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Control and stabilization of 2 × 2 hyperbolic systems on graphs
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 |
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.
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, 3rd 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, 2nd 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, 3rd 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, 2nd 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. |


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