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

Decompositions and bang-bang properties

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.

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

F. Ammar KhodjaA. BenabdallahM. 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.  Google Scholar

[2]

F. Ammar KhodjaA. BenabdallahM. 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.  Google Scholar

[3]

F. Ammar KhodjaA. BenabdallahM. 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.  Google Scholar

[4]

J. ApraizL. EscauriazaG. Wang and C. Zhang, Observability inequalities and measurable sets, J. Eur. Math. Soc., 16 (2014), 2433-2475.  doi: 10.4171/JEMS/490.  Google Scholar

[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.  Google Scholar

[6]

V. Barbu, Analysis and Control of Nonlinear Infinite-dimensional Systems, Academic Press, Boston, 1993.  Google Scholar

[7]

O. Cârjǎ, On continuity of the minimal time function for distributed control systems, Boll. Un. Mat. Ital. -A, 4 (1985), 293-302.   Google Scholar

[8]

J. M. Coron, Control and Nonlinearity, American Mathematical Society, 2007.  Google Scholar

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H. O. Fattorini, Infinite Dimensional Linear Control Systems: The Time Optimal and Norm Optimal Problems, ELSEVIER, 2005.  Google Scholar

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H. O. Fattorini, Time-optimal control of solutions of operational differential equations, J. SIAM Control -A, 2 (1964), 54-59.  doi: 10.1137/0302005.  Google Scholar

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H. O. Fattorini, The time-optimal control problem in Banach spaces, Appl. Math. Optim., 1 (1974/75), 163-188.  doi: 10.1007/BF01449028.  Google Scholar

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H. O. Fattorini, Some remarks on the time optimal control problem in infinite dimension, Calculus of Variations and Optimal Control, 411 (2000), 77-96.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[23]

J. Lohéac and E. Zuazua, Norm saturating property of time optimal controls for wave-type equations, 2016, <hal-01258878>. Google Scholar

[24]

H. LouJ. 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.  Google Scholar

[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.  Google Scholar

[26]

S. MicuI. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

N. V. Petrov, The Bellman problem for a time-optimality problem, Prikl. Mat. Meh., 34 (1970), 820-826.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

K. D. PhungL. 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.  Google Scholar

[34]

K. D. PhungG. 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.  Google Scholar

[35]

W. Rudin, Real and Complex Analysis, 3rd edition, McGraw-Hill Companies, 1987.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

E. D. Sontag, Mathematical Control Theory: Deterministic Finite-Dimensional Systems, 2nd edition, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0577-7.  Google Scholar

[39]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Verlag AG, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

G. WangY. 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.  Google Scholar

[44]

G. Wang and C. Zhang, Observability inequalities from measurable sets for some evolution equations, preprint, arXiv: 1406.3422v1. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

F. Ammar KhodjaA. BenabdallahM. 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.  Google Scholar

[2]

F. Ammar KhodjaA. BenabdallahM. 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.  Google Scholar

[3]

F. Ammar KhodjaA. BenabdallahM. 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.  Google Scholar

[4]

J. ApraizL. EscauriazaG. Wang and C. Zhang, Observability inequalities and measurable sets, J. Eur. Math. Soc., 16 (2014), 2433-2475.  doi: 10.4171/JEMS/490.  Google Scholar

[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.  Google Scholar

[6]

V. Barbu, Analysis and Control of Nonlinear Infinite-dimensional Systems, Academic Press, Boston, 1993.  Google Scholar

[7]

O. Cârjǎ, On continuity of the minimal time function for distributed control systems, Boll. Un. Mat. Ital. -A, 4 (1985), 293-302.   Google Scholar

[8]

J. M. Coron, Control and Nonlinearity, American Mathematical Society, 2007.  Google Scholar

[9]

H. O. Fattorini, Infinite Dimensional Linear Control Systems: The Time Optimal and Norm Optimal Problems, ELSEVIER, 2005.  Google Scholar

[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.  Google Scholar

[11]

H. O. Fattorini, The time-optimal control problem in Banach spaces, Appl. Math. Optim., 1 (1974/75), 163-188.  doi: 10.1007/BF01449028.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[20]

X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[23]

J. Lohéac and E. Zuazua, Norm saturating property of time optimal controls for wave-type equations, 2016, <hal-01258878>. Google Scholar

[24]

H. LouJ. 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.  Google Scholar

[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.  Google Scholar

[26]

S. MicuI. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[30]

N. V. Petrov, The Bellman problem for a time-optimality problem, Prikl. Mat. Meh., 34 (1970), 820-826.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

K. D. PhungL. 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.  Google Scholar

[34]

K. D. PhungG. 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.  Google Scholar

[35]

W. Rudin, Real and Complex Analysis, 3rd edition, McGraw-Hill Companies, 1987.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

E. D. Sontag, Mathematical Control Theory: Deterministic Finite-Dimensional Systems, 2nd edition, Springer-Verlag, New York, 1998. doi: 10.1007/978-1-4612-0577-7.  Google Scholar

[39]

M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Verlag AG, 2009. doi: 10.1007/978-3-7643-8994-9.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

G. WangY. 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.  Google Scholar

[44]

G. Wang and C. Zhang, Observability inequalities from measurable sets for some evolution equations, preprint, arXiv: 1406.3422v1. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  The BBP decomposition for $(NP)^{T,y_0}$
Figure 2.  The BBP decomposition for $(TP)^{M,y_0}$
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