### A class of delay evolution hemivariational inequalities and optimal feedback controls

#### Abstract

#### Keywords

#### References

J.P. Aubin and H. Frankowska, Set Valued Analysis. Springer, Berlin, 2009.

A.V. Balakrishnan, Optimal control problem in Banach spaces, J. Soc. Indust. Appl. Math. A: Control 3 (1965), 152–180.

J. Banaś and K. Goebel, Measures of Noncompactness in Banach Spaces, Lecture Notes in Pure and Applied Mathematics, vol. 60, Marcel Dekker, New York, 1980.

M. Benchohra, J. Henderson and S. Ntouyas, Impulsive Differential Equations and Inclusions, Hindawi Publishing Corporation, New York, 2006.

F.H. Clarke, Optimization and Nonsmooth Analysis, Wiley, New York, 1983.

G.F. Franklin, J.D. Powell and A. Emami-Naeini, Feedback Control of Dynamic Systems, Vol. 3, Reading, Addison–Wesley, 1994.

B.Z. Guo and B. Sun, Numerical solution to the optimal birth feedback control of a population dynamics: viscosity solution approach, Optimal Control Appl. Methods 26 (2005), 229–254.

J. Haslinger and P.D. Panagiotopoulos, Optimal control of systems governed by hemivariational inequalities. Existence and approximation results, Nonlinear Anal. 24 (1995), 105–119.

S. Hu and N.S. Papageorgiou, Handbook of Multivalued Analysis, Vol. I, Kluwer Academic Publishers, Dordrecht, Boston, London, 1997.

Y. Huang, Z.H. Liu and B. Zeng, Optimal control of feedback control systems governed by hemivariational inequalities, Comput. Math. Appl. 70 (2015) 2125–2136.

J.M. Jeong and S.J. Son, Time optimal control of semilinear control systems involving time delays, J. Optim. Theory Appl. 165 (2015), 793–811.

M.I. Kamenskiı̆, P. Nistri, V.V. Obukhovskiı̆ and P. Zecca, Optimal feedback control for a semilinear evolution equation, J. Optim. Theory Appl. 82 (1994), 503–517.

J. Klamka, Stochastic controllability of linear systems with state delays, Int. J. Appl. Math. Comput. Sci. 17 (2007), 5–13.

J. Klamka, Stochastic controllability of systems with variable delay in control, Bull. Pol. Acad. Sci. Math. 56 (2008), 279–284.

S. Kumar, Mild solution and fractional optimal control of semilinear system with fixed delay, J. Optim. Theory Appl. 174 (2017), 1–14.

S. Kumar and N. Sukavanam, Approximate controllability of fractional order semilinear systems with bounded delay, J. Differential Equations 252 (2012), 6163–6174.

X. Li and J. Yong, Optimal Control Theory for Infinite Dimensional Systems. Birkhäuser, Boston, 1995.

Q. Lin, R. Loxton, K.L. Teo et al., Optimal feedback control for dynamic systems with state constraints: an exact penalty approach, Optim. Lett. 8 (2014), 1535–1551.

Z.H. Liu, Existence results for quasilinear parabolic hemivariational inequalities, J. Differential Equations 244 (2008), 1395–1409.

Z.H. Liu and X.W. Li, Approximate controllability for a class of hemivariational inequalities, Nonlinear Anal. Real World Appl. 22 (2015), 581–591.

Z.H. Liu and B. Zeng, Existence and controllability for fractional evolution inclusions of Clarke’s subdifferential type, Appl. Math. Comput. 257 (2015), 178–189.

Z.H. Liu, S.D. Zeng and D. Motreanu, Partial differential hemivariational inequalities, Adv. Nonlinear Anal., DOI: 10.1515/anona-2016-0102.

L. Lu, Z.H. Liu, W. Jiang and J.L. Luo, Solvability and optimal controls for semilinear fractional evolution hemivariational inequalities, Math. Methods Appl. Sci. 39 (2016), 5452–5464.

M. Martelli, A Rothe’s type theorem for non-compact acyclic-valued maps, Boll. Un. Mat. Ital. 4 (1975), 70–76.

A.I. Mees, Dynamics of Feedback Systems, Wiley, New York, 1981.

S. Migórski and A. Ochal, Quasi-static hemivariational inequality via vanishing acceleration approach, SIAM J. Math. Anal. 41 (2009), 1415–1435.

S. Migórski and A. Ochal, Optimal control of parabolic hemivariational inequalities, J. Global Optim. 17 (2000), 285–300.

S. Migórski, A. Ochal and M. Sofonea, A dynamic frictional contact problem for piezoelectric materials, J. Math. Anal. Appl. 361 (2010), 161–176.

S. Migórski, A. Ochal and M. Sofonea, Analysis of a dynamic contact problem for electro-viscoelastic cylinders, Nonlinear Anal. 73 (2010), 1221–1238.

S. Migórski, A. Ochal and M. Sofonea, Nonlinear Inclusions and Hemivariational Inequalities, Models and Analysis of Contact Problems. Advances in Mechanics and Mathematics, Vol. 26, Springer, New York, 2013.

P.D. Panagiotopoulos, Nonconvex superpotentials in sense of F.H. Clarke and applications, Mech. Res. Comm. 8 (1981), 335–340.

P.D. Panagiotopoulos, Hemivariational Inequalities, Applications in Mechanics and Engineering, Springer, Berlin, 1993.

J.Y. Park and S.H. Park, Existence of solutions and optimal control problems for hyperbolic hemivariational inequalities, ANZIAM J. 47 (2005), 51–63.

J.Y. Park and S.H. Park, Optimal control problems for anti-periodic quasilinear hemivariational inequalities, Optimal Control Appl. Methods 28 (2007), 275–287.

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.

Y. Ren, L. Hu and R. Sakthivel, Controllability of impulsive neutral stochastic functional differential inclusions with infinite delay, J. Comput. Appl. Math. 235 (2011), 2603–2614.

R. Sakthivel, R. Ganesh and N.I. Mahmudov, Approximate controllability of fractional functional equations with infinite delay, Topol. Methods Nonlinear Anal. 43 (2014), 345–364.

S. Sönmez and S. Ayasun, Stability region in the parameter space of PI controller for a single-area load frequency control system with time delay, IEEE Trans. Power Syst. 31 (2016), 829–830.

C. Tan, L. Li and H. Zhang, Stabilization of networked control systems with both network-induced delay and packet dropout, Automatica 59 (2015), 194–199.

A.A. Tolstonogov, Control systems of subdifferential type depending on a parameter, Izv. Math. 72 (2008), 985–1022.

A.A. Tolstonogov, Relaxation in nonconvex optimal control problems with subdifferential operators, J. Math. Sci. 140 (2007), 850–872.

J.R. Wang, Y. Zhou and W. Wei, Optimal feedback control for semilinear fractional evolution equations in Banach spaces, Systems Control Lett. 61 (2012), 472–476.

W. Wei and X. Xiang, Optimal feedback control for a class of nonlinear impulsive evolution equations, Chinese J. Engrg. Math. 23 (2006), 333–342.

Y. Yang, D. Yue and Y. Xue, Decentralized adaptive neural output feedback control of a class of large-scale time-delay systems with input saturation, J. Franklin Inst. 352 (2015), 2129–2151.

J. Yong, Time optimal controls for semilinear distributed parameter systems — existence theory and necessary conditions, Kodai Math. J. 14 (1991), 239–253.

Y.S. Zhou and Z.H. Wang, Optimal feedback control for linear systems with input delays revisited, J. Optim. Theory Appl. 163 (2014), 989–1017.

### Refbacks

- There are currently no refbacks.