcdsb.2013.18.2427.
[43]	G. Wang, Y. Xu and Y. Zhang, Attainable subspaces and the bang-bang property of time optimal controls for heat equations, SIAM J. Control Optim., 53 (2015), 592-621. doi: 10.1137/140966022.
[44]	G. Wang and C. Zhang, Observability inequalities from measurable sets for some evolution equations, preprint, arXiv: 1406.3422v1.
[45]	G. Wang and G. Zheng, An approach to the optimal time for a time optimal control problem of an internally controlled heat equation, SIAM J. Control Optim., 50 (2012), 601-628. doi: 10.1137/100793645.
[46]	G. Wang and E. Zuazua, On the equivalence of minimal time and minimal norm controls for internally controlled heat equations, SIAM J. Control Optim., 50 (2012), 2938-2958. doi: 10.1137/110857398.
[47]	H. Yu, Approximation of time optimal controls for heat equations with perturbations in the system potential, SIAM J. Control Optim., 52 (2014), 1663-1692. doi: 10.1137/120904251.
[48]	C. Zhang, An observability estimate for the heat equation from a product of two measurable sets, J. Math. Anal. Appl., 396 (2012), 7-12. doi: 10.1016/j.jmaa.2012.05.082.
[49]	C. Zhang, The time optimal control with constraints of the rectangular type for linear time-varying ODEs, SIAM J. Control Optim., 51 (2013), 1528-1542. doi: 10.1137/110858999.
[50]	G. Zheng and B. Ma, A time optimal control problem of some linear switching controlled ordinary differential equations, Adv. Difference Equ., 2012 (2012), 1-7. doi: 10.1186/1687-1847-2012-52.

Figure 1. The BBP decomposition for $(NP)^{T,y_0}$

Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2. The BBP decomposition for $(TP)^{M,y_0}$

Figure Options

Download full-size image

Download as PowerPoint slide

[1]	Karl Kunisch, Lijuan Wang. The bang-bang property of time optimal controls for the Burgers equation. Discrete and Continuous Dynamical Systems, 2014, 34 (9) : 3611-3637. doi: 10.3934/dcds.2014.34.3611
[2]	Karl Kunisch, Lijuan Wang. Bang-bang property of time optimal controls of semilinear parabolic equation. Discrete and Continuous Dynamical Systems, 2016, 36 (1) : 279-302. doi: 10.3934/dcds.2016.36.279
[3]	Helmut Maurer, Tanya Tarnopolskaya, Neale Fulton. Computation of bang-bang and singular controls in collision avoidance. Journal of Industrial and Management Optimization, 2014, 10 (2) : 443-460. doi: 10.3934/jimo.2014.10.443
[4]	M. S. Lee, H. G. Harno, B. S. Goh, K. H. Lim. On the bang-bang control approach via a component-wise line search strategy for unconstrained optimization. Numerical Algebra, Control and Optimization, 2021, 11 (1) : 45-61. doi: 10.3934/naco.2020014
[5]	Walter Alt, Robert Baier, Matthias Gerdts, Frank Lempio. Error bounds for Euler approximation of linear-quadratic control problems with bang-bang solutions. Numerical Algebra, Control and Optimization, 2012, 2 (3) : 547-570. doi: 10.3934/naco.2012.2.547
[6]	Daniel Glasscock, Andreas Koutsogiannis, Florian Karl Richter. Multiplicative combinatorial properties of return time sets in minimal dynamical systems. Discrete and Continuous Dynamical Systems, 2019, 39 (10) : 5891-5921. doi: 10.3934/dcds.2019258
[7]	Kim Dang Phung, Gengsheng Wang, Xu Zhang. On the existence of time optimal controls for linear evolution equations. Discrete and Continuous Dynamical Systems - B, 2007, 8 (4) : 925-941. doi: 10.3934/dcdsb.2007.8.925
[8]	Shulin Qin, Gengsheng Wang, Huaiqiang Yu. On switching properties of time optimal controls for linear ODEs. Mathematical Control and Related Fields, 2021, 11 (2) : 329-351. doi: 10.3934/mcrf.2020039
[9]	Jiongmin Yong. Optimality conditions for controls of semilinear evolution systems with mixed constraints. Discrete and Continuous Dynamical Systems, 1995, 1 (3) : 371-388. doi: 10.3934/dcds.1995.1.371
[10]	Donghui Yang, Jie Zhong. Optimal actuator location of the minimum norm controls for stochastic heat equations. Mathematical Control and Related Fields, 2018, 8 (3&4) : 1081-1095. doi: 10.3934/mcrf.2018046
[11]	Matúš Dirbák. Minimal skew products with hypertransitive or mixing properties. Discrete and Continuous Dynamical Systems, 2012, 32 (5) : 1657-1674. doi: 10.3934/dcds.2012.32.1657
[12]	N. Arada, J.-P. Raymond. Time optimal problems with Dirichlet boundary controls. Discrete and Continuous Dynamical Systems, 2003, 9 (6) : 1549-1570. doi: 10.3934/dcds.2003.9.1549
[13]	Monica Motta. Minimum time problem with impulsive and ordinary controls. Discrete and Continuous Dynamical Systems, 2018, 38 (11) : 5781-5809. doi: 10.3934/dcds.2018252
[14]	Y. Peng, X. Xiang. Second order nonlinear impulsive time-variant systems with unbounded perturbation and optimal controls. Journal of Industrial and Management Optimization, 2008, 4 (1) : 17-32. doi: 10.3934/jimo.2008.4.17
[15]	M. Soledad Aronna, J. Frédéric Bonnans, Andrei V. Dmitruk, Pablo A. Lotito. Quadratic order conditions for bang-singular extremals. Numerical Algebra, Control and Optimization, 2012, 2 (3) : 511-546. doi: 10.3934/naco.2012.2.511
[16]	El Hassan Zerrik, Nihale El Boukhari. Optimal bounded controls problem for bilinear systems. Evolution Equations and Control Theory, 2015, 4 (2) : 221-232. doi: 10.3934/eect.2015.4.221
[17]	Yong He. Switching controls for linear stochastic differential systems. Mathematical Control and Related Fields, 2020, 10 (2) : 443-454. doi: 10.3934/mcrf.2020005
[18]	Ovidiu Cârjă, Alina Lazu. On the minimal time null controllability of the heat equation. Conference Publications, 2009, 2009 (Special) : 143-150. doi: 10.3934/proc.2009.2009.143
[19]	Lydia Ouaili. Minimal time of null controllability of two parabolic equations. Mathematical Control and Related Fields, 2020, 10 (1) : 89-112. doi: 10.3934/mcrf.2019031
[20]	Marc-Auréle Lagache, Ulysse Serres, Vincent Andrieu. Minimal time synthesis for a kinematic drone model. Mathematical Control and Related Fields, 2017, 7 (2) : 259-288. doi: 10.3934/mcrf.2017009

2020 Impact Factor: 1.284

Tools

Metrics

Tools

Article outline

Figures and Tables

2020 Impact Factor: 1.284

Tools

Figures and Tables

2020 Impact Factor: 1.284

American Institute of Mathematical Sciences

Decompositions and bang-bang properties

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Decompositions and bang-bang properties

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors