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The central idea of our approach consists of introducing a parametric variational problem that defines a variational selection associated with (QVHVI). We prove the solvability of the parametric variational problem by employing a surjectivity theorem for the sum of operators, combined with Minty\u2019s formulation and techniques from the nonsmooth analysis. Then, an existence theorem for (QVHVI) is established by using Kluge\u2019s fixed point theorem for set-valued operators. As an application, an abstract optimal control problem for the (QVHVI) is investigated. We prove the existence of solutions for the optimal control problem and the weak sequential compactness of the solution set via the Weierstrass minimization theorem and the Kuratowski-type continuity properties.<\/jats:p>","DOI":"10.1007\/s10957-021-01963-3","type":"journal-article","created":{"date-parts":[[2021,11,2]],"date-time":"2021-11-02T17:51:20Z","timestamp":1635875480000},"page":"950-970","update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":20,"title":["Evolutionary Quasi-Variational-Hemivariational Inequalities I: Existence and Optimal Control"],"prefix":"10.1007","volume":"193","author":[{"given":"Shengda","family":"Zeng","sequence":"first","affiliation":[]},{"given":"Dumitru","family":"Motreanu","sequence":"additional","affiliation":[]},{"given":"Akhtar A.","family":"Khan","sequence":"additional","affiliation":[]}],"member":"297","published-online":{"date-parts":[[2021,11,2]]},"reference":[{"key":"1963_CR1","doi-asserted-by":"publisher","first-page":"527","DOI":"10.1137\/140969737","volume":"53","author":"M Barboteu","year":"2015","unstructured":"Barboteu, M., Bartosz, K., Han, W., Janiczko, T.: Numerical analysis of a hyperbolic hemivariational inequality arising in dynamic contact. 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