March 2018, 8(1): 1-34. doi: 10.3934/mcrf.2018001

Second order optimality conditions for optimal control of quasilinear parabolic equations

1. 

Technische Universität München, Fakultät für Mathematik, Boltzmannstr. 3, 85748 Garching, Germany

2. 

Rheinische Friedrich-Wilhelms-Universität Bonn, Institut für Numerische Simulation, Wegelerstr. 6, 53115 Bonn, Germany

* Corresponding author: Ira Neitzel

Received  March 2017 Revised  September 2017 Published  January 2018

Fund Project: The first author is supported by the International Research Training Group IGDK, funded by the German Science Foundation (DFG) and the Austrian Science Fund (FWF)

We discuss an optimal control problem governed by a quasilinear parabolic PDE including mixed boundary conditions and Neumann boundary control, as well as distributed control. Second order necessary and sufficient optimality conditions are derived. The latter leads to a quadratic growth condition without two-norm discrepancy. Furthermore, maximal parabolic regularity of the state equation in Bessel-potential spaces $H_D^{-\zeta,p}$ with uniform bound on the norm of the solution operator is proved and used to derive stability results with respect to perturbations of the nonlinear differential operator.

Citation: Lucas Bonifacius, Ira Neitzel. Second order optimality conditions for optimal control of quasilinear parabolic equations. Mathematical Control & Related Fields, 2018, 8 (1) : 1-34. doi: 10.3934/mcrf.2018001
References:
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P. Acquistapace, Evolution operators and strong solutions of abstract linear parabolic equations, Differential Integral Equations, 1 (1988), 433-457.

[2]

P. Acquistapace and B. Terreni, A unified approach to abstract linear nonautonomous parabolic equations, Rend. Sem. Mat. Univ. Padova, 78 (1987), 47-107, URL http://www.numdam.org/item?id=RSMUP_1987__78__47_0.

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N. U. Ahmed, Optimal control of a class of strongly nonlinear parabolic systems, J. Math. Anal. Appl., 61 (1977), 188-207.

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H. Amann, Maximal regularity for nonautonomous evolution equations, Adv. Nonlinear Stud., 4 (2004), 417-430.

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H. Amann, Nonautonomous parabolic equations involving measures, Journal of Mathematical Sciences, 130 (2005), 4780-4802. doi: 10.1007/s10958-005-0376-8.

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H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I, vol. 89 of Monographs in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 1995, Abstract linear theory. doi: 10.1007/978-3-0348-9221-6.

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W. Arendt and A. F. M. ter, Gaussian estimates for second order elliptic operators with boundary conditions, J. Operator Theory, 38 (1997), 87-130.

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W. Arendt and S. Bu, Tools for maximal regularity, Math. Proc. Cambridge Philos. Soc., 134 (2003), 317-336. doi: 10.1017/S0305004102006345.

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W. Arendt and A. F. M. ter Elst, From forms to semigroups, in Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations, vol. 221 of Oper. Theory Adv. Appl., Birkhäuser/Springer Basel AG, Basel, 2012, 47-69. doi: 10.1007/978-3-0348-0297-0_4.

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E. Casas and V. Dhamo, Optimality conditions for a class of optimal boundary control problems with quasilinear elliptic equations, Control Cybernet., 40 (2011), 457-490.

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E. CasasL. A. Fernández and J. Yong, Optimal control of quasilinear parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 545-565. doi: 10.1017/S0308210500032674.

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E. Casas and F. Tröltzsch, Error estimates for the finite-element approximation of a semilinear elliptic control problem, Control Cybernet., 31 (2002), 695-712.

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E. Casas and F. Tröltzsch, First- and second-order optimality conditions for a class of optimal control problems with quasilinear elliptic equations, SIAM J. Control Optim., 48 (2009), 688-718. doi: 10.1137/080720048.

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E. Casas and F. Tröltzsch, A general theorem on error estimates with application to a quasilinear elliptic optimal control problem, Comput. Optim. Appl., 53 (2012), 173-206. doi: 10.1007/s10589-011-9453-8.

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R. HallerC. MeyerJ. Rehberg and A. Schiela, Hölder continuity and optimal control for nonsmooth elliptic problems, Appl. Math. Optim., 60 (2009), 397-428. doi: 10.1007/s00245-009-9077-x.

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R. HallerA. JonssonD. Knees and J. Rehberg, Elliptic and parabolic regularity for second- order divergence operators with mixed boundary conditions, Mathematical Methods in the Applied Sciences, 39 (2016), 5007-5026. doi: 10.1002/mma.3484.

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H. Meinlschmidt and J. Rehberg, Hölder-estimates for non-autonomous parabolic problems with rough data, Evolution Equations and Control Theory, 5 (2016), 147-184. doi: 10.3934/eect.2016.5.147.

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S. Nababan and E. S. Noussair, Existence theorems for optimal control of quasilinear parabolic partial differential equations, J. Austral. Math. Soc. Ser. B, 21 (1979/80), 90-101. doi: 10.1017/S0334270000001958.

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show all references

References:
[1]

P. Acquistapace, Evolution operators and strong solutions of abstract linear parabolic equations, Differential Integral Equations, 1 (1988), 433-457.

[2]

P. Acquistapace and B. Terreni, A unified approach to abstract linear nonautonomous parabolic equations, Rend. Sem. Mat. Univ. Padova, 78 (1987), 47-107, URL http://www.numdam.org/item?id=RSMUP_1987__78__47_0.

[3]

N. U. Ahmed, Optimal control of a class of strongly nonlinear parabolic systems, J. Math. Anal. Appl., 61 (1977), 188-207.

[4]

H. Amann, Linear parabolic problems involving measures, RACSAM. Rev. R. Acad. Cienc. Exactas Fis. Nat. Ser. A Mat., 95 (2001), 85-119.

[5]

H. Amann, Maximal regularity for nonautonomous evolution equations, Adv. Nonlinear Stud., 4 (2004), 417-430.

[6]

H. Amann, Nonautonomous parabolic equations involving measures, Journal of Mathematical Sciences, 130 (2005), 4780-4802. doi: 10.1007/s10958-005-0376-8.

[7]

H. Amann, Linear and Quasilinear Parabolic Problems. Vol. I, vol. 89 of Monographs in Mathematics, Birkhäuser Boston, Inc., Boston, MA, 1995, Abstract linear theory. doi: 10.1007/978-3-0348-9221-6.

[8]

W. Arendt and A. F. M. ter, Gaussian estimates for second order elliptic operators with boundary conditions, J. Operator Theory, 38 (1997), 87-130.

[9]

W. Arendt and S. Bu, Tools for maximal regularity, Math. Proc. Cambridge Philos. Soc., 134 (2003), 317-336. doi: 10.1017/S0305004102006345.

[10]

W. Arendt and A. F. M. ter Elst, From forms to semigroups, in Spectral Theory, Mathematical System Theory, Evolution Equations, Differential and Difference Equations, vol. 221 of Oper. Theory Adv. Appl., Birkhäuser/Springer Basel AG, Basel, 2012, 47-69. doi: 10.1007/978-3-0348-0297-0_4.

[11]

P. AuscherN. BadrR. Haller and J. Rehberg, The square root problem for second-order, divergence form operators with mixed boundary conditions on $L^p$, J. Evol. Equ., 15 (2015), 165-208. doi: 10.1007/s00028-014-0255-1.

[12]

E. Casas and K. Chrysafinos, Analysis and optimal control of some quasilinear parabolic equations, Submitted.

[13]

E. Casas and V. Dhamo, Optimality conditions for a class of optimal boundary control problems with quasilinear elliptic equations, Control Cybernet., 40 (2011), 457-490.

[14]

E. CasasL. A. Fernández and J. Yong, Optimal control of quasilinear parabolic equations, Proc. Roy. Soc. Edinburgh Sect. A, 125 (1995), 545-565. doi: 10.1017/S0308210500032674.

[15]

E. Casas and F. Tröltzsch, Error estimates for the finite-element approximation of a semilinear elliptic control problem, Control Cybernet., 31 (2002), 695-712.

[16]

E. Casas and F. Tröltzsch, First- and second-order optimality conditions for a class of optimal control problems with quasilinear elliptic equations, SIAM J. Control Optim., 48 (2009), 688-718. doi: 10.1137/080720048.

[17]

E. Casas and F. Tröltzsch, A general theorem on error estimates with application to a quasilinear elliptic optimal control problem, Comput. Optim. Appl., 53 (2012), 173-206. doi: 10.1007/s10589-011-9453-8.

[18]

E. Casas and F. Tröltzsch, Second order analysis for optimal control problems: improving results expected from abstract theory, SIAM J. Optim., 22 (2012), 261-279. doi: 10.1137/110840406.

[19]

E. Casas and F. Tröltzsch, Second order optimality conditions and their role in PDE control, Jahresber. Dtsch. Math.-Ver., 117 (2015), 3-44. doi: 10.1365/s13291-014-0109-3.

[20]

J. C. de Los ReyesP. MerinoJ. Rehberg and F. Tröltzsch, Optimality conditions for state-constrained PDE control problems with time-dependent controls, Control Cybernet., 37 (2008), 5-38.

[21]

R. Denk, M. Hieber and J. Prüss, $\mathcal{R}$-boundedness, Fourier multipliers and problems of elliptic and parabolic type, Mem. Amer. Math. Soc., 166 (2003), viii+114pp. doi: 10.1090/memo/0788.

[22]

D. DiA. Lunardi and R. Schnaubelt, Optimal regularity and Fredholm properties of abstract parabolic operators in $L^p$ spaces on the real line, Proc. London Math. Soc. (3), 91 (2005), 703-737. doi: 10.1112/S0024611505015406.

[23]

K. DisserA. F. M. ter Elst and J. Rehberg, Hölder estimates for parabolic operators on domains with rough boundary, Ann. Sc. Norm. Sup. Pisa Cl. Sci. (5), 17 (2017), 65-79. doi: 10.2422/2036-2145/201503-013.

[24]

K. DisserH.-C. Kaiser and J. Rehberg, Optimal Sobolev regularity for linear second-order divergence elliptic operators occurring in real-world problems, SIAM J. Math. Anal., 47 (2015), 1719-1746. doi: 10.1137/140982969.

[25]

K. DisserA. F. M. ter Elst and J. Rehberg, On maximal parabolic regularity for non-autonomous parabolic operators, J. Differential Equations, 262 (2017), 2039-2072. doi: 10.1016/j.jde.2016.10.033.

[26]

X. T. Duong and D. W. Robinson, Semigroup kernels, Poisson bounds, and holomorphic functional calculus, J. Funct. Anal., 142 (1996), 89-128. doi: 10.1006/jfan.1996.0145.

[27]

M. Egert, Lp-estimates for the square root of elliptic systems with mixed boundary conditions, arXiv: 1712.09851.

[28]

M. EgertR. Haller and P. Tolksdorf, The Kato square root problem for mixed boundary conditions, J. Funct. Anal., 267 (2014), 1419-1461. doi: 10.1016/j.jfa.2014.06.003.

[29]

J. ElschnerJ. Rehberg and G. Schmidt, Optimal regularity for elliptic transmission problems including $C^1$ interfaces, Interfaces Free Bound., 9 (2007), 233-252. doi: 10.4171/IFB/163.

[30]

L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions, Studies in Advanced Mathematics, CRC Press, Boca Raton, FL, 1992.

[31]

L. A. Fernández, Integral state-constrained optimal control problems for some quasilinear parabolic equations, Nonlinear Anal., 39 (2000), 977-996. doi: 10.1016/S0362-546X(98)00264-8.

[32]

H. Gajewski, K. Gröger and K. Zacharias, Nichtlineare Operatorgleichungen und Operatordifferentialgleichungen, Akademie-Verlag, Berlin, 1974, Mathematische Lehrbücher und Monographien, Ⅱ. Abteilung, Mathematische Monographien, Band 38.

[33]

J. A. GriepentrogK. GrögerH.-C. Kaiser and J. Rehberg, Interpolation for function spaces related to mixed boundary value problems, Math. Nachr., 241 (2002), 110-120. doi: 10.1002/1522-2616(200207)241:1<110::AID-MANA110>3.0.CO;2-R.

[34]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, vol. 24 of Monographs and Studies in Mathematics, Pitman (Advanced Publishing Program), Boston, MA, 1985. doi: 10.1137/1.9781611972030.

[35]

K. Gröger, A $W^{1,p}$-estimate for solutions to mixed boundary value problems for second order elliptic differential equations, Math. Ann., 283 (1989), 679-687. doi: 10.1007/BF01442860.

[36]

M. Haase, The Functional Calculus for Sectorial Operators, vol. 169 of Operator Theory: Advances and Applications, Birkhäuser Verlag, Basel, 2006. doi: 10.1007/3-7643-7698-8.

[37]

R. HallerC. MeyerJ. Rehberg and A. Schiela, Hölder continuity and optimal control for nonsmooth elliptic problems, Appl. Math. Optim., 60 (2009), 397-428. doi: 10.1007/s00245-009-9077-x.

[38]

R. HallerA. JonssonD. Knees and J. Rehberg, Elliptic and parabolic regularity for second- order divergence operators with mixed boundary conditions, Mathematical Methods in the Applied Sciences, 39 (2016), 5007-5026. doi: 10.1002/mma.3484.

[39]

R. Haller and J. Rehberg, Maximal parabolic regularity for divergence operators including mixed boundary conditions, J. Differential Equations, 247 (2009), 1354-1396. doi: 10.1016/j.jde.2009.06.001.

[40]

M. Hieber and S. Monniaux, Pseudo-differential operators and maximal regularity results for non-autonomous parabolic equations, Proc. Amer. Math. Soc., 128 (2000), 1047-1053. doi: 10.1090/S0002-9939-99-05145-X.

[41]

A. Jonsson and H. Wallin, Function spaces on subsets of ${\mathbb{R}}^n$, Math. Rep., 2 (1984), xiv+221 pp.

[42]

D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, vol. 31 of Classics in Applied Mathematics, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2000, Reprint of the 1980 original. doi: 10.1137/1.9780898719451.

[43]

K. Krumbiegel and J. Rehberg, Second order sufficient optimality conditions for parabolic optimal control problems with pointwise state constraints, SIAM J. Control Optim., 51 (2013), 304-331. doi: 10.1137/120871687.

[44]

M. Krízek and P. Neittaanmäki, Mathematical and Numerical Modelling in Electrical Engineering, vol. 1 of Mathematical Modelling: Theory and Applications, Kluwer Academic Publishers, Dordrecht, 1996, Theory and applications, With a foreword by Ivo Babuška.

[45]

P. C. Kunstmann and L. Weis, Maximal $L_p$-regularity for parabolic equations, Fourier multiplier theorems and $H^\infty$-functional calculus, in Functional Analytic Methods for Evolution Equations, vol. 1855 of Lecture Notes in Math., Springer, Berlin, 2004, 65-311. doi: 10.1007/978-3-540-44653-8_2.

[46]

J.-L. Lions, Optimisation pour certaines classes d'équations d'évolution non linéaires, Ann. Mat. Pura Appl. (4), 72 (1966), 275-293.

[47]

A. Lunardi, Interpolation Theory, 2nd edition, Appunti. Scuola Normale Superiore di Pisa (Nuova Serie). [Lecture Notes. Scuola Normale Superiore di Pisa (New Series)], Edizioni della Normale, Pisa, 2009.

[48]

V. G. Maz'ja, Sobolev Spaces, Springer Series in Soviet Mathematics, Springer-Verlag, Berlin, 1985, Translated from the Russian by T. O. Shaposhnikova. doi: 10.1007/978-3-662-09922-3.

[49]

H. Meinlschmidt and J. Rehberg, Hölder-estimates for non-autonomous parabolic problems with rough data, Evolution Equations and Control Theory, 5 (2016), 147-184. doi: 10.3934/eect.2016.5.147.

[50]

S. Nababan and E. S. Noussair, Existence theorems for optimal control of quasilinear parabolic partial differential equations, J. Austral. Math. Soc. Ser. B, 21 (1979/80), 90-101. doi: 10.1017/S0334270000001958.

[51]

E. S. NoussairS. Nababan and K. L. Teo, On the existence of optimal controls for quasilinear parabolic partial differential equations, J. Optim. Theory Appl., 34 (1981), 99-115. doi: 10.1007/BF00933359.

[52]

E. M. Ouhabaz, Analysis of Heat Equations on Domains, vol. 31 of London Mathematical Society Monographs Series, Princeton University Press, Princeton, NJ, 2005.

[53]

N. S. Papageorgiou, On the optimal control of strongly nonlinear evolution equations, J. Math. Anal. Appl., 164 (1992), 83-103. doi: 10.1016/0022-247X(92)90146-5.

[54]

P. Portal and Ž. Štrkalj, Pseudodifferential operators on Bochner spaces and an application, Math. Z., 253 (2006), 805-819. doi: 10.1007/s00209-006-0934-x.

[55]

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Table 1.  Summary of differentiability and integrability exponents
Variable Description
$p$ Given by isomorphism $-\nabla\cdot\mu\nabla + 1 \colon W_D^{1,p}\rightarrow W_D^{-1,p}$ and close to the spatial dimension $d$ ; see Assumption 3.
$\zeta$ Differentiability exponent close to one defining $H_D^{-\zeta,p}$ .
$s$ Integrability exponent for the controls determined by $p$ , $\zeta$ , and $d$ , possibly large; see Assumption 4.
$r$ , $r'$ Integrability exponents for linearized and adjoint state equation introduced in Section 4.
Variable Description
$p$ Given by isomorphism $-\nabla\cdot\mu\nabla + 1 \colon W_D^{1,p}\rightarrow W_D^{-1,p}$ and close to the spatial dimension $d$ ; see Assumption 3.
$\zeta$ Differentiability exponent close to one defining $H_D^{-\zeta,p}$ .
$s$ Integrability exponent for the controls determined by $p$ , $\zeta$ , and $d$ , possibly large; see Assumption 4.
$r$ , $r'$ Integrability exponents for linearized and adjoint state equation introduced in Section 4.
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