# American Institute of Mathematical Sciences

September & December  2018, 8(3&4): 607-623. doi: 10.3934/mcrf.2018025

## Analysis and optimal control of some quasilinear parabolic equations

 1 Departamento de Matemática Aplicada y Ciencias de la Computación, E.T.S.I. Industriales y de Telecomunicación, Universidad de Cantabria, 39005 Santander, Spain 2 Department of Mathematics, School of Applied Mathematics and Physical Sciences, National Technical University of Athens, Zografou Campus, 15780 Athens, Greece

Corresponding author: Eduardo Casas

Dedicated to Prof. Jiongmin Yong on the occasion of his 60th birthday

Received  May 2017 Revised  January 2018 Published  September 2018

Fund Project: The first author was partially supported by the Spanish Ministerio de Economía, Industria y Competitividad under projects MTM2014-57531-P and MTM2017-83185-P.

In this paper, we consider optimal control problems associated with a class of quasilinear parabolic equations, where the coefficients of the elliptic part of the operator depend on the state function. We prove existence, uniqueness and regularity for the solution of the state equation. Then, we analyze the control problem. The goal is to get first and second order optimality conditions. To this aim we prove the necessary differentiability properties of the relation control-to-state and of the cost functional.

Citation: Eduardo Casas, Konstantinos Chrysafinos. Analysis and optimal control of some quasilinear parabolic equations. Mathematical Control and Related Fields, 2018, 8 (3&4) : 607-623. doi: 10.3934/mcrf.2018025
##### References:
 [1] H. Amann, Linear and Quasilinear Parabolic Problems, Birkhäuser, Boston, 1995. doi: 10.1007/978-3-0348-9221-6. [2] N. Arada and J.-P. Raymond, Dirichlet boundary control of semilinear parabolic equations. I. Problems with no state constraints, Appl. Math. Optim., 45 (2002), 125-143.  doi: 10.1007/s00245-001-0035-5. [3] N. Arada and J.-P. Raymond, Dirichlet boundary control of semilinear parabolic equations. Ⅱ. Problems with pointwise state constraints, Appl. Math. Optim., 45 (2002), 145-167.  doi: 10.1007/s00245-001-0036-4. [4] T. Bayen and F. Silva, Second order analysis for strong solutions in the optimal control of parabolic equations, SIAM J. Control Optim., 54 (2016), 819-844.  doi: 10.1137/141000415. [5] L. Bonifacius and I. Neitzel, Second order optimality conditions for boundary control of quasilinear parabolic equations, Math. Control Relat. Fields, 8 (2018), 1-34.  doi: 10.3934/mcrf.2018001. [6] J. Bonnans, Optimal control of a semilinear parabolic equation with singular arcs, Optim. Methods Softw., 29 (2014), 964-978.  doi: 10.1080/10556788.2013.830220. [7] J. Bonnans, X. Dupuis and L. Pfeiffer, Second-order necessary conditions in Pontryagin form for optimal control problems, SIAM J. Control Optim., 52 (2014), 3887-3916.  doi: 10.1137/130923452. [8] S. C. Brenner and L. R. Scott, The mathematical Theory of Finite Element Methods, vol. 15 of Texts in Applied Mathematics, 3rd edition, Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0. [9] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. [10] E. Casas, Boundary control problems of quasilinear elliptic equations: A Pontryagin's principle, Appl. Math. Optim., 33 (1996), 265-291.  doi: 10.1007/BF01204705. [11] E. Casas, Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations, SIAM J. Control Optim., 35 (1997), 1297-1327.  doi: 10.1137/S0363012995283637. [12] E. Casas and V. Dhamo, Optimality conditions for a class of optimal boundary control problems with quasilinear elliptic equations, Control & Cybernetics, 40 (2011), 457-490. [13] E. Casas and V. Dhamo, Error estimates for the numerical approximation of Neumann control problems governed by a class of quasilinear elliptic equations, Comput. Optim. Appl., 52 (2012), 719-756.  doi: 10.1007/s10589-011-9440-0. [14] E. Casas and L. Fernández, Distributed control of systems governed by a general class of quasilinear elliptic equations, J. Diff. Equat., 104 (1993), 20-47.  doi: 10.1006/jdeq.1993.1062. [15] E. Casas and L. Fernández, Dealing with integral state constraints in control problems of quasilinear elliptic equations, SIAM J. Control Optim., 33 (1995), 568-589.  doi: 10.1137/S0363012992234633. [16] E. Casas, L. Fernández and J. Yong, Optimal control of quasilinear parabolic equations, Proc. R. Soc. Edinb. Sect. A-Math, 125 (1995), 545-565.  doi: 10.1017/S0308210500032674. [17] E. Casas, R. Herzog and G. Wachsmuth, Analysis of spatio-temporally sparse optimal control problems of semilinear parabolic equations, ESAIM Control Optim. Calc. Var., 23 (2017), 263-295.  doi: 10.1051/cocv/2015048. [18] E. Casas, P. Kogut and G. Leugering, Approximation of optimal control problems in the coefficient for the p-Laplace equation. Ⅰ Convergence result, SIAM J. Control Optim., 54 (2016), 1406-1422.  doi: 10.1137/15M1028108. [19] E. Casas, F. Kruse and K. Kunisch, Optimal control of semilinear parabolic equations by BV-functions, SIAM J. Control Optim., 55 (2017), 1752-1788.  doi: 10.1137/16M1056511. [20] E. Casas, J. Raymond and H. Zidani, Pontryagin's principle for local solutions of control problems with mixed control-state constraints, SIAM J. Control Optim., 39 (2000), 1182-1203.  doi: 10.1137/S0363012998345627. [21] E. Casas and F. Tröltzsch, 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. [22] E. Casas and F. Tröltzsch, Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations, ESAIM:COCV, 17 (2010), 771-800.  doi: 10.1051/cocv/2010025. [23] E. Casas and F. Tröltzsch, Second order analysis for optimal control problems: Improving results expected from from abstract theory, SIAM J. Optim., 22 (2012), 261-279.  doi: 10.1137/110840406. [24] 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. [25] E. Casas and F. Tröltzsch, Second order optimality conditions for weak and strong local solutions of parabolic optimal control problems, Vietnam J. Math., 44 (2016), 181-202.  doi: 10.1007/s10013-015-0175-6. [26] E. Casas and J. Yong, Maximum principle for state-constrained optimal control problems governed by quasilinear elliptic equations, Differential Integral Equations, 8 (1995), 1-18. [27] M. Dauge, Neumann and mixed problems on curvilinear polyhedra, Integr. Equ. Oper. Theory, 15 (1992), 227-261.  doi: 10.1007/BF01204238. [28] E. Di Benedetto, On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13 (1986), 487-535. [29] K. Disser, H.-C. Kaiser and J. Rehberg, Optimal Sobolev regularity for linear second-order divergence elliptic operators occurring in real-wordl problems, SIAM J. Math. Anal., 47 (2015), 1719-1746.  doi: 10.1137/140982969. [30] H. Fattorini, Infinite Dimensional Optimization and Control Theory, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9780511574795. [31] L. 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] M. Giaquinta, Introduction to Regularity Theory for Nonlinear Elliptic Systems, Birkhäuser, Basel, 1993, Lecture Notes in Mathematics. [33] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston-London-Melbourne, 1985. [34] R. Haller-Dintelmann 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. [35] M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, vol. 23 of Mathematical Modelling: Theory and Applications, Springer, New York, 2009. [36] D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130 (1995), 161-219.  doi: 10.1006/jfan.1995.1067. [37] K. Kunisch, K. Pieper and B. Vexler, Measure valued directional sparsity for parabolic optimal control problems, SIAM J. Control Optim., 52 (2014), 3078-3108.  doi: 10.1137/140959055. [38] O. Ladyzhenskaya, V. Solonnikov and N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, 1988. [39] X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4. [40] J. Lions, Contrôle Optimal de Systèmes Gouvernés par des Equations aux Dérivées Partielles, Dunod, Paris, 1968. [41] H. Lou, Maximum principle of optimal control for degenerate quasilinear elliptic equations, SIAM J. Control Optim., 42 (2003), 1-23.  doi: 10.1137/S0363012902401664. [42] C. Morrey, Multiple Integrals in the Calculus of Variations, Springer-Verlag, New York, 1966. [43] J. Prüss and R. Schnaubelt, Solvability and maximal regularity of parabolic evolution equations with coefficients continuous in time, J. Math. Anal. Appl., 256 (2001), 405-430.  doi: 10.1006/jmaa.2000.7247. [44] J. Raymond and F. Tröltzsch, Second order sufficient optimality conditions for nonlinear parabolic control problems with state-constraints, Discrete Contin. Dynam. Systems, 6 (2000), 431-450.  doi: 10.3934/dcds.2000.6.431. [45] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Notrh-Holland, Berlin, 1978. [46] F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications, vol. 112 of Graduate Studies in Mathematics, American Mathematical Society, Philadelphia, 2010. doi: 10.1090/gsm/112.

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##### References:
 [1] H. Amann, Linear and Quasilinear Parabolic Problems, Birkhäuser, Boston, 1995. doi: 10.1007/978-3-0348-9221-6. [2] N. Arada and J.-P. Raymond, Dirichlet boundary control of semilinear parabolic equations. I. Problems with no state constraints, Appl. Math. Optim., 45 (2002), 125-143.  doi: 10.1007/s00245-001-0035-5. [3] N. Arada and J.-P. Raymond, Dirichlet boundary control of semilinear parabolic equations. Ⅱ. Problems with pointwise state constraints, Appl. Math. Optim., 45 (2002), 145-167.  doi: 10.1007/s00245-001-0036-4. [4] T. Bayen and F. Silva, Second order analysis for strong solutions in the optimal control of parabolic equations, SIAM J. Control Optim., 54 (2016), 819-844.  doi: 10.1137/141000415. [5] L. Bonifacius and I. Neitzel, Second order optimality conditions for boundary control of quasilinear parabolic equations, Math. Control Relat. Fields, 8 (2018), 1-34.  doi: 10.3934/mcrf.2018001. [6] J. Bonnans, Optimal control of a semilinear parabolic equation with singular arcs, Optim. Methods Softw., 29 (2014), 964-978.  doi: 10.1080/10556788.2013.830220. [7] J. Bonnans, X. Dupuis and L. Pfeiffer, Second-order necessary conditions in Pontryagin form for optimal control problems, SIAM J. Control Optim., 52 (2014), 3887-3916.  doi: 10.1137/130923452. [8] S. C. Brenner and L. R. Scott, The mathematical Theory of Finite Element Methods, vol. 15 of Texts in Applied Mathematics, 3rd edition, Springer, New York, 2008. doi: 10.1007/978-0-387-75934-0. [9] H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Springer, New York, 2011. [10] E. Casas, Boundary control problems of quasilinear elliptic equations: A Pontryagin's principle, Appl. Math. Optim., 33 (1996), 265-291.  doi: 10.1007/BF01204705. [11] E. Casas, Pontryagin's principle for state-constrained boundary control problems of semilinear parabolic equations, SIAM J. Control Optim., 35 (1997), 1297-1327.  doi: 10.1137/S0363012995283637. [12] E. Casas and V. Dhamo, Optimality conditions for a class of optimal boundary control problems with quasilinear elliptic equations, Control & Cybernetics, 40 (2011), 457-490. [13] E. Casas and V. Dhamo, Error estimates for the numerical approximation of Neumann control problems governed by a class of quasilinear elliptic equations, Comput. Optim. Appl., 52 (2012), 719-756.  doi: 10.1007/s10589-011-9440-0. [14] E. Casas and L. Fernández, Distributed control of systems governed by a general class of quasilinear elliptic equations, J. Diff. Equat., 104 (1993), 20-47.  doi: 10.1006/jdeq.1993.1062. [15] E. Casas and L. Fernández, Dealing with integral state constraints in control problems of quasilinear elliptic equations, SIAM J. Control Optim., 33 (1995), 568-589.  doi: 10.1137/S0363012992234633. [16] E. Casas, L. Fernández and J. Yong, Optimal control of quasilinear parabolic equations, Proc. R. Soc. Edinb. Sect. A-Math, 125 (1995), 545-565.  doi: 10.1017/S0308210500032674. [17] E. Casas, R. Herzog and G. Wachsmuth, Analysis of spatio-temporally sparse optimal control problems of semilinear parabolic equations, ESAIM Control Optim. Calc. Var., 23 (2017), 263-295.  doi: 10.1051/cocv/2015048. [18] E. Casas, P. Kogut and G. Leugering, Approximation of optimal control problems in the coefficient for the p-Laplace equation. Ⅰ Convergence result, SIAM J. Control Optim., 54 (2016), 1406-1422.  doi: 10.1137/15M1028108. [19] E. Casas, F. Kruse and K. Kunisch, Optimal control of semilinear parabolic equations by BV-functions, SIAM J. Control Optim., 55 (2017), 1752-1788.  doi: 10.1137/16M1056511. [20] E. Casas, J. Raymond and H. Zidani, Pontryagin's principle for local solutions of control problems with mixed control-state constraints, SIAM J. Control Optim., 39 (2000), 1182-1203.  doi: 10.1137/S0363012998345627. [21] E. Casas and F. Tröltzsch, 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. [22] E. Casas and F. Tröltzsch, Numerical analysis of some optimal control problems governed by a class of quasilinear elliptic equations, ESAIM:COCV, 17 (2010), 771-800.  doi: 10.1051/cocv/2010025. [23] E. Casas and F. Tröltzsch, Second order analysis for optimal control problems: Improving results expected from from abstract theory, SIAM J. Optim., 22 (2012), 261-279.  doi: 10.1137/110840406. [24] 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. [25] E. Casas and F. Tröltzsch, Second order optimality conditions for weak and strong local solutions of parabolic optimal control problems, Vietnam J. Math., 44 (2016), 181-202.  doi: 10.1007/s10013-015-0175-6. [26] E. Casas and J. Yong, Maximum principle for state-constrained optimal control problems governed by quasilinear elliptic equations, Differential Integral Equations, 8 (1995), 1-18. [27] M. Dauge, Neumann and mixed problems on curvilinear polyhedra, Integr. Equ. Oper. Theory, 15 (1992), 227-261.  doi: 10.1007/BF01204238. [28] E. Di Benedetto, On the local behaviour of solutions of degenerate parabolic equations with measurable coefficients, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 13 (1986), 487-535. [29] K. Disser, H.-C. Kaiser and J. Rehberg, Optimal Sobolev regularity for linear second-order divergence elliptic operators occurring in real-wordl problems, SIAM J. Math. Anal., 47 (2015), 1719-1746.  doi: 10.1137/140982969. [30] H. Fattorini, Infinite Dimensional Optimization and Control Theory, Cambridge University Press, Cambridge, 1999. doi: 10.1017/CBO9780511574795. [31] L. 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] M. Giaquinta, Introduction to Regularity Theory for Nonlinear Elliptic Systems, Birkhäuser, Basel, 1993, Lecture Notes in Mathematics. [33] P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston-London-Melbourne, 1985. [34] R. Haller-Dintelmann 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. [35] M. Hinze, R. Pinnau, M. Ulbrich and S. Ulbrich, Optimization with PDE Constraints, vol. 23 of Mathematical Modelling: Theory and Applications, Springer, New York, 2009. [36] D. Jerison and C. Kenig, The inhomogeneous Dirichlet problem in Lipschitz domains, J. Funct. Anal., 130 (1995), 161-219.  doi: 10.1006/jfan.1995.1067. [37] K. Kunisch, K. Pieper and B. Vexler, Measure valued directional sparsity for parabolic optimal control problems, SIAM J. Control Optim., 52 (2014), 3078-3108.  doi: 10.1137/140959055. [38] O. Ladyzhenskaya, V. Solonnikov and N. Ural'tseva, Linear and Quasilinear Equations of Parabolic Type, American Mathematical Society, 1988. [39] X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems, Birkhäuser, Boston, 1995. doi: 10.1007/978-1-4612-4260-4. [40] J. Lions, Contrôle Optimal de Systèmes Gouvernés par des Equations aux Dérivées Partielles, Dunod, Paris, 1968. [41] H. Lou, Maximum principle of optimal control for degenerate quasilinear elliptic equations, SIAM J. Control Optim., 42 (2003), 1-23.  doi: 10.1137/S0363012902401664. [42] C. Morrey, Multiple Integrals in the Calculus of Variations, Springer-Verlag, New York, 1966. [43] J. Prüss and R. Schnaubelt, Solvability and maximal regularity of parabolic evolution equations with coefficients continuous in time, J. Math. Anal. Appl., 256 (2001), 405-430.  doi: 10.1006/jmaa.2000.7247. [44] J. Raymond and F. Tröltzsch, Second order sufficient optimality conditions for nonlinear parabolic control problems with state-constraints, Discrete Contin. Dynam. Systems, 6 (2000), 431-450.  doi: 10.3934/dcds.2000.6.431. [45] H. Triebel, Interpolation Theory, Function Spaces, Differential Operators, Notrh-Holland, Berlin, 1978. [46] F. Tröltzsch, Optimal Control of Partial Differential Equations: Theory, Methods and Applications, vol. 112 of Graduate Studies in Mathematics, American Mathematical Society, Philadelphia, 2010. doi: 10.1090/gsm/112.
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