The cost of controlling weakly degenerate parabolic equations by boundary controls

Piermarco Cannarsa; Patrick Martinez; Judith Vancostenoble

doi:10.3934/mcrf.2017006

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June 2017, 7(2): 171-211. doi: 10.3934/mcrf.2017006

The cost of controlling weakly degenerate parabolic equations by boundary controls

Piermarco Cannarsa ^1,, , Patrick Martinez ^2, and Judith Vancostenoble ^2,

1.	Dipartimento di Matematica, Universitá di Roma "Tor Vergata", Via della Ricerca Scientifica, 00133 Roma, Italy
2.	Institut de Mathématiques de Toulouse, UMR CNRS 5219, Université Paul Sabatier Toulouse Ⅲ, 118 route de Narbonne, 31 062 Toulouse Cedex 4, France

* Corresponding author: Piermarco Cannarsa

Received May 2015 Revised July 2016 Published April 2017

Fund Project: This research was partly supported by the Institut Mathematique de Toulouse and Istituto Nazionale di Alta Matematica through funds provided by the national group GNAMPA and the GDRE CONEDP. Moreover, this work was completed while the first author was visiting the Institut Henri Poincaré and Institut des Hautes Études Scientifiques on a CARMIN senior position.

We consider the one-dimensional degenerate parabolic equation

$u_t - (x^α u_x)_x =0 \;\;x∈(0,1),\ t ∈ (0,T) ,$

controlled by a boundary force acting at the degeneracy point

$ x=0$

We study the reachable targets at some given time

$T$

using

$ H^1$

controls, studying the influence of the degeneracy parameter

$ α ∈ [0,1)$

. First we obtain precise upper and lower bounds for the null controllability cost, proving that the cost blows up rationnally as

$ \mathit{\alpha } \to {{\rm{1}}^{\rm{ - }}}$

and exponentially fast when

$ \mathit{T} \to {{\rm{0}}^{\rm{ + }}}$

.
Next, thanks to the special structure of the eigenfunctions of the problem, we investigate and obtain (partial) results concerning the structure of the reachable states.
Our approach is based on the moment method developed by Fattorini and Russell [19,20]. To achieve our goals, we extend some of their general results concerning biorthogonal families, using complex analysis techniques developped by Seidman [48], Guichal [26], Tenenbaum-Tucsnak [49] and Lissy [35,36].

Keywords: Controllability, reachable set, degenerate parabolic equation.

Mathematics Subject Classification: 35K65, 93B05, 33C10, 30B10.

Citation: Piermarco Cannarsa, Patrick Martinez, Judith Vancostenoble. The cost of controlling weakly degenerate parabolic equations by boundary controls. Mathematical Control & Related Fields, 2017, 7 (2) : 171-211. doi: 10.3934/mcrf.2017006

References:

[1]	F. Alabau-Boussouira, P. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., 6 (2006), 161-204. doi: 10.1007/s00028-006-0222-6. Google Scholar
[2]	F. Ammar Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials, J. Math. Pures Appl., 96 (2011), 555-590. doi: 10.1016/j.matpur.2011.06.005. Google Scholar
[3]	K. Beauchard, P. Cannarsa and R. Guglielmi, Null controllability of Grushin-type operators in dimension two, J. Eur. Math. Soc. (JEMS), 16 (2014), 67-101. doi: 10.4171/JEMS/428. Google Scholar
[4]	M. Campiti, G. Metafune and D. Pallara, Degenerate self-adjoint evolution equations on the unit interval, Semigroup Forum, 57 (1998), 1-36. doi: 10.1007/PL00005959. Google Scholar
[5]	P. Cannarsa, G. Fragnelli and D. Rocchetti, Null controllability of degenerate parabolic operators with drift, Netw. Heterog. Media, 2 (2007), 695-715. doi: 10.3934/nhm.2007.2.695. Google Scholar
[6]	P. Cannarsa, G. Fragnelli and D. Rocchetti, Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form, J. Evol. Equ., 8 (2008), 583-616. doi: 10.1007/s00028-008-0353-34. Google Scholar
[7]	P. Cannarsa, P. Martinez and J. Vancostenoble, Null Controllability of degenerate heat equations, Adv. Differential Equations, 10 (2005), 153-190. Google Scholar
[8]	P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., 47 (2008), 1-19. doi: 10.1137/04062062X. Google Scholar
[9]	P. Cannarsa, P. Martinez and J. Vancostenoble, Global Carleman estimates for degenerate parabolic operators with applications Memoirs of the American Mathematical Society, 239 (2016), ix+209 pp. doi: 10.1090/memo/1133. Google Scholar
[10]	P. Cannarsa, P. Martinez and J. Vancostenoble, {The cost of controlling degenerate parabolic equations by locally distributed controls}, in preparation. Google Scholar
[11]	P. Cannarsa, J. Tort and M. Yamamoto, Unique continuation and approximate controllability for a degenerate parabolic equation, Appl. Anal., 91 (2012), 1409-1425. doi: 10.1080/00036811.2011.639766. Google Scholar
[12]	J. M. Coron and S. Guerrero, Singular optimal control: A linear 1-D parabolic-hyperbolic example, Asymp. Anal., 44 (2005), 237-257. Google Scholar
[13]	J. Dardé and S. Ervedoza, On the reachable set for the one-dimensional heat equation (2016), arXiv: 1609.02692. Google Scholar
[14]	S. Ervedoza and E. Zuazua, Sharp observability estimates for heat equations, Arch. Ration. Mech. Anal., 202 (2011), 975-1017. doi: 10.1007/s00205-011-0445-8. Google Scholar
[15]	S. Ervedoza and E. Zuazua, Observability of heat processes by transmutation without geometric restrictions, Math. Control Relat. Fields, 1 (2011), 177-187. doi: 10.3934/mcrf.2011.1.177. Google Scholar
[16]	L. Escauriaza, G. Seregin and V. Šverák, Backward uniqueness for the heat operator in half-space, St. Petersburg Math. J., 15 (2004), 139-148. doi: 10.1090/S1061-0022-03-00806-9. Google Scholar
[17]	W. N. Everitt and A. Zettl, On a class of integral inequalities, J. London Math. Soc., 17 (1978), 291-303. doi: 10.1112/jlms/s2-17.2.291. Google Scholar
[18]	W. N. Everitt, A catalogue of Sturm-Liouville differential equations, Sturm-Liouville Theory, Birkhäuser, Basel, (2005), 271-331. doi: 10.1007/3-7643-7359-8_12. Google Scholar
[19]	H. O. Fattorini and D. L. Russel, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rat. Mech. Anal., 43 (1971), 272-292. doi: 10.1007/BF00250466. Google Scholar
[20]	H. O. Fattorini and D. L. Russel, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Quart. Appl. Math., 13 (1974/75), 1-13. doi: 10.1090/qam/510972. Google Scholar
[21]	H. O. Fattorini, Boundary control of temperature distributions in a parallelepipedon, SIAM J. Control, 13 (1975), 1-13. doi: 10.1137/0313001. Google Scholar
[22]	E. Fernandez-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Adv. Differential equations, 5 (2000), 465-514. Google Scholar
[23]	A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Ser. , 34, Seoul National University, Seoul, Korea, 1996. Google Scholar
[24]	O. Glass, A complex-analytic approach to the problem of uniform controllability of transport equation in the vanishing viscosity limit, J. Funct. Anal., 258 (2010), 852-868. doi: 10.1016/j.jfa.2009.06.035. Google Scholar
[25]	M. Gueye, Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations, SIAM J. Control Optim, 52 (2014), 2037-2054. doi: 10.1137/120901374. Google Scholar
[26]	E. N. Güichal, A lower bound of the norm of the control operator for the heat equation, Journal of Mathematical Analysis and Applications, 110 (1985), 519-527. doi: 10.1016/0022-247X(85)90313-0. Google Scholar
[27]	S. Hansen, Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems, Journal of Math. Anal. and Appl., 158 (1991), 487-508. doi: 10.1016/0022-247X(91)90252-U. Google Scholar
[28]	E. Kamke, Differentialgleichungen: Lösungsmethoden und Lösungen. Band 1: Gewöhnliche Differentialgleichungen, Neunte Auflage. Mit einem Vorwort von Detlef Kamke. B. G. Teubner, Stuttgart, 1977. Google Scholar
[29]	V. Komornik, Functional Analysis, Springer editions, 2016. Google Scholar
[30]	V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer, Berlin, 2005. Google Scholar
[31]	J. Lagnese, Control of wave processes with distributed controls supported on a subregion, SIAM J. Control Optim., 21 (1983), 68-85. doi: 10.1137/0321004. Google Scholar
[32]	L. J. Landau, Bessel functions: monotonicity and bounds, Journal of the London Mathematical Society, 61 (2000), 197-215. doi: 10.1112/S0024610799008352. Google Scholar
[33]	N. N. Lebedev, Special Functions and their Applications, Dover Publications, New York, 1972. Google Scholar
[34]	J.-L. Lions and E. Zuazua, On the cost of controlling unstable systems: The case of boundary controls, J. Anal. Math., 73 (1997), 225-249. doi: 10.1007/BF02788145. Google Scholar
[35]	P. Lissy, On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension, SIAM J. Control Optim., 52 (2014), 2651-2676. doi: 10.1137/140951746. Google Scholar
[36]	P. Lissy, Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation, J. Differential Equations, 259 (2015), 5331-5352. doi: 10.1016/j.jde.2015.06.031. Google Scholar
[37]	L. Lorch and M. E. Muldoon, Monotonic sequences related to zeros of Bessel functions, Numer. Algor, 49 (2008), 221-233. doi: 10.1007/s11075-008-9189-4. Google Scholar
[38]	P. Martin, L. Rosier and P. Rouchon, Null controllability of one-dimensional parabolic equations using flatness, Automatica J. IFAC, 50 (2014), 3067-3076. doi: 10.1016/j.automatica.2014.10.049. Google Scholar
[39]	P. Martin, L. Rosier and P. Rouchon, On the reachable states for the boundary control of the heat equation, Applied Mathematics Research eXpress, (2016), 181-216. doi: 10.1093/amrx/abv013. Google Scholar
[40]	P. Martinez and J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Eq, 6 (2006), 325-362. doi: 10.1007/s00028-006-0214-6. Google Scholar
[41]	L. Miller, Geometric bounds on the growth rate of null controllability cost for the heat equation in small time, J. Differential Equations, 204 (2004), 202-226. doi: 10.1016/j.jde.2004.05.007. Google Scholar
[42]	F. W. Olver, Asymptotics and Special Functions, New York, Academic press, 1974. Google Scholar
[43]	C. K. Qu and R. Wong, "Best possible" upper and lower bounds for the zeros of the Bessel function $ J_ν(x)$, Trans. Amer. Math. Soc., 351 (1999), 2833-2859. doi: 10.1090/S0002-9947-99-02165-0. Google Scholar
[44]	R. M. Redheffer, Elementary remarks on completeness, Duke Math. Journal, 35 (1968), 103-116. doi: 10.1215/S0012-7094-68-03511-4. Google Scholar
[45]	L. Schwartz, Étude des Sommes D'exponentielles, deuxiéme édition. Paris, Hermann, 1959. Google Scholar
[46]	T. Seidman, Time invarinace of the reachable set for linear control problems, J. Math. Annal. Appl., 72 (1979), 17-20. doi: 10.1016/0022-247X(79)90271-3. Google Scholar
[47]	T. Seidman, Two results on exact boundary control of parabolic equations, Appl. Math. Optim., 11 (1984), 145-152. doi: 10.1007/BF01442174. Google Scholar
[48]	T. I. Seidman, S. A. Avdonin and S. A. Ivanov, The "window problem" for series of complex exponentials, J. Fourier Anal. Appl., 6 (2000), 233-254. doi: 10.1007/BF02511154. Google Scholar
[49]	G. Tenenbaum and M. Tucsnak, New blow-up rates for fast controls of Schrodinger and heat equations, J. Differential Equations, 243 (2007), 70-100. doi: 10.1016/j.jde.2007.06.019. Google Scholar
[50]	G. N. Watson, A Treatise on the Theory of Bessel Functions second edition, Cambridge University Press, Cambridge, England, 1944. Google Scholar
[51]	R. M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, 1980. Google Scholar

show all references

References:

[1]	F. Alabau-Boussouira, P. Cannarsa and G. Fragnelli, Carleman estimates for degenerate parabolic operators with applications to null controllability, J. Evol. Equ., 6 (2006), 161-204. doi: 10.1007/s00028-006-0222-6. Google Scholar
[2]	F. Ammar Khodja, A. Benabdallah, M. González-Burgos and L. de Teresa, The Kalman condition for the boundary controllability of coupled parabolic systems. Bounds on biorthogonal families to complex matrix exponentials, J. Math. Pures Appl., 96 (2011), 555-590. doi: 10.1016/j.matpur.2011.06.005. Google Scholar
[3]	K. Beauchard, P. Cannarsa and R. Guglielmi, Null controllability of Grushin-type operators in dimension two, J. Eur. Math. Soc. (JEMS), 16 (2014), 67-101. doi: 10.4171/JEMS/428. Google Scholar
[4]	M. Campiti, G. Metafune and D. Pallara, Degenerate self-adjoint evolution equations on the unit interval, Semigroup Forum, 57 (1998), 1-36. doi: 10.1007/PL00005959. Google Scholar
[5]	P. Cannarsa, G. Fragnelli and D. Rocchetti, Null controllability of degenerate parabolic operators with drift, Netw. Heterog. Media, 2 (2007), 695-715. doi: 10.3934/nhm.2007.2.695. Google Scholar
[6]	P. Cannarsa, G. Fragnelli and D. Rocchetti, Controllability results for a class of one-dimensional degenerate parabolic problems in nondivergence form, J. Evol. Equ., 8 (2008), 583-616. doi: 10.1007/s00028-008-0353-34. Google Scholar
[7]	P. Cannarsa, P. Martinez and J. Vancostenoble, Null Controllability of degenerate heat equations, Adv. Differential Equations, 10 (2005), 153-190. Google Scholar
[8]	P. Cannarsa, P. Martinez and J. Vancostenoble, Carleman estimates for a class of degenerate parabolic operators, SIAM J. Control Optim., 47 (2008), 1-19. doi: 10.1137/04062062X. Google Scholar
[9]	P. Cannarsa, P. Martinez and J. Vancostenoble, Global Carleman estimates for degenerate parabolic operators with applications Memoirs of the American Mathematical Society, 239 (2016), ix+209 pp. doi: 10.1090/memo/1133. Google Scholar
[10]	P. Cannarsa, P. Martinez and J. Vancostenoble, {The cost of controlling degenerate parabolic equations by locally distributed controls}, in preparation. Google Scholar
[11]	P. Cannarsa, J. Tort and M. Yamamoto, Unique continuation and approximate controllability for a degenerate parabolic equation, Appl. Anal., 91 (2012), 1409-1425. doi: 10.1080/00036811.2011.639766. Google Scholar
[12]	J. M. Coron and S. Guerrero, Singular optimal control: A linear 1-D parabolic-hyperbolic example, Asymp. Anal., 44 (2005), 237-257. Google Scholar
[13]	J. Dardé and S. Ervedoza, On the reachable set for the one-dimensional heat equation (2016), arXiv: 1609.02692. Google Scholar
[14]	S. Ervedoza and E. Zuazua, Sharp observability estimates for heat equations, Arch. Ration. Mech. Anal., 202 (2011), 975-1017. doi: 10.1007/s00205-011-0445-8. Google Scholar
[15]	S. Ervedoza and E. Zuazua, Observability of heat processes by transmutation without geometric restrictions, Math. Control Relat. Fields, 1 (2011), 177-187. doi: 10.3934/mcrf.2011.1.177. Google Scholar
[16]	L. Escauriaza, G. Seregin and V. Šverák, Backward uniqueness for the heat operator in half-space, St. Petersburg Math. J., 15 (2004), 139-148. doi: 10.1090/S1061-0022-03-00806-9. Google Scholar
[17]	W. N. Everitt and A. Zettl, On a class of integral inequalities, J. London Math. Soc., 17 (1978), 291-303. doi: 10.1112/jlms/s2-17.2.291. Google Scholar
[18]	W. N. Everitt, A catalogue of Sturm-Liouville differential equations, Sturm-Liouville Theory, Birkhäuser, Basel, (2005), 271-331. doi: 10.1007/3-7643-7359-8_12. Google Scholar
[19]	H. O. Fattorini and D. L. Russel, Exact controllability theorems for linear parabolic equations in one space dimension, Arch. Rat. Mech. Anal., 43 (1971), 272-292. doi: 10.1007/BF00250466. Google Scholar
[20]	H. O. Fattorini and D. L. Russel, Uniform bounds on biorthogonal functions for real exponentials with an application to the control theory of parabolic equations, Quart. Appl. Math., 13 (1974/75), 1-13. doi: 10.1090/qam/510972. Google Scholar
[21]	H. O. Fattorini, Boundary control of temperature distributions in a parallelepipedon, SIAM J. Control, 13 (1975), 1-13. doi: 10.1137/0313001. Google Scholar
[22]	E. Fernandez-Cara and E. Zuazua, The cost of approximate controllability for heat equations: The linear case, Adv. Differential equations, 5 (2000), 465-514. Google Scholar
[23]	A. V. Fursikov and O. Yu. Imanuvilov, Controllability of Evolution Equations, Lecture Notes Ser. , 34, Seoul National University, Seoul, Korea, 1996. Google Scholar
[24]	O. Glass, A complex-analytic approach to the problem of uniform controllability of transport equation in the vanishing viscosity limit, J. Funct. Anal., 258 (2010), 852-868. doi: 10.1016/j.jfa.2009.06.035. Google Scholar
[25]	M. Gueye, Exact boundary controllability of 1-D parabolic and hyperbolic degenerate equations, SIAM J. Control Optim, 52 (2014), 2037-2054. doi: 10.1137/120901374. Google Scholar
[26]	E. N. Güichal, A lower bound of the norm of the control operator for the heat equation, Journal of Mathematical Analysis and Applications, 110 (1985), 519-527. doi: 10.1016/0022-247X(85)90313-0. Google Scholar
[27]	S. Hansen, Bounds on functions biorthogonal to sets of complex exponentials; control of damped elastic systems, Journal of Math. Anal. and Appl., 158 (1991), 487-508. doi: 10.1016/0022-247X(91)90252-U. Google Scholar
[28]	E. Kamke, Differentialgleichungen: Lösungsmethoden und Lösungen. Band 1: Gewöhnliche Differentialgleichungen, Neunte Auflage. Mit einem Vorwort von Detlef Kamke. B. G. Teubner, Stuttgart, 1977. Google Scholar
[29]	V. Komornik, Functional Analysis, Springer editions, 2016. Google Scholar
[30]	V. Komornik and P. Loreti, Fourier Series in Control Theory, Springer, Berlin, 2005. Google Scholar
[31]	J. Lagnese, Control of wave processes with distributed controls supported on a subregion, SIAM J. Control Optim., 21 (1983), 68-85. doi: 10.1137/0321004. Google Scholar
[32]	L. J. Landau, Bessel functions: monotonicity and bounds, Journal of the London Mathematical Society, 61 (2000), 197-215. doi: 10.1112/S0024610799008352. Google Scholar
[33]	N. N. Lebedev, Special Functions and their Applications, Dover Publications, New York, 1972. Google Scholar
[34]	J.-L. Lions and E. Zuazua, On the cost of controlling unstable systems: The case of boundary controls, J. Anal. Math., 73 (1997), 225-249. doi: 10.1007/BF02788145. Google Scholar
[35]	P. Lissy, On the cost of fast controls for some families of dispersive or parabolic equations in one space dimension, SIAM J. Control Optim., 52 (2014), 2651-2676. doi: 10.1137/140951746. Google Scholar
[36]	P. Lissy, Explicit lower bounds for the cost of fast controls for some 1-D parabolic or dispersive equations, and a new lower bound concerning the uniform controllability of the 1-D transport-diffusion equation, J. Differential Equations, 259 (2015), 5331-5352. doi: 10.1016/j.jde.2015.06.031. Google Scholar
[37]	L. Lorch and M. E. Muldoon, Monotonic sequences related to zeros of Bessel functions, Numer. Algor, 49 (2008), 221-233. doi: 10.1007/s11075-008-9189-4. Google Scholar
[38]	P. Martin, L. Rosier and P. Rouchon, Null controllability of one-dimensional parabolic equations using flatness, Automatica J. IFAC, 50 (2014), 3067-3076. doi: 10.1016/j.automatica.2014.10.049. Google Scholar
[39]	P. Martin, L. Rosier and P. Rouchon, On the reachable states for the boundary control of the heat equation, Applied Mathematics Research eXpress, (2016), 181-216. doi: 10.1093/amrx/abv013. Google Scholar
[40]	P. Martinez and J. Vancostenoble, Carleman estimates for one-dimensional degenerate heat equations, J. Evol. Eq, 6 (2006), 325-362. doi: 10.1007/s00028-006-0214-6. Google Scholar
[41]	L. Miller, Geometric bounds on the growth rate of null controllability cost for the heat equation in small time, J. Differential Equations, 204 (2004), 202-226. doi: 10.1016/j.jde.2004.05.007. Google Scholar
[42]	F. W. Olver, Asymptotics and Special Functions, New York, Academic press, 1974. Google Scholar
[43]	C. K. Qu and R. Wong, "Best possible" upper and lower bounds for the zeros of the Bessel function $ J_ν(x)$, Trans. Amer. Math. Soc., 351 (1999), 2833-2859. doi: 10.1090/S0002-9947-99-02165-0. Google Scholar
[44]	R. M. Redheffer, Elementary remarks on completeness, Duke Math. Journal, 35 (1968), 103-116. doi: 10.1215/S0012-7094-68-03511-4. Google Scholar
[45]	L. Schwartz, Étude des Sommes D'exponentielles, deuxiéme édition. Paris, Hermann, 1959. Google Scholar
[46]	T. Seidman, Time invarinace of the reachable set for linear control problems, J. Math. Annal. Appl., 72 (1979), 17-20. doi: 10.1016/0022-247X(79)90271-3. Google Scholar
[47]	T. Seidman, Two results on exact boundary control of parabolic equations, Appl. Math. Optim., 11 (1984), 145-152. doi: 10.1007/BF01442174. Google Scholar
[48]	T. I. Seidman, S. A. Avdonin and S. A. Ivanov, The "window problem" for series of complex exponentials, J. Fourier Anal. Appl., 6 (2000), 233-254. doi: 10.1007/BF02511154. Google Scholar
[49]	G. Tenenbaum and M. Tucsnak, New blow-up rates for fast controls of Schrodinger and heat equations, J. Differential Equations, 243 (2007), 70-100. doi: 10.1016/j.jde.2007.06.019. Google Scholar
[50]	G. N. Watson, A Treatise on the Theory of Bessel Functions second edition, Cambridge University Press, Cambridge, England, 1944. Google Scholar
[51]	R. M. Young, An Introduction to Nonharmonic Fourier Series, Academic Press, 1980. Google Scholar

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[20]	Xin Li, Chunyou Sun, Na Zhang. Dynamics for a non-autonomous degenerate parabolic equation in $\mathfrak{D}_{0}^{1}(\Omega, \sigma)$. Discrete & Continuous Dynamical Systems - A, 2016, 36 (12) : 7063-7079. doi: 10.3934/dcds.2016108

American Institute of Mathematical Sciences

The cost of controlling weakly degenerate parabolic equations by boundary controls

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American Institute of Mathematical Sciences

The cost of controlling weakly degenerate parabolic equations by boundary controls

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