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Izv. RAN. Ser. Mat., 2008, Volume 72, Issue 5, Pages 149–188 (Mi izv2603)  

This article is cited in 14 scientific papers (total in 14 papers)

Control systems of subdifferential type depending on a parameter

A. A. Tolstonogov

Institute of System Dynamics and Control Theory, Siberian Branch of the Russian Academy of Sciences

Abstract: In a separable Hilbert space, we consider a control system with a subdifferential operator and a non-linear perturbation of monotonic type. The control is subject to a restriction that is a multi-valued map depending on the phase variables with closed non-convex values in a reflexive separable Banach space. The subdifferential operator, the perturbation, the restriction on the control and the initial condition depend on a parameter. Along with this system we consider a control system with convexified restrictions on the control. By a solution of such a system we mean a pair ‘trajectory–control’. We prove theorems on the existence of selectors that are continuous with respect to the parameter and whose values are solutions of the control system. We establish relations between the sets of selectors continuous with respect to the parameter whose values are solutions of the original system and solutions of the system with convexified restrictions on the control. We deduce from these relations various topological properties of the sets of solutions. We apply the results obtained to a control system described by a vector parabolic equation with a small diffusion coefficient in the elliptic term. We prove that solutions of the control system converge to solutions of the limit singular system as the diffusion coefficient tends to zero.

DOI: https://doi.org/10.4213/im2603

Full text: PDF file (780 kB)
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English version:
Izvestiya: Mathematics, 2008, 72:5, 985–1022

Bibliographic databases:

UDC: 517.988
MSC: 49J45, 35F25, 49J24
Received: 26.12.2006
Revised: 24.09.2007

Citation: A. A. Tolstonogov, “Control systems of subdifferential type depending on a parameter”, Izv. RAN. Ser. Mat., 72:5 (2008), 149–188; Izv. Math., 72:5 (2008), 985–1022

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    Citing articles on Google Scholar: Russian citations, English citations
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    This publication is cited in the following articles:
    1. A. A. Tolstonogov, “Variational stability of optimal control problems involving subdifferential operators”, Sb. Math., 202:4 (2011), 583–619  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    2. Tolstonogov A.A., “Continuity in the parameter of the minimum value of an integral functional over the solutions of an evolution control system”, Nonlinear Anal., 75:12 (2012), 4711–4727  crossref  mathscinet  zmath  isi  elib  scopus
    3. Zhenhai Liu, Xiuwen Li, “Approximate controllability for a class of hemivariational inequalities”, Nonlinear Analysis: Real World Applications, 2014  crossref  mathscinet  scopus
    4. Liu Z.H., Migorski S., “Analysis and Control of Differential Inclusions With Anti-Periodic Conditions”, Proc. R. Soc. Edinb. Sect. A-Math., 144:3 (2014), 591–602  crossref  mathscinet  zmath  isi  scopus
    5. Huang Y., Liu Zh., Zeng B., “Optimal Control of Feedback Control Systems Governed By Hemivariational Inequalities”, Comput. Math. Appl., 70:8 (2015), 2125–2136  crossref  mathscinet  isi  elib  scopus
    6. Li Yu., Lu L., “Existence and Controllability For Stochastic Evolution Inclusions of Clarke'S Subdifferential Type”, Electron. J. Qual. Theory Differ., 2015, no. 59  mathscinet  isi
    7. Liu Zh., Li X., Motreanu D., “Approximate Controllability For Nonlinear Evolution Hemivariational Inequalities in Hilbert Spaces”, SIAM J. Control Optim., 53:5 (2015), 3228–3244  crossref  mathscinet  zmath  isi  elib  scopus
    8. Li X., Liu Zh., Migorski S., “Approximate controllability for second order nonlinear evolution hemivariational inequalities”, Electron. J. Qual. Theory Differ., 2015, no. 100, 100  crossref  mathscinet  isi  elib  scopus
    9. Lu L., Liu Zh., Bin M., “Approximate controllability for stochastic evolution inclusions of Clarke's subdifferential type”, Appl. Math. Comput., 286 (2016), 201–212  crossref  mathscinet  isi  elib  scopus
    10. Li Yu., Li X., Liu Y., “On the approximate controllability for fractional evolution hemivariational inequalities”, Math. Meth. Appl. Sci., 39:11 (2016), 3088–3101  crossref  mathscinet  zmath  isi  elib  scopus
    11. Tolstonogov A.A., “Relaxation in Nonconvex Optimal Control Problems Containing the Difference of Two Subdifferentials”, SIAM J. Control Optim., 54:1 (2016), 175–197  crossref  mathscinet  zmath  isi  elib  scopus
    12. Tolstonogov A.A., “Existence and relaxation of solutions for a subdifferential inclusion with unbounded perturbation”, J. Math. Anal. Appl., 447:1 (2017), 269–288  crossref  mathscinet  zmath  isi  elib  scopus
    13. Lu L., Liu Zh., Zhao J., “A Class of Delay Evolution Hemivariational Inequalities and Optimal Feedback Controls”, Topol. Methods Nonlinear Anal., 51:1 (2018), 1–22  crossref  mathscinet  zmath  isi  scopus
    14. Ceng L.-Ch., Liu Zh., Yao J.-Ch., Yao Y., “Optimal Control of Feedback Control Systems Governed By Systems of Evolution Hemivariational Inequalities”, Filomat, 32:15 (2018), 5205–5220  crossref  mathscinet  isi  scopus
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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