- Previous Article
- DCDS-B Home
- This Issue
-
Next Article
On the optimization problems of the principal eigenvalues of measure differential equations with indefinite measures
Null controllability for distributed systems with time-varying constraint and applications to parabolic-like equations
ANLIMAD, Département de Mathématiques, ENS, Université Mohammed V, BP 5118, Rabat, Morocco |
We consider the null controllability problem fo linear systems of the form $ y'(t) = Ay(t)+Bu(t) $ on a Hilbert space $ Y $. We suppose that the control operator $ B $ is bounded from the control space $ U $ to a larger extrapolation space containing $ Y $. The control $ u $ is constrained to lie in a time-varying bounded subset $ \Gamma(t) \subset U $. From a general existence result based on a selection theorem, we obtain various properties on local and global constrained null controllability. The existence of the time optimal control is established in a general framework. When the constraint set $ \Gamma (t) $ contains the origin in its interior at each $ t>0 $, the local constrained property turns out to be equivalent to a weighted dual observability inequality of $ L^{1} $ type with respect to the time variable. We treat also the problem of determining a steering control for general constraint sets $ \Gamma (t) $ in nonsmooth convex analysis context. Applications to the heat equation are treated for distributed and boundary controls under the assumptions that $ \Gamma (t) $ is a closed ball centered at the origin and its radius is time-varying.
References:
[1] |
N. U. Ahmed,
Finite-time null controllability for a class of linear evolution equations on a Banach space with control constraints, J. Optim. Theory Appl., 47 (1985), 129-158.
doi: 10.1007/BF00940766. |
[2] |
J. Apraiz, L. Escauriaza, G. Wang and C. Zhang,
Observability inequalities and measurable sets, J. Eur. Math. Soc. (JEMS), 16 (2014), 2433-2475.
doi: 10.4171/JEMS/490. |
[3] |
J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley & Sons, Inc., New York, 1984.
doi: 10.1112/blms/17.5.487. |
[4] |
J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2, Birkhäuser Boston, Inc., Boston, MA, 1990. |
[5] |
L. Berrahmoune,
A varational approach to constrained controllability for distributed systems, J. Math. Anal. Appl., 416 (2014), 805-823.
doi: 10.1016/j.jmaa.2014.03.004. |
[6] |
L. Berrahmoune, Constrained null controllability for distributed systems and applications to hyperbolic-like equations, ESAIM Control Optim. Calc. Var., 25 (2019), 40pp.
doi: 10.1051/cocv/2018018. |
[7] |
C. Carthel, R. Glowinski and J.-L. Lions,
On exact and approximate boundary controllability for the heat equation: A numerical approach, J. Optim. Theory Appl., 82 (1994), 429-484.
doi: 10.1007/BF02192213. |
[8] |
N. Chen, Y. Wang and D.-H. Yang,
Time-varying bang-bang property of time optimal controls for heat equation and its application, Systems Control Lett., 112 (2018), 18-23.
doi: 10.1016/j.sysconle.2017.12.008. |
[9] |
F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics, 264, Springer, London, 2013.
doi: 10.1007/978-1-4471-4820-3. |
[10] |
R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Lecture Notes in Information Sciences, 8, Springer-Verlag, Berlin-New York, 1978.
doi: 10.1007/BFb0006761. |
[11] |
S. Dolecki and D. L. Russell,
A general theory of observation and control, SIAM J. Control Optim., 15 (1977), 185-220.
doi: 10.1137/0315015. |
[12] |
I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Studies in Mathematics and its Applications, 1, North-Holland Publishing Co., Amsterdam-Oxford, 1976. |
[13] |
H. O. Fattorini, Infinite Dimensional Linear Control Systems. The Time Optimal and Norm Optimal Problems, North-Holland Mathematics Studies, 201, Elsevier Science B.V., Amsterdam, 2005. |
[14] |
H. O. Fattorini,
Time and norm optimal controls: A survey of recent results and open problems, Acta Math. Sci. Ser. B, 31 (2011), 2203-2218.
doi: 10.1016/S0252-9602(11)60394-9. |
[15] |
R. Glowinski, J.-L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems. A Numerical Approach, Encyclopedia of Mathematics and its Applications, 117, Cambridge University Press, Cambridge, 2008.
doi: 10.1017/CBO9780511721595.![]() ![]() ![]() |
[16] |
E. B. Lee and L. Markus, Foundations of Optimal Control Theory, John Wiley & Sons, Inc., New York-London-Sydney, 1967. |
[17] |
J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York-Berlin, 1971. |
[18] |
J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, 2, Dunod, Paris, 1968. |
[19] |
J. Lohéac, E. Trélat and E. Zuazua,
Minimal controllability time for the heat equation under unilateral state or control constraints, Math. Models Methods Appl. Sci., 27 (2017), 1587-1644.
doi: 10.1142/S0218202517500270. |
[20] |
J. Lohéac, E. Trélat and E. Zuazua,
Minimal controllability time for finite-dimensional control systems under state constraints, Automatica J. IFAC, 96 (2018), 380-392.
doi: 10.1016/j.automatica.2018.07.010. |
[21] |
D. Maity, M. Tucsnak and E. Zuazua,
Controllability and positivity constraints in population dynamics with age structuring and diffusion, J. Math. Pures Appl. (9), 129 (2019), 153-179.
doi: 10.1016/j.matpur.2018.12.006. |
[22] |
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. |
[23] |
J. Mizel and T. I. 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. |
[24] |
A. Münch and P. Pedregal,
Numerical null controllability of the heat equation through a least squares and variational approach, European J. Appl. Math., 25 (2014), 277-306.
doi: 10.1017/S0956792514000023. |
[25] |
A. Münch and E. Zuazua, Numerical approximation of null controls for the heat equation: Ill-posedness and remedies, Inverse Problems, 26 (2010), 39pp.
doi: 10.1088/0266-5611/26/8/085018. |
[26] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[27] |
G. Peichel and W. Schapacher,
Constrained controllability in Banach spaces, Siam J. Control Optim., 24 (1986), 1261-1275.
doi: 10.1137/0324076. |
[28] |
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. (JEMS), 15 (2013), 681-703.
doi: 10.4171/JEMS/371. |
[29] |
K. D. Phung, G. Wang and X. Zhang,
On the existence of time optimal controls for linear evolution equations, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 925-941.
doi: 10.3934/dcdsb.2007.8.925. |
[30] |
R. T. Rockafellar,
Duality and stability in extremum problems involving convex function, Pacific J. Math., 21 (1967), 167-187.
doi: 10.2140/pjm.1967.21.167. |
[31] |
R. T. Rockafellar, Conjugate Duality and Optimization, CBMS Regional Conference Series in Applied Mathematics, 16, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1974.
doi: 10.1137/1.9781611970524. |
[32] |
D. Salamon,
Infinite dimensional systems with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431.
doi: 10.2307/2000351. |
[33] |
T. Schanbacher,
Aspects of positivity in control theory, SIAM J. Control Optim., 27 (1989), 457-475.
doi: 10.1137/0327024. |
[34] |
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. |
[35] |
W. E. Schmittendorf and B. R. Barmish,
Null controllability of linear systems with constrained controls, Siam J. Control Optim., 18 (1980), 327-345.
doi: 10.1137/0318025. |
[36] |
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. |
[37] |
N. K. Son,
Local controllability of linear systems with restrained controls in Banach space, Acta Math. Vietnam, 5 (1980), 78-87.
|
[38] |
N. K. Son,
A unified approach to constrained approximate controllability for the heat equations and the retarded equations, J. Math. Anal. Appl., 150 (1990), 1-19.
doi: 10.1016/0022-247X(90)90192-I. |
[39] |
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts, Birkhäuser Verlag, Basel, 2009.
doi: 10.1007/978-3-7643-8994-9. |
[40] |
A. Vieru,
On null controllability of linear systems in Banach spaces, Systems Control Lett., 54 (2005), 331-337.
doi: 10.1016/j.sysconle.2004.09.004. |
[41] |
G. Wang,
$L^{\infty}$-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. |
[42] |
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. |
[43] |
G. Wang and C. Zhang,
Observability inequalities from measurable sets for some abstract evolution equations, SIAM J. Control Optim., 55 (2017), 1862-1886.
doi: 10.1137/15M1051907. |
[44] |
G. Wang and Y. Zhang,
Decompositions and bang-bang properties, Math. Control Relat. Fields, 7 (2017), 73-170.
doi: 10.3934/mcrf.2017005. |
[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] |
D. Washburn,
A bound on the boundary input map for parabolic equations with applications to time-optimal control, SIAM J. Control Optim., 17 (1979), 652-671.
doi: 10.1137/0317046. |
[48] |
G. Weiss,
Admissibility of unbounded control operators, SIAM J. Control Optim., 27 (1989), 527-545.
doi: 10.1137/0327028. |
[49] |
G. Weiss,
Admissible observation operators for linear semigroups, Israel J. Math., 65 (1989), 17-43.
doi: 10.1007/BF02788172. |
show all references
References:
[1] |
N. U. Ahmed,
Finite-time null controllability for a class of linear evolution equations on a Banach space with control constraints, J. Optim. Theory Appl., 47 (1985), 129-158.
doi: 10.1007/BF00940766. |
[2] |
J. Apraiz, L. Escauriaza, G. Wang and C. Zhang,
Observability inequalities and measurable sets, J. Eur. Math. Soc. (JEMS), 16 (2014), 2433-2475.
doi: 10.4171/JEMS/490. |
[3] |
J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley & Sons, Inc., New York, 1984.
doi: 10.1112/blms/17.5.487. |
[4] |
J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2, Birkhäuser Boston, Inc., Boston, MA, 1990. |
[5] |
L. Berrahmoune,
A varational approach to constrained controllability for distributed systems, J. Math. Anal. Appl., 416 (2014), 805-823.
doi: 10.1016/j.jmaa.2014.03.004. |
[6] |
L. Berrahmoune, Constrained null controllability for distributed systems and applications to hyperbolic-like equations, ESAIM Control Optim. Calc. Var., 25 (2019), 40pp.
doi: 10.1051/cocv/2018018. |
[7] |
C. Carthel, R. Glowinski and J.-L. Lions,
On exact and approximate boundary controllability for the heat equation: A numerical approach, J. Optim. Theory Appl., 82 (1994), 429-484.
doi: 10.1007/BF02192213. |
[8] |
N. Chen, Y. Wang and D.-H. Yang,
Time-varying bang-bang property of time optimal controls for heat equation and its application, Systems Control Lett., 112 (2018), 18-23.
doi: 10.1016/j.sysconle.2017.12.008. |
[9] |
F. Clarke, Functional Analysis, Calculus of Variations and Optimal Control, Graduate Texts in Mathematics, 264, Springer, London, 2013.
doi: 10.1007/978-1-4471-4820-3. |
[10] |
R. F. Curtain and A. J. Pritchard, Infinite Dimensional Linear Systems Theory, Lecture Notes in Information Sciences, 8, Springer-Verlag, Berlin-New York, 1978.
doi: 10.1007/BFb0006761. |
[11] |
S. Dolecki and D. L. Russell,
A general theory of observation and control, SIAM J. Control Optim., 15 (1977), 185-220.
doi: 10.1137/0315015. |
[12] |
I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Studies in Mathematics and its Applications, 1, North-Holland Publishing Co., Amsterdam-Oxford, 1976. |
[13] |
H. O. Fattorini, Infinite Dimensional Linear Control Systems. The Time Optimal and Norm Optimal Problems, North-Holland Mathematics Studies, 201, Elsevier Science B.V., Amsterdam, 2005. |
[14] |
H. O. Fattorini,
Time and norm optimal controls: A survey of recent results and open problems, Acta Math. Sci. Ser. B, 31 (2011), 2203-2218.
doi: 10.1016/S0252-9602(11)60394-9. |
[15] |
R. Glowinski, J.-L. Lions and J. He, Exact and Approximate Controllability for Distributed Parameter Systems. A Numerical Approach, Encyclopedia of Mathematics and its Applications, 117, Cambridge University Press, Cambridge, 2008.
doi: 10.1017/CBO9780511721595.![]() ![]() ![]() |
[16] |
E. B. Lee and L. Markus, Foundations of Optimal Control Theory, John Wiley & Sons, Inc., New York-London-Sydney, 1967. |
[17] |
J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York-Berlin, 1971. |
[18] |
J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, 2, Dunod, Paris, 1968. |
[19] |
J. Lohéac, E. Trélat and E. Zuazua,
Minimal controllability time for the heat equation under unilateral state or control constraints, Math. Models Methods Appl. Sci., 27 (2017), 1587-1644.
doi: 10.1142/S0218202517500270. |
[20] |
J. Lohéac, E. Trélat and E. Zuazua,
Minimal controllability time for finite-dimensional control systems under state constraints, Automatica J. IFAC, 96 (2018), 380-392.
doi: 10.1016/j.automatica.2018.07.010. |
[21] |
D. Maity, M. Tucsnak and E. Zuazua,
Controllability and positivity constraints in population dynamics with age structuring and diffusion, J. Math. Pures Appl. (9), 129 (2019), 153-179.
doi: 10.1016/j.matpur.2018.12.006. |
[22] |
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. |
[23] |
J. Mizel and T. I. 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. |
[24] |
A. Münch and P. Pedregal,
Numerical null controllability of the heat equation through a least squares and variational approach, European J. Appl. Math., 25 (2014), 277-306.
doi: 10.1017/S0956792514000023. |
[25] |
A. Münch and E. Zuazua, Numerical approximation of null controls for the heat equation: Ill-posedness and remedies, Inverse Problems, 26 (2010), 39pp.
doi: 10.1088/0266-5611/26/8/085018. |
[26] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Applied Mathematical Sciences, 44, Springer-Verlag, New York, 1983.
doi: 10.1007/978-1-4612-5561-1. |
[27] |
G. Peichel and W. Schapacher,
Constrained controllability in Banach spaces, Siam J. Control Optim., 24 (1986), 1261-1275.
doi: 10.1137/0324076. |
[28] |
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. (JEMS), 15 (2013), 681-703.
doi: 10.4171/JEMS/371. |
[29] |
K. D. Phung, G. Wang and X. Zhang,
On the existence of time optimal controls for linear evolution equations, Discrete Contin. Dyn. Syst. Ser. B, 8 (2007), 925-941.
doi: 10.3934/dcdsb.2007.8.925. |
[30] |
R. T. Rockafellar,
Duality and stability in extremum problems involving convex function, Pacific J. Math., 21 (1967), 167-187.
doi: 10.2140/pjm.1967.21.167. |
[31] |
R. T. Rockafellar, Conjugate Duality and Optimization, CBMS Regional Conference Series in Applied Mathematics, 16, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1974.
doi: 10.1137/1.9781611970524. |
[32] |
D. Salamon,
Infinite dimensional systems with unbounded control and observation: A functional analytic approach, Trans. Amer. Math. Soc., 300 (1987), 383-431.
doi: 10.2307/2000351. |
[33] |
T. Schanbacher,
Aspects of positivity in control theory, SIAM J. Control Optim., 27 (1989), 457-475.
doi: 10.1137/0327024. |
[34] |
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. |
[35] |
W. E. Schmittendorf and B. R. Barmish,
Null controllability of linear systems with constrained controls, Siam J. Control Optim., 18 (1980), 327-345.
doi: 10.1137/0318025. |
[36] |
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. |
[37] |
N. K. Son,
Local controllability of linear systems with restrained controls in Banach space, Acta Math. Vietnam, 5 (1980), 78-87.
|
[38] |
N. K. Son,
A unified approach to constrained approximate controllability for the heat equations and the retarded equations, J. Math. Anal. Appl., 150 (1990), 1-19.
doi: 10.1016/0022-247X(90)90192-I. |
[39] |
M. Tucsnak and G. Weiss, Observation and Control for Operator Semigroups, Birkhäuser Advanced Texts, Birkhäuser Verlag, Basel, 2009.
doi: 10.1007/978-3-7643-8994-9. |
[40] |
A. Vieru,
On null controllability of linear systems in Banach spaces, Systems Control Lett., 54 (2005), 331-337.
doi: 10.1016/j.sysconle.2004.09.004. |
[41] |
G. Wang,
$L^{\infty}$-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. |
[42] |
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. |
[43] |
G. Wang and C. Zhang,
Observability inequalities from measurable sets for some abstract evolution equations, SIAM J. Control Optim., 55 (2017), 1862-1886.
doi: 10.1137/15M1051907. |
[44] |
G. Wang and Y. Zhang,
Decompositions and bang-bang properties, Math. Control Relat. Fields, 7 (2017), 73-170.
doi: 10.3934/mcrf.2017005. |
[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] |
D. Washburn,
A bound on the boundary input map for parabolic equations with applications to time-optimal control, SIAM J. Control Optim., 17 (1979), 652-671.
doi: 10.1137/0317046. |
[48] |
G. Weiss,
Admissibility of unbounded control operators, SIAM J. Control Optim., 27 (1989), 527-545.
doi: 10.1137/0327028. |
[49] |
G. Weiss,
Admissible observation operators for linear semigroups, Israel J. Math., 65 (1989), 17-43.
doi: 10.1007/BF02788172. |
[1] |
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 |
[2] |
Shu Zhang, Jian Xu. Time-varying delayed feedback control for an internet congestion control model. Discrete and Continuous Dynamical Systems - B, 2011, 16 (2) : 653-668. doi: 10.3934/dcdsb.2011.16.653 |
[3] |
Lijuan Wang, Yashan Xu. Admissible controls and controllable sets for a linear time-varying ordinary differential equation. Mathematical Control and Related Fields, 2018, 8 (3&4) : 1001-1019. doi: 10.3934/mcrf.2018043 |
[4] |
Qi Lü, Enrique Zuazua. Robust null controllability for heat equations with unknown switching control mode. Discrete and Continuous Dynamical Systems, 2014, 34 (10) : 4183-4210. doi: 10.3934/dcds.2014.34.4183 |
[5] |
Roberta Fabbri, Russell Johnson, Sylvia Novo, Carmen Núñez. On linear-quadratic dissipative control processes with time-varying coefficients. Discrete and Continuous Dynamical Systems, 2013, 33 (1) : 193-210. doi: 10.3934/dcds.2013.33.193 |
[6] |
Le Viet Cuong, Thai Son Doan. Assignability of dichotomy spectra for discrete time-varying linear control systems. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3597-3607. doi: 10.3934/dcdsb.2020074 |
[7] |
Bing Sun. Optimal control of transverse vibration of a moving string with time-varying lengths. Mathematical Control and Related Fields, 2021 doi: 10.3934/mcrf.2021042 |
[8] |
Di Wu, Yanqin Bai, Fusheng Xie. Time-scaling transformation for optimal control problem with time-varying delay. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1683-1695. doi: 10.3934/dcdss.2020098 |
[9] |
K. Aruna Sakthi, A. Vinodkumar. Stabilization on input time-varying delay for linear switched systems with truncated predictor control. Numerical Algebra, Control and Optimization, 2020, 10 (2) : 237-247. doi: 10.3934/naco.2019050 |
[10] |
Serge Nicaise, Julie Valein, Emilia Fridman. Stability of the heat and of the wave equations with boundary time-varying delays. Discrete and Continuous Dynamical Systems - S, 2009, 2 (3) : 559-581. doi: 10.3934/dcdss.2009.2.559 |
[11] |
Mokhtari Yacine. Boundary controllability and boundary time-varying feedback stabilization of the 1D wave equation in non-cylindrical domains. Evolution Equations and Control Theory, 2022, 11 (2) : 373-397. doi: 10.3934/eect.2021004 |
[12] |
Abdelaziz Khoutaibi, Lahcen Maniar. Null controllability for a heat equation with dynamic boundary conditions and drift terms. Evolution Equations and Control Theory, 2020, 9 (2) : 535-559. doi: 10.3934/eect.2020023 |
[13] |
Abdelaziz Khoutaibi, Lahcen Maniar, Omar Oukdach. Null controllability for semilinear heat equation with dynamic boundary conditions. Discrete and Continuous Dynamical Systems - S, 2022, 15 (6) : 1525-1546. doi: 10.3934/dcdss.2022087 |
[14] |
Jérome Lohéac, Jean-François Scheid. Time optimal control for a nonholonomic system with state constraint. Mathematical Control and Related Fields, 2013, 3 (2) : 185-208. doi: 10.3934/mcrf.2013.3.185 |
[15] |
Serge Nicaise, Cristina Pignotti, Julie Valein. Exponential stability of the wave equation with boundary time-varying delay. Discrete and Continuous Dynamical Systems - S, 2011, 4 (3) : 693-722. doi: 10.3934/dcdss.2011.4.693 |
[16] |
Larissa Fardigola, Kateryna Khalina. Controllability problems for the heat equation on a half-axis with a bounded control in the Neumann boundary condition. Mathematical Control and Related Fields, 2021, 11 (1) : 211-236. doi: 10.3934/mcrf.2020034 |
[17] |
Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control and Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011 |
[18] |
Idriss Boutaayamou, Lahcen Maniar, Omar Oukdach. Stackelberg-Nash null controllability of heat equation with general dynamic boundary conditions. Evolution Equations and Control Theory, 2021 doi: 10.3934/eect.2021044 |
[19] |
Yong-Kui Chang, Xiaojing Liu. Time-varying integro-differential inclusions with Clarke sub-differential and non-local initial conditions: existence and approximate controllability. Evolution Equations and Control Theory, 2020, 9 (3) : 845-863. doi: 10.3934/eect.2020036 |
[20] |
Shihchung Chiang. Numerical optimal unbounded control with a singular integro-differential equation as a constraint. Conference Publications, 2013, 2013 (special) : 129-137. doi: 10.3934/proc.2013.2013.129 |
2020 Impact Factor: 1.327
Tools
Metrics
Other articles
by authors
[Back to Top]