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

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

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English version:
Izvestiya: Mathematics, 2008, 72:5, 985–1022

Bibliographic databases:

UDC: 517.988
MSC: 49J45, 35F25, 49J24
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

Citation in format AMSBIB
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• http://mi.mathnet.ru/eng/izv2603
• https://doi.org/10.4213/im2603
• http://mi.mathnet.ru/eng/izv/v72/i5/p149

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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

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
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
3. Zhenhai Liu, Xiuwen Li, “Approximate controllability for a class of hemivariational inequalities”, Nonlinear Analysis: Real World Applications, 2014
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
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
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
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
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
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
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
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
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
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
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
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