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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show all references

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

[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.  Google Scholar

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[13]

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[14]

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[15]

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[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.  Google Scholar

[17]

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[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.  Google Scholar

[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.   Google Scholar

[21]

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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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[27]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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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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