On the Approximate Controllability of Second-Order Evolution Hemivariational Inequalities
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Results in Mathematics
On the Approximate Controllability of Second-Order Evolution Hemivariational Inequalities N. I. Mahmudov , R. Udhayakumar, and V. Vijayakumar Abstract. In our manuscript, we organize a group of sufficient conditions for the approximate controllability of second-order evolution hemivariational inequalities. By applying a suitable fixed-point theorem for multivalued maps, we prove our results. Lastly, we present an example to illustrate the obtained theory. Mathematics Subject Classification. 34K30, 35R10, 93B05. Keywords. Approximate controllability, second-order system, hemivariational inequality, clarke subdifferential, multivalued map.
1. Introduction Several partial differential equations that arise in many problems connected with the transverse motion of an extensible beam, the vibration of hinged bars, and many other physical phenomena can be formulated as second-order abstract differential equations in infinite-dimensional spaces. A useful tool for the study of second-order abstract differential equations is the theory of strongly continuous cosine families of operators. The existence and uniqueness of the solutions of second-order nonlinear systems in Banach spaces have been investigated extensively by many authors [4,10–12,15,34,35,40,42]. As is well known, mathematical control theory has many fundamental concepts, including that of controllability. Roughly speaking, controllability means being capable of steering the state of a dynamical system to a suitable state by employing a control function. A detailed discussion of theory and applications related to controllability can be found in the research articles [2,3,20–24,33,36–39,41,43]. 0123456789().: V,-vol
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Results Math
In many complicated physical processes and engineering applications, mathematical models of problems are formulated as inequalities instead of the more commonly seen equations. Many problems are focused on the study of variational inequalities and hemivariational inequalities. Generally speaking, variational inequalities are used in inequality problems with a convex framework, while hemivariational inequalities are involved in systems with nonconvex and nonsmooth structure. In recent years, the study of variational and hemivariational inequalities has been considered extensively in a variety fields of mathematical analysis and engineering applications; see [5–7,9,13,16,19,25, 27–31]. Nonetheless, it appears that the investigation of approximate controllability discussed here has not been contemplated, and that gives the inspiration for this paper. Consider the second-order evolution hemivariational inequalities in the following form −(Kz (t)) + A(t)z(t) + Bx(t), vX + E 0 (t, z(t); v) ≥ 0, t ∈ I = [0, T ], z(0) = z0 , z (0) = z1 , (1.1) where ·, ·X denotes the scalar product of the separable Hilbert space X, A : D(A) ⊂ X → X and K : D(A) ⊂ X → X are linear operators on X. The notation E 0 (t, ·, ·) stands for the generalized Clarke directional derivative (cf. [7]) of a l
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