August  2020, 25(8): 3275-3303. doi: 10.3934/dcdsb.2020062

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

Received  January 2019 Revised  August 2019 Published  April 2020

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.

Citation: Larbi Berrahmoune. Null controllability for distributed systems with time-varying constraint and applications to parabolic-like equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (8) : 3275-3303. doi: 10.3934/dcdsb.2020062
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.  Google Scholar

[2]

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

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J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley & Sons, Inc., New York, 1984. doi: 10.1112/blms/17.5.487.  Google Scholar

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J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2, Birkhäuser Boston, Inc., Boston, MA, 1990.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[7]

C. CarthelR. 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.  Google Scholar

[8]

N. ChenY. 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.  Google Scholar

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I. Ekeland and R. Temam, Convex Analysis and Variational Problems, Studies in Mathematics and its Applications, 1, North-Holland Publishing Co., Amsterdam-Oxford, 1976.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

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J. LohéacE. 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.  Google Scholar

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J. LohéacE. 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.  Google Scholar

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D. MaityM. 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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

G. Peichel and W. Schapacher, Constrained controllability in Banach spaces, Siam J. Control Optim., 24 (1986), 1261-1275.  doi: 10.1137/0324076.  Google Scholar

[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.  Google Scholar

[29]

K. D. PhungG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

T. Schanbacher, Aspects of positivity in control theory, SIAM J. Control Optim., 27 (1989), 457-475.  doi: 10.1137/0327024.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

N. K. Son, Local controllability of linear systems with restrained controls in Banach space, Acta Math. Vietnam, 5 (1980), 78-87.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

G. Wang and Y. Zhang, Decompositions and bang-bang properties, Math. Control Relat. Fields, 7 (2017), 73-170.  doi: 10.3934/mcrf.2017005.  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]

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.  Google Scholar

[48]

G. Weiss, Admissibility of unbounded control operators, SIAM J. Control Optim., 27 (1989), 527-545.  doi: 10.1137/0327028.  Google Scholar

[49]

G. Weiss, Admissible observation operators for linear semigroups, Israel J. Math., 65 (1989), 17-43.  doi: 10.1007/BF02788172.  Google Scholar

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.  Google Scholar

[2]

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

[3]

J.-P. Aubin and I. Ekeland, Applied Nonlinear Analysis, John Wiley & Sons, Inc., New York, 1984. doi: 10.1112/blms/17.5.487.  Google Scholar

[4]

J.-P. Aubin and H. Frankowska, Set-Valued Analysis, Systems & Control: Foundations & Applications, 2, Birkhäuser Boston, Inc., Boston, MA, 1990.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

C. CarthelR. 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.  Google Scholar

[8]

N. ChenY. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[15] R. GlowinskiJ.-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.  Google Scholar
[16]

E. B. Lee and L. Markus, Foundations of Optimal Control Theory, John Wiley & Sons, Inc., New York-London-Sydney, 1967.  Google Scholar

[17]

J.-L. Lions, Optimal Control of Systems Governed by Partial Differential Equations, Springer-Verlag, New York-Berlin, 1971.  Google Scholar

[18]

J.-L. Lions and E. Magenes, Problèmes aux Limites non Homogènes et Applications, 2, Dunod, Paris, 1968.  Google Scholar

[19]

J. LohéacE. 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.  Google Scholar

[20]

J. LohéacE. 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.  Google Scholar

[21]

D. MaityM. 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.  Google Scholar

[22]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

G. Peichel and W. Schapacher, Constrained controllability in Banach spaces, Siam J. Control Optim., 24 (1986), 1261-1275.  doi: 10.1137/0324076.  Google Scholar

[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.  Google Scholar

[29]

K. D. PhungG. 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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[33]

T. Schanbacher, Aspects of positivity in control theory, SIAM J. Control Optim., 27 (1989), 457-475.  doi: 10.1137/0327024.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[37]

N. K. Son, Local controllability of linear systems with restrained controls in Banach space, Acta Math. Vietnam, 5 (1980), 78-87.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[44]

G. Wang and Y. Zhang, Decompositions and bang-bang properties, Math. Control Relat. Fields, 7 (2017), 73-170.  doi: 10.3934/mcrf.2017005.  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]

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.  Google Scholar

[48]

G. Weiss, Admissibility of unbounded control operators, SIAM J. Control Optim., 27 (1989), 527-545.  doi: 10.1137/0327028.  Google Scholar

[49]

G. Weiss, Admissible observation operators for linear semigroups, Israel J. Math., 65 (1989), 17-43.  doi: 10.1007/BF02788172.  Google Scholar

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