doi: 10.3934/eect.2021048
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The method of extremal shift in control problems for evolution variational inequalities under disturbances

Krasovskii Institute of Mathematics and Mechanics of UB RAS, Yekaterinburg 620990, Russia

Received  April 2021 Revised  June 2021 Early access August 2021

The problems of designing feedback control algorithms for parabolic and hyperbolic variational inequalities are considered. These algorithms should preserve given properties of solutions of inequalities under the action of unknown disturbances. Solving algorithms that are stable with respect to informational noises are constructed. The algorithms are based on the method of extremal shift well-known in the theory of guaranteed control.

Citation: Vyacheslav Maksimov. The method of extremal shift in control problems for evolution variational inequalities under disturbances. Evolution Equations and Control Theory, doi: 10.3934/eect.2021048
References:
[1]

D. R. Adams and S. Lenhart, Optimal control of the obstacle for a parabolic variational inequality, J. Math. Anal. Appl., 268 (2002), 602-614.  doi: 10.1006/jmaa.2001.7833.

[2]

B. AllecheV. D. Rădulescu and M. Sebaoui, The Tikhonov regularization for equilibrium problems and applications to quasi-hemivariational inequalities, Optim. Lett., 9 (2015), 483-503.  doi: 10.1007/s11590-014-0765-3.

[3]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976.

[4]

V. Barbu, Optimal Control of Variational Inequalities, Research Notes in Mathematics, 100, Pitman (Advanced Publishing Program), Boston, MA, 1984.

[5]

M. Bergounioux and H. Zidani, Pontryagin maximum principle for optimal control of variational inequalities, SIAM J. Control Optim., 37 (1999), 1273-1290.  doi: 10.1137/S0363012997328087.

[6]

M. Boukrouche and D. A. Tarzia, Convergence of optimal control problems governed by second kind parabolic variational inequalities, J. Control Theory Appl., 11 (2013), 422-427.  doi: 10.1007/s11768-013-2155-2.

[7]

H. Brézis, Problèmes unilatéraux, J. Math. Pures Appl. (9), 51 (1972), 1-168. 

[8] H. O. Fattorini, Infinite-Dimensional Optimization and Control Theory, Encyclopedia of Mathematics and its Applications, 62, Cambridge University Press, Cambridge, 1999.  doi: 10.1017/CBO9780511574795.
[9]

A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, Translations of Mathematical Monographs, 187, American Mathematical Society, Providence, RI, 2000. doi: 10.1090/mmono/187.

[10]

D. Goeleven and D. Motreanu, Variational and Hemivariational Inequalities: Theory, Methods and Applications. Vol. Ⅱ: Unilateral Problems, Nonconvex Optimization and its Applications, 70, Kluwer Academic Publishers, Boston, MA, 2003. doi: 10.1007/978-1-4419-8758-7.

[11]

K. Ito and K. Kunisch, Optimal control of parabolic variational inequalities, J. Math. Pures Appl. (9), 93 (2010), 329-360.  doi: 10.1016/j.matpur.2009.10.005.

[12]

E.-Y. Ju and J.-M. Jeong, Optimal control problems for nonlinear variational evolution inequalities, Abstr. Appl. Anal., (2013), 10pp. doi: 10.1155/2013/724190.

[13] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics, 88, Academic Press, Inc., New York-London, 1980. 
[14]

N. N. Krasovski\u{i} and A. I. Subbotin, Game-Theoretical Control Problems, Springer Series in Soviet Mathematics, Springer-Verlag, New York, 1988.

[15] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems, Encyclopedia of Mathematics and its Applications, 74, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9781107340848.
[16]

V. Maksimov, Feedback robust control for a parabolic variational inequality, in System Modeling and Optimization, IFIP Int. Fed. Inf. Process., 166, Kluwer Acad. Publ., Boston, MA, 2005,123–134. doi: 10.1007/0-387-23467-5_7.

[17]

V. Maksimov, Some problems of guaranteed control of the Schlögl and FitzHugh–-Nagumo systems, Evol. Equ. Control Theory, 6 (2017), 559-586.  doi: 10.3934/eect.2017028.

[18]

V. Maksimov and L. Pandolfi, The problem of dynamical reconstruction of Dirichlet boundary control in semilinear hyperbolic equation, J. Inverse Ill-Posed Probl., 8 (2000), 399-420.  doi: 10.1515/jiip.2000.8.4.399.

[19]

V. I. Maksimov, Dynamical Inverse Problems of Distributed Systems, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2002. doi: 10.1515/9783110944839.

[20]

V. I. Maksimov, Feedback minimax control for parabolic variational inequality, C. R. Acad. Sci. Paris, Series IIB, 328 (2000), 105-108.  doi: 10.1016/S1287-4620(00)88424-0.

[21]

V. Maksimov, Method of extremal shift in problems of reconstruction of an input for parabolic variational inequalities, in Analysis and Optimization of Differential Systems, Kluwer Acad. Publ., Boston, MA, 2003,259–268.

[22]

F. Mignot and J.-P. Puel, Optimal control in some variational inequalities, SIAM J. Control Optim., 22 (1984), 466-476.  doi: 10.1137/0322028.

[23]

L. Pandolfi, Adaptive recursive deconvolution and adaptive noise cancellation, Internat. J. Control., 80 (2007), 403-415.  doi: 10.1080/00207170601042346.

[24]

L. Pandolfi, Distributed Systems with Persistent Memory. Control and Moment Problems, SpringerBriefs in Electrical and Computer Engineering, SpringerBriefs in Control, Automation and Robotics, Springer, Cham, 2014. doi: 10.1007/978-3-319-12247-2.

[25]

A. A. Samarski\u{i}, Introduction to the Theory of Difference Schemes, Izdat. "Nauka", Moscow, 1971.

[26] M. Sofonea and S. Migórski, Variational-Hemivariational Inequalities with Applications, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2018. 
[27]

D. Tiba, Optimal Control of Nonsmooth Distributed Parameter Systems, Lecture Notes in Mathematics, 1459, Springer-Verlag, Berlin, 1990. doi: 10.1007/BFb0085564.

[28]

F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, Graduate Studies in Mathematics, 112, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.

[29]

M. Tucshak and G. Weiss, Observation and Control For Operator Semigroups, Birkhäuser Advanced Texts: Basel Textbooks, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

show all references

References:
[1]

D. R. Adams and S. Lenhart, Optimal control of the obstacle for a parabolic variational inequality, J. Math. Anal. Appl., 268 (2002), 602-614.  doi: 10.1006/jmaa.2001.7833.

[2]

B. AllecheV. D. Rădulescu and M. Sebaoui, The Tikhonov regularization for equilibrium problems and applications to quasi-hemivariational inequalities, Optim. Lett., 9 (2015), 483-503.  doi: 10.1007/s11590-014-0765-3.

[3]

V. Barbu, Nonlinear Semigroups and Differential Equations in Banach Spaces, Editura Academiei Republicii Socialiste România, Bucharest; Noordhoff International Publishing, Leiden, 1976.

[4]

V. Barbu, Optimal Control of Variational Inequalities, Research Notes in Mathematics, 100, Pitman (Advanced Publishing Program), Boston, MA, 1984.

[5]

M. Bergounioux and H. Zidani, Pontryagin maximum principle for optimal control of variational inequalities, SIAM J. Control Optim., 37 (1999), 1273-1290.  doi: 10.1137/S0363012997328087.

[6]

M. Boukrouche and D. A. Tarzia, Convergence of optimal control problems governed by second kind parabolic variational inequalities, J. Control Theory Appl., 11 (2013), 422-427.  doi: 10.1007/s11768-013-2155-2.

[7]

H. Brézis, Problèmes unilatéraux, J. Math. Pures Appl. (9), 51 (1972), 1-168. 

[8] H. O. Fattorini, Infinite-Dimensional Optimization and Control Theory, Encyclopedia of Mathematics and its Applications, 62, Cambridge University Press, Cambridge, 1999.  doi: 10.1017/CBO9780511574795.
[9]

A. V. Fursikov, Optimal Control of Distributed Systems. Theory and Applications, Translations of Mathematical Monographs, 187, American Mathematical Society, Providence, RI, 2000. doi: 10.1090/mmono/187.

[10]

D. Goeleven and D. Motreanu, Variational and Hemivariational Inequalities: Theory, Methods and Applications. Vol. Ⅱ: Unilateral Problems, Nonconvex Optimization and its Applications, 70, Kluwer Academic Publishers, Boston, MA, 2003. doi: 10.1007/978-1-4419-8758-7.

[11]

K. Ito and K. Kunisch, Optimal control of parabolic variational inequalities, J. Math. Pures Appl. (9), 93 (2010), 329-360.  doi: 10.1016/j.matpur.2009.10.005.

[12]

E.-Y. Ju and J.-M. Jeong, Optimal control problems for nonlinear variational evolution inequalities, Abstr. Appl. Anal., (2013), 10pp. doi: 10.1155/2013/724190.

[13] D. Kinderlehrer and G. Stampacchia, An Introduction to Variational Inequalities and Their Applications, Pure and Applied Mathematics, 88, Academic Press, Inc., New York-London, 1980. 
[14]

N. N. Krasovski\u{i} and A. I. Subbotin, Game-Theoretical Control Problems, Springer Series in Soviet Mathematics, Springer-Verlag, New York, 1988.

[15] I. Lasiecka and R. Triggiani, Control Theory for Partial Differential Equations: Continuous and Approximation Theories. I. Abstract Parabolic Systems, Encyclopedia of Mathematics and its Applications, 74, Cambridge University Press, Cambridge, 2000.  doi: 10.1017/CBO9781107340848.
[16]

V. Maksimov, Feedback robust control for a parabolic variational inequality, in System Modeling and Optimization, IFIP Int. Fed. Inf. Process., 166, Kluwer Acad. Publ., Boston, MA, 2005,123–134. doi: 10.1007/0-387-23467-5_7.

[17]

V. Maksimov, Some problems of guaranteed control of the Schlögl and FitzHugh–-Nagumo systems, Evol. Equ. Control Theory, 6 (2017), 559-586.  doi: 10.3934/eect.2017028.

[18]

V. Maksimov and L. Pandolfi, The problem of dynamical reconstruction of Dirichlet boundary control in semilinear hyperbolic equation, J. Inverse Ill-Posed Probl., 8 (2000), 399-420.  doi: 10.1515/jiip.2000.8.4.399.

[19]

V. I. Maksimov, Dynamical Inverse Problems of Distributed Systems, Inverse and Ill-posed Problems Series, VSP, Utrecht, 2002. doi: 10.1515/9783110944839.

[20]

V. I. Maksimov, Feedback minimax control for parabolic variational inequality, C. R. Acad. Sci. Paris, Series IIB, 328 (2000), 105-108.  doi: 10.1016/S1287-4620(00)88424-0.

[21]

V. Maksimov, Method of extremal shift in problems of reconstruction of an input for parabolic variational inequalities, in Analysis and Optimization of Differential Systems, Kluwer Acad. Publ., Boston, MA, 2003,259–268.

[22]

F. Mignot and J.-P. Puel, Optimal control in some variational inequalities, SIAM J. Control Optim., 22 (1984), 466-476.  doi: 10.1137/0322028.

[23]

L. Pandolfi, Adaptive recursive deconvolution and adaptive noise cancellation, Internat. J. Control., 80 (2007), 403-415.  doi: 10.1080/00207170601042346.

[24]

L. Pandolfi, Distributed Systems with Persistent Memory. Control and Moment Problems, SpringerBriefs in Electrical and Computer Engineering, SpringerBriefs in Control, Automation and Robotics, Springer, Cham, 2014. doi: 10.1007/978-3-319-12247-2.

[25]

A. A. Samarski\u{i}, Introduction to the Theory of Difference Schemes, Izdat. "Nauka", Moscow, 1971.

[26] M. Sofonea and S. Migórski, Variational-Hemivariational Inequalities with Applications, Monographs and Research Notes in Mathematics, CRC Press, Boca Raton, FL, 2018. 
[27]

D. Tiba, Optimal Control of Nonsmooth Distributed Parameter Systems, Lecture Notes in Mathematics, 1459, Springer-Verlag, Berlin, 1990. doi: 10.1007/BFb0085564.

[28]

F. Tröltzsch, Optimal Control of Partial Differential Equations. Theory, Methods and Applications, Graduate Studies in Mathematics, 112, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/112.

[29]

M. Tucshak and G. Weiss, Observation and Control For Operator Semigroups, Birkhäuser Advanced Texts: Basel Textbooks, Birkhäuser Verlag, Basel, 2009. doi: 10.1007/978-3-7643-8994-9.

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